RRB Group D Official Paper (Held On: 22 Dec, 2025 Shift 2) (Previous Year Paper)

p data-end="57" data-start="0">The correct answer is Salt (NaCl).

 

Key Points

• A compound is a substance formed when two or more chemical elements are chemically bonded together in a fixed ratio by mass.
• Salt (NaCl), also known as Sodium Chloride, is a classic example of a compound. It is formed by the ionic bonding of Sodium (Na) and Chlorine (Cl). Hence, option 1 is correct.
• In a compound, the properties of the resulting substance are entirely different from its constituent elements. For instance, Sodium is a highly reactive metal and Chlorine is a toxic gas, but their compound, Salt, is a stable, edible crystalline solid used in food preservation and seasoning.
• The constituents of a compound cannot be separated by physical methods such as filtration, decantation, or centrifugation. They can only be separated into their elements by chemical or electrochemical reactions like electrolysis.
• Compounds always follow the Law of Constant Proportions, meaning every sample of NaCl will always contain Sodium and Chlorine in the same 1:1 ratio of atoms.

Additional Information

• Soil:
• Soil is a heterogeneous mixture. It consists of varying amounts of minerals, organic matter (humus), gases, liquids, and countless organisms.
• Because its composition is not uniform and the components are not chemically bonded in a fixed ratio, it does not qualify as a compound. Hence, option 2 is incorrect.
• Lemon Juice:
• Lemon juice is a mixture (specifically a solution) containing water (H2O), citric acid, Vitamin C, and sugars.
• The ingredients in lemon juice retain their individual chemical identities and can vary in concentration depending on the fruit. Hence, option 3 is incorrect.
• Air:
• Air is a homogeneous mixture of several gases. It primarily consists of Nitrogen (78%), Oxygen (21%), Argon (0.9%), and Carbon Dioxide (0.04%).
• Since these gases are physically mixed and not chemically combined, and their proportions can vary slightly based on location or altitude, air is classified as a mixture rather than a compound. Hence, option 4 is incorrect.

Important Points

• Characteristics of Compounds:
• A compound has a fixed melting point and boiling point.
• The formation of a compound usually involves the absorption or release of energy in the form of heat or light.
• Difference between Mixtures and Compounds:
• Mixtures can be separated by physical means (like using a magnet to separate iron filings from sulfur), whereas compounds (like Iron Sulfide) cannot.
• Common examples of other compounds include Water (H2O), Carbon Dioxide (CO2), and Methane (CH4).

The correct answer is The object is placed between 20 cm and 40 cm from the pole of the mirror.

Key Points

• A concave mirror is a spherical mirror with an inwardly curved reflecting surface. The nature of the image formed depends on the distance of the object from the pole (P) of the mirror.
• In this problem, the focal length (f) is given as 20 cm. The center of curvature (C), which is the point at a distance of radius of curvature (R), is calculated as R = 2f. Thus, C is at 40 cm from the pole.
• For a concave mirror to produce a real, inverted, and enlarged image, the object must be positioned between the principal focus (F) and the center of curvature (C).
• Since F is at 20 cm and C is at 40 cm, the object must be placed at a distance greater than 20 cm but less than 40 cm. Hence, statement 3 is correct.
• If the object is placed at exactly 20 cm (at F), the reflected rays become parallel, and the image is formed at infinity, being highly enlarged. Hence, statement 1 is incorrect.
• If the object is placed at 10 cm (between P and F), the image formed is virtual, erect, and enlarged behind the mirror. Hence, statement 2 is incorrect.
• If the object is placed at 40 cm (at C), the image formed is real, inverted, and of the same size as the object, located at C. Hence, statement 4 is incorrect.

Additional Information

• Image Formation by Concave Mirror:
• At Infinity: The image is formed at the focus (F). It is real, inverted, and highly diminished (point-sized).
• Beyond C: When the object is beyond 40 cm, the image is formed between F and C. It is real, inverted, and diminished.
• At C: When the object is at 40 cm, the image is formed at C. It is real, inverted, and same size.
• Between F and C: When the object is between 20 cm and 40 cm, the image is formed beyond C. It is real, inverted, and enlarged.
• At F: When the object is at 20 cm, the image is formed at infinity. It is real, inverted, and highly enlarged.
• Between P and F: When the object is within 20 cm, the image is virtual, erect, and enlarged.
• Mirror Formula and Magnification:
• The relationship between object distance (u), image distance (v), and focal length (f) is given by the mirror formula: 1/v + 1/u = 1/f.
• Magnification (m) is the ratio of the height of the image to the height of the object, expressed as m = h'/h = -v/u.
• For a real and inverted image, the magnification is negative. If the image is enlarged, the absolute value of m is greater than 1.

Important Points

• Sign Convention: According to the Cartesian sign convention, the focal length of a concave mirror is always negative, while that of a convex mirror is positive.
• Real vs Virtual Image: A real image can be obtained on a screen because the light rays actually intersect. A virtual image cannot be obtained on a screen as the rays only appear to diverge from a point.
• Common Applications: Concave mirrors are used in reflecting telescopes, shaving mirrors to see an enlarged view of the face, and by dentists to see enlarged images of teeth. They are also used in headlights of vehicles and searchlights to obtain powerful parallel beams of light when the source is kept at the focus.

The correct answer is A saturated alicyclic compound composed of six carbon atoms in a ring.

Key Points

• Cyclohexane is a cycloalkane with the molecular formula C6H12. It consists of six carbon atoms arranged in a ring, with each carbon atom bonded to two hydrogen atoms.
• The term saturated indicates that all the carbon-carbon bonds in the molecule are single bonds, and it contains the maximum possible number of hydrogen atoms for its cyclic structure. Hence, statement 4 is correct.
• It is classified as an alicyclic compound because it is both aliphatic (non-aromatic) and cyclic. It does not possess a linear chain of atoms. Hence, statement 3 is incorrect.
• Unlike benzene, cyclohexane does not have a delocalised π-electron ring system and is not aromatic. Each carbon in cyclohexane is sp3 hybridized. Hence, statement 2 is incorrect.
• Cyclohexane is a simple ring structure and is not classified as a branched hydrocarbon, nor does it contain unsaturated double or triple bonds. Hence, statement 1 is incorrect.

Additional Information

• Structure and Conformations:
• Due to the tetrahedral bond angles of sp3 hybridized carbon, cyclohexane is not a planar molecule. It adopts various conformations to minimize torsional strain and angle strain.
• The most stable arrangement is the chair conformation, followed by the twist-boat, boat, and half-chair conformations.
• Physical Properties:
• Cyclohexane is a colorless, flammable liquid at room temperature with a detergent-like odor.
• It has a molar mass of approximately 84.16 g/mol, a density of 0.779 g/cm³, and a boiling point of 80.7 °C.
• Industrial Production and Uses:
• On a large scale, it is produced by the hydrogenation of benzene in the presence of a nickel or platinum catalyst.
• It serves as a critical intermediate in the production of nylon. Specifically, it is oxidized to produce cyclohexanone and cyclohexanol, which are then used to manufacture adipic acid and caprolactam.
• It is also widely used as a non-polar solvent in chemical laboratories and industrial processes.

Important Points

• Alicyclic vs Aromatic:
• Alicyclic compounds (like cyclohexane) behave more like open-chain aliphatic compounds in their chemical reactions, such as free radical substitution.
• Aromatic compounds (like benzene, C6H6) follow Huckel's Rule (4n + 2 π electrons) and undergo electrophilic aromatic substitution rather than addition.
• Hybridization: In cyclohexane, every carbon atom uses four sp3 hybrid orbitals to form sigma (σ) bonds, resulting in a bond angle of approximately 109.5°.
• Cycloalkanes General Formula: The general formula for a monocyclic cycloalkane like cyclohexane is CnH2n, which differs from the open-chain alkane formula CnH2n+2 because of the ring closure.

The correct answer is Ethene.

Key Points

• Hydrocarbons are organic compounds consisting entirely of Hydrogen and Carbon atoms. They are classified into two main categories: saturated and unsaturated based on the types of bonds present between the carbon atoms.
• Unsaturated hydrocarbons are those hydrocarbons that contain at least one double bond or triple bond between adjacent carbon atoms. These compounds are more reactive than saturated ones.
• Ethene (C2H4), also known as ethylene, belongs to the alkene group. It contains a carbon-carbon double bond (C=C), which makes it an unsaturated hydrocarbon. Hence, option 1 is the correct answer.
• Methane (CH4) is the simplest alkane. It consists of a single carbon atom bonded to four hydrogen atoms through single covalent bonds, making it a saturated hydrocarbon. Hence, option 3 is incorrect.
• Propane (C3H8) and Butane (C4H10) are also members of the alkane series. They consist of carbon chains where all carbon-carbon bonds are single bonds. These are classified as saturated hydrocarbons. Hence, options 2 and 4 are incorrect.

Additional Information

• Alkanes (Saturated Hydrocarbons):
• The general formula for alkanes is CnH2n+2.
• In these molecules, every carbon atom is sp3 hybridized, forming four sigma bonds.
• Common examples include Ethane, Pentane, and Hexane. They are generally used as fuels due to their high combustion energy.
• Alkenes and Alkynes (Unsaturated Hydrocarbons):
• Alkenes contain at least one double bond and follow the general formula CnH2n. Carbon atoms in the double bond are sp2 hybridized.
• Alkynes contain at least one triple bond and follow the general formula CnH2n-2 (e.g., Ethyne or Acetylene). Carbon atoms in the triple bond are sp hybridized.
• Unsaturated hydrocarbons undergo addition reactions (like hydrogenation or bromination) to become saturated.
• Chemical Tests:
• The Bromine Water Test is used to distinguish between saturated and unsaturated compounds. An unsaturated hydrocarbon will decolorize the reddish-brown bromine water, while a saturated one will not react under normal conditions.

Important Points

• Methane is the primary component of Natural Gas and Biogas. It is a potent greenhouse gas but is widely used as a clean-burning fuel.
• Propane and Butane are the main constituents of Liquefied Petroleum Gas (LPG), which is used globally for cooking and heating.
• Ethene plays a vital biological role as a plant hormone responsible for the ripening of fruits. Industrially, it is the precursor for polyethylene (polythene), the most common plastic in use today.
• The number of carbon atoms determines the prefix of the name: Meth- (1), Eth- (2), Prop- (3), But- (4), Pent- (5), etc.

The correct answer is Higher pressure reduces the space between particles, leading to liquefaction.

Key Points

• Liquefaction of gases is the physical process by which a substance in the gaseous state is converted into the liquid state. This is typically achieved through the manipulation of pressure and temperature.
• According to the Kinetic Molecular Theory, the particles in a gas are separated by large distances and possess high kinetic energy, which allows them to move randomly. When high pressure is applied to a gas, the intermolecular space between the particles is significantly reduced. Hence, the primary explanation for liquefaction under pressure is the reduction of space between particles.
• As the particles are forced closer together by external pressure, the intermolecular forces of attraction (such as Van der Waals forces) become strong enough to bind the particles together, causing the gas to condense into a liquid. Hence, statement 1 is correct.
• The molecular size of a substance does not change during this process; rather, it is the void volume or the "empty space" between the molecules that decreases. Hence, statement 2 is incorrect.
• While gases do have the property of expansibility and can fill any container, pressure has a profound effect on their state. Increasing pressure decreases volume (as per Boyle’s Law), making liquefaction possible. Hence, statement 3 is incorrect.
• Applying pressure can sometimes lead to an increase in temperature (adiabatic compression), but this increase actually hinders liquefaction because higher temperatures increase the kinetic energy of the particles, helping them resist attractive forces. For liquefaction to be successful, the heat generated must be removed by cooling. Hence, statement 4 is incorrect.

Additional Information

• Critical Temperature: This is the maximum temperature at which a gas can be liquefied by the application of pressure alone. Above this temperature, the kinetic energy of the molecules is so high that no amount of pressure can force them into a liquid state. For example, the critical temperature of Carbon Dioxide (CO2) is 31.1 degrees Celsius.
• Critical Pressure: This refers to the minimum pressure required to liquefy a gas exactly at its critical temperature. Every gas has a unique set of critical constants that determine its behavior under compression.
• Thomas Andrews’ Experiments: The relationship between pressure, volume, and temperature (PVT) for gases was famously studied by Thomas Andrews using Carbon Dioxide. His isotherms demonstrated that liquefaction is a gradual transition where gas and liquid exist in equilibrium until the process is complete.
• Applications of Liquefaction: This scientific principle is used in several industrial applications, such as the production of Liquid Petroleum Gas (LPG) for domestic fuel, Liquid Oxygen for medical and aerospace purposes, and Liquid Nitrogen which is used in cryogenics and food preservation at temperatures around -196 degrees Celsius.

Important Points

• Joule-Thomson Effect: In many industrial processes, gases are liquefied using the Joule-Thomson effect, which involves the cooling of a real gas as it expands from a region of high pressure to low pressure through a porous plug or a valve.
• Intermolecular Forces: The ease with which a gas can be liquefied depends on the strength of its intermolecular forces. Gases like Ammonia (NH3) and Sulfur Dioxide (SO2) are easier to liquefy than Permanent Gases like Hydrogen (H2) or Helium (He) because they have stronger attractive forces.
• Phase Transitions: Liquefaction is an exothermic process, meaning it releases latent heat. To convert a gas to a liquid, the system must lose energy to the surroundings, which is why cooling is almost always paired with compression in practical setups.

The correct answer is ciliated columnar epithelium.

Key Points

• Ciliated columnar epithelium is a specialized type of epithelial tissue consisting of tall, pillar-like cells that possess hair-like protoplasmic outgrowths called cilia on their free surface.
• These cilia are capable of rhythmic beating, which creates a current to move mucus, eggs, or other particles in a specific direction along the surface of the cell layer.
• In the respiratory tract, specifically the trachea and bronchi, this tissue plays a vital role in the muco-ciliary escalator, where it moves trapped dust particles and microbes towards the throat to be expelled. Hence, the statement regarding mucus movement is correct.
• The nucleus of these cells is typically oval-shaped and located near the basal region of the cell, providing enough space for the ciliary apparatus at the apical end.
• Apart from the respiratory tract, this tissue is also found in the fallopian tubes (oviducts) of the female reproductive system, where it helps in the propulsion of the ovum towards the uterus.

Additional Information

• Simple Squamous Epithelium:
• It consists of a single layer of extremely thin and flat cells, forming a delicate lining that looks like floor tiles.
• Its primary function is to facilitate the transport of substances through a selectively permeable surface via diffusion or filtration.
• It is found in locations where rapid exchange is required, such as the alveoli of the lungs, the lining of blood vessels (endothelium), and Bowman's capsule in the kidneys.
• Cuboidal Epithelium:
• As the name suggests, the cells are cube-shaped in appearance with centrally located spherical nuclei.
• This tissue provides mechanical support and is primarily involved in secretion and absorption.
• Common locations include the lining of kidney tubules, ducts of salivary glands, and the surface of the ovaries (germinal epithelium).
• Stratified Squamous Epithelium:
• This tissue is composed of multiple layers of cells arranged in a pattern to prevent wear and tear and mechanical injury.
• The deepest layers are often columnar or cuboidal, while the superficial layers are flat (squamous).
• It is found in the skin (where it is keratinized to prevent water loss) and the lining of the mouth and esophagus (non-keratinized).

Important Points

• Goblet Cells: These are specialized unicellular glands often interspersed within the ciliated columnar epithelium of the respiratory tract to secrete mucus, which traps foreign particles.
• Basement Membrane: Like all epithelial tissues, these cells rest on a non-cellular fibrous layer called the basement membrane, which separates them from the underlying connective tissue.
• Clinical Relevance: Damage to the cilia (e.g., due to smoking or pollution) impairs the movement of mucus, leading to respiratory infections and conditions like chronic bronchitis.
• Classification: Epithelial tissues are classified based on the number of layers (Simple vs. Stratified) and the shape of the cells (Squamous, Cuboidal, or Columnar).

The correct answer is the weight of water displaced equals the weight of the ship.

Key Points

• A ship floats on water based on Archimedes' Principle, which states that any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
• According to the Law of Flotation, for an object to float, the weight of the liquid displaced by its submerged part must be exactly equal to the total weight of the object. Hence, statement 2 is correct.
• The Upthrust or Buoyant Force acts in the upward direction, opposing the weight of the ship (gravity) that acts downwards. When these two forces are in equilibrium, the ship stays afloat.
• Atmospheric pressure acts uniformly on the surface of the water and the ship from all sides; it does not provide the net upward force required for flotation. Hence, statement 1 is incorrect.
• While surface tension allows very light objects like needles or insects to stay on the water's surface, it is a molecular force that is far too weak to support the massive weight of a steel ship. Hence, statement 3 is incorrect.
• Most ships are made of steel or iron, which have a density much higher than that of water (7.8 g/cm³ compared to 1 g/cm³). Therefore, the "material" itself is denser than water, but the average density of the ship (including the large volume of air trapped in the hull) is less than that of water. Hence, statement 4 is incorrect.

Additional Information

• Buoyancy: This is the upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus, the pressure at the bottom of a submerged object is greater than at the top, resulting in a net upward force.
• Density and Volume: The reason a heavy iron ship floats while a small iron nail sinks is due to volume. The ship is designed with a hollow hull, which increases its total volume without significantly increasing its mass, thereby reducing its average density to a value lower than the density of water.
• Relative Density: Also known as Specific Gravity, it is the ratio of the density of a substance to the density of a reference material (usually water at 4 degrees Celsius). If an object's relative density is less than 1, it will float.
• Displacement: When a ship is placed in water, it pushes the water aside. The amount of water "moved" is called displacement. For a ship to remain stable, the Center of Gravity must be vertically aligned with the Center of Buoyancy (the geometric center of the displaced water).

Important Points

• Plimsoll Line: This is a marking on a ship's hull that indicates the maximum depth to which the vessel may be safely immersed when loaded with cargo. This depth varies based on the water's salinity (density) and temperature.
• Saltwater vs. Freshwater: Objects float higher in saltwater (like the ocean) than in freshwater because saltwater is denser (approx. 1025 kg/m³) and provides a greater buoyant force for the same volume of displacement.
• The Dead Sea: Due to its extremely high salt concentration (about 34%), the density of the water is so high that humans can float effortlessly on its surface without swimming.

The correct answer is It will increase.

Key Points

• In a series connection, electrical components or resistors are joined end-to-end in a single path, meaning the current has only one route to flow through the circuit.
• The equivalent resistance (also known as total resistance) of resistors connected in series is calculated by the arithmetic sum of the individual resistances. The formula is expressed as R_total = R1 + R2 + ... + Rn.
• When a second resistor is added in series to an existing one, the total opposition to the flow of electric charge is increased. Consequently, the equivalent resistance of the circuit will increase. Hence, the statement that it will increase is correct.
• In a series circuit, the current remains constant at every point in the loop, while the total voltage provided by the source is distributed across each resistor based on its resistance value.
• If any single component in a series circuit breaks or is disconnected, the entire current flow stops, causing the whole circuit to fail.

Additional Information

• Parallel Connection:
• In a parallel circuit, resistors are connected across the same two nodes, providing multiple paths for the current.
• The equivalent resistance in parallel is always less than the smallest individual resistor in the group. The formula used is 1/R_total = 1/R1 + 1/R2.
• In parallel, the voltage remains the same across all branches, but the current divides among them.
• Ohm's Law:
• Formulated by Georg Simon Ohm in 1827, it states that the current (I) passing through a conductor is directly proportional to the voltage (V) across it, provided physical conditions like temperature remain constant (V = IR).
• The SI unit of resistance is the Ohm, symbolized by the Greek letter Omega (Ω).
• Practical Applications:
• Series circuits are used in low-power applications like decorative fairy lights or flashlights.
• Parallel circuits are standard for household wiring, ensuring that turning off one appliance does not shut down the entire home's electricity.

Important Points

• Factors Affecting Resistance:
• Length: The resistance of a conductor is directly proportional to its length. Connecting resistors in series is physically equivalent to increasing the total length of the resistive material.
• Cross-sectional Area: Resistance is inversely proportional to the area of the conductor. A thicker wire has less resistance.
• Nature of Material: Different materials have different resistivity values; for instance, Silver and Copper are excellent conductors with very low resistance.
• Temperature: For most metallic conductors, resistance increases with an increase in temperature.
• Energy and Power:
• In a series circuit, the resistor with the highest numerical value of resistance will consume the most power and dissipate the most heat, according to the formula P = I^2 * R.

The correct answer is CaO + CO2 → CaCO3.

Key Points

• A combination reaction (also known as a synthesis reaction) is a reaction where two or more reactants combine to form a single product.
• The specific condition for this question is that two compounds must react to form a new compound. A compound is a substance formed when two or more chemical elements are chemically bonded together in a fixed ratio.
• In the reaction CaO + CO2 → CaCO3, Calcium Oxide (CaO) is a compound consisting of calcium and oxygen, and Carbon Dioxide (CO2) is a compound consisting of carbon and oxygen. They combine to form Calcium Carbonate (CaCO3). Hence, statement 1 is correct.
• In the reaction C + O2 → CO2, Carbon (C) and Oxygen (O2) are elements in their standard states. While it is a combination reaction, it involves elements rather than two compounds. Hence, statement 2 is incorrect.
• In the reaction Na + Cl2 → NaCl (unbalanced form), Sodium (Na) and Chlorine (Cl2) are elements. They combine to form the compound sodium chloride, but the reactants are not compounds. Hence, statement 3 is incorrect.
• In the reaction H2 + Cl2 → HCl (unbalanced form), Hydrogen (H2) and Chlorine (Cl2) are diatomic elements. They react to form hydrogen chloride, which is a compound, but since the reactants are elements, it does not meet the "two compounds" criteria. Hence, statement 4 is incorrect.

Additional Information

• Chemical Compound: A compound is a material formed by chemically bonding two or more chemical elements. The bonds in a compound can be ionic or covalent. Examples include water (H2O), table salt (NaCl), and glucose (C6H12O6).
• Chemical Element: An element is a pure substance consisting only of atoms that all have the same number of protons in their atomic nuclei. Elements cannot be broken down into simpler substances by chemical means. Examples include Iron (Fe), Gold (Au), and Nitrogen (N2).
• Classification of Combination Reactions:
• Between two elements: Example: 2Mg + O2 → 2MgO (Magnesium and Oxygen are elements).
• Between an element and a compound: Example: 2CO + O2 → 2CO2 (Carbon monoxide is a compound, Oxygen is an element).
• Between two compounds: Example: NH3 + HCl → NH4Cl (Ammonia and Hydrogen Chloride are both compounds).
• Calcium Oxide (CaO): Commonly known as quicklime or burnt lime, it is a widely used chemical compound. It is a white, caustic, alkaline, crystalline solid at room temperature. It is often used in the manufacture of cement and in metallurgy.
• Calcium Carbonate (CaCO3): This compound is found naturally as limestone, chalk, and marble. It is also the main component of pearls and the shells of marine organisms and eggs.
• Nature of Combination Reactions: Most combination reactions are exothermic in nature because they involve the formation of new bonds, which releases energy to the surroundings.

The correct answer is Green.

Key Points

• When copper objects are exposed to moist air for a long period, they lose their shiny brown surface and acquire a dull green coating.
• This green material is a mixture of copper hydroxide [Cu(OH)₂] and copper carbonate [CuCO₃], which is formed due to the reaction of copper with oxygen, water, and carbon dioxide in the atmosphere.
• The chemical reaction for this process is represented as: 2Cu + H₂O + CO₂ + O₂ → Cu(OH)₂ + CuCO₃.
• This process is a type of corrosion. Unlike iron rust, this green layer, often called a patina, actually protects the underlying metal from further deep-seated corrosion.
• The patina forms naturally over time, usually taking several years to fully develop depending on the environmental conditions such as humidity and air pollution levels.

Additional Information

• Statue of Liberty:
• The most famous example of copper corrosion is the Statue of Liberty in New York.
• Originally made of shining brown copper, it turned entirely green by the year 1906, roughly 20 years after its dedication, due to exposure to the elements.
• Corrosion of Other Metals:
• Iron: Undergoes rusting when exposed to air and moisture, forming a reddish-brown flaky substance known as hydrated ferric oxide.
• Silver: Develops a black coating when exposed to air because it reacts with sulphur compounds to form silver sulphide (Ag₂S).
• Aluminium: Forms a thin, transparent layer of aluminium oxide which, like copper patina, protects the metal from further oxidation.
• Prevention of Corrosion:
• Common methods to prevent corrosion include galvanization (coating with zinc), electroplating, painting, oiling, or greasing.
• Creating alloys, such as stainless steel (a mixture of iron, carbon, chromium, and nickel), is also an effective way to stop metals from corroding.
• Factors Affecting Corrosion:
• Reactivity: Highly reactive metals like sodium and potassium corrode very quickly.
• Presence of Electrolytes: The presence of salt in water (like seawater) accelerates the rate of electrochemical corrosion.
• Pollution: Presence of oxides of sulphur and nitrogen in the air increases the acidity of moisture, speeding up the corrosion process.

Important Points

• Basic Copper Carbonate: The specific chemical name for the green coating on copper is basic copper carbonate.
• Reddish-Brown: This is the natural color of pure, unoxidized copper. The option "Red" in the question refers to the initial state rather than the corroded state.
• Aesthetic Uses: In architecture, architects often use copper for roofing specifically because they desire the green patina look that develops over decades.
• Environmental Indicator: The speed at which copper turns green can be an indicator of acid rain or industrial pollution in a specific location.

The correct answer is The SI unit of linear velocity is m/s2.

Key Points

• Linear velocity is defined as the rate of change of displacement with respect to time. It indicates how fast an object is changing its position in a specific direction. Hence, statement 3 is correct.
• The mathematical formula for velocity is v = s / t, where s represents displacement and t represents the time interval. This shows that the value of velocity is directly dependent on both the change in position and the duration of the motion. Hence, statement 2 is correct.
• In uniform motion, an object covers equal displacements in equal intervals of time along a straight line. Because neither the magnitude (speed) nor the direction changes, the linear velocity remains constant throughout the motion. Hence, statement 1 is correct.
• The SI unit of linear velocity is meters per second (m/s). The unit m/s2 (meters per second squared) is the SI unit of acceleration, which measures the rate at which velocity changes over time. Hence, statement 4 is incorrect.
• Velocity is a vector quantity, meaning it is characterized by both magnitude and direction. A change in either the speed or the direction of an object will result in a change in its velocity.

Additional Information

• Difference between Speed and Velocity: While speed is a scalar quantity that only accounts for how fast an object is moving, velocity is the vector counterpart that specifies the direction of motion. For example, 50 km/h is speed, while 50 km/h towards the North is velocity.
• Average Velocity: This is calculated as the total displacement divided by the total time taken for a journey. Unlike average speed, the average velocity can be zero if the starting and ending points are the same (zero displacement).
• Instantaneous Velocity: This refers to the velocity of an object at a specific instant in time or at a specific point in its path. It is often measured by the speedometer of a vehicle combined with a compass for direction.
• Dimensional Formula: In physics, the dimensional representation of linear velocity is [M0 L1 T-1], where M is mass, L is length, and T is time.

Important Points

• Acceleration: It is the rate of change of velocity. If an object's velocity increases, decreases, or changes direction, the object is said to be accelerating. Its unit is m/s2.
• Negative Velocity: Velocity can be negative, positive, or zero, depending on the chosen coordinate system and the direction of the displacement.
• Uniform Circular Motion: Even if an object moves at a constant speed in a circle, its linear velocity is not constant because its direction is continuously changing at every point along the circular path.
• Relative Velocity: This is the velocity of an object with respect to another observer or reference frame. For instance, the velocity of a moving train appears different to a person standing on the platform compared to a person in another moving train.

The correct answer is four.

Key Points

• Joule's Law of Heating describes the rate at which resistance in an electrical circuit converts electrical energy into heat energy.
• The mathematical expression for Joule's law is H = I^2Rt, where H represents the heat produced, I is the electric current, R is the resistance of the conductor, and t is the time for which the current flows.
• According to the law, the heat produced (H) is directly proportional to the square of the current (I^2) flowing through the resistor.
• If the current is doubled (from I to 2I), the new heat produced becomes H' = (2I)^2 * R * t, which simplifies to H' = 4 * I^2 * R * t.
• Comparing the new heat to the original, H' = 4H. Therefore, the heat produced increases by a factor of four when the current is doubled and other factors remain constant. Hence, the statement is correct.

Additional Information

• James Prescott Joule: The law is named after the English physicist James Prescott Joule, who studied the relationship between mechanical work and heat transfer in the 1840s.
• SI Unit of Heat: The standard SI unit for heat and energy is the Joule (J). One Joule is equal to the work done by a force of one newton acting through a distance of one meter.
• Practical Applications:
• Electric Heaters and Irons: These devices use high-resistance coils to maximize the heat produced via Joule heating.
• Electric Bulbs: The filament (usually Tungsten) is heated to such a high temperature by the current that it begins to glow and emit light.
• Electric Fuse: A safety device that protects circuits from overloading. When the current exceeds a certain limit, the heat produced melts the fuse wire, breaking the circuit.
• Dependence on Resistance: For a constant current and time, the heat produced is directly proportional to the resistance (R). If resistance is doubled, the heat is also doubled.
• Dependence on Time: Heat produced is directly proportional to the duration (t) of current flow. Extending the time the appliance is "on" linearly increases the total heat generated.

Important Points

• Relationship with Power: Electrical Power (P) is defined as the rate at which electrical energy is dissipated as heat, expressed as P = H / t = I^2R.
• Ohm’s Law Integration: By substituting V = IR (Ohm’s Law) into the heating formula, heat can also be expressed as H = VIt or H = (V^2 / R) * t, where V is the potential difference.
• Energy Conservation: Joule heating is an illustration of the Law of Conservation of Energy, where electrical energy is transformed into thermal energy.
• Unwanted Heating: In many electronic components like computers and transformers, Joule heating is considered a "loss" of energy and can cause damage if not managed by cooling fans or heat sinks.

The correct answer is Binary fission in Amoeba.

Key Points

• Asexual reproduction is a mode of reproduction where offspring are produced from a single parent without the involvement of gamete formation or fertilization.
• Binary fission in Amoeba is a classic example of asexual reproduction where the parent unicellular organism divides into two equal daughter cells. Since it involves only one parent and no gametes, it fits the description perfectly. Hence, statement 3 is correct.
• Fertilisation in frogs is a type of sexual reproduction that involves the fusion of male gametes (sperm) and female gametes (ova), typically occurring in water (external fertilization). Hence, statement 1 is incorrect.
• Seed formation in flowering plants is the result of sexual reproduction, involving the process of pollination followed by the fusion of pollen grains and ovules to form a zygote. Hence, statement 2 is incorrect.
• Internal fertilisation in mammals involves the union of gametes inside the body of the female parent, requiring two parents and resulting in genetic variation. Hence, statement 4 is incorrect.

Additional Information

• Types of Asexual Reproduction:
• Budding: A small outgrowth or bud develops on the parent body, which eventually detaches to become a new individual. Common examples include Hydra and Yeast.
• Fragmentation: The parent body breaks into distinct pieces, and each piece grows into a complete new organism. This is seen in Spirogyra and Planaria.
• Spore Formation: Organisms produce microscopic reproductive units called spores that can survive harsh conditions and germinate when environments become favorable. Example: Rhizopus (bread mould).
• Vegetative Propagation: In plants, new individuals are produced from vegetative parts like roots, stems, or leaves rather than seeds. Examples include potato tubers, onion bulbs, and Bryophyllum leaves.
• Characteristics of Asexual Offspring:
• The offspring produced are genetically identical to the parent and to each other, often referred to as clones.
• This process is generally faster and more energy-efficient than sexual reproduction because it does not require finding a mate.

Important Points

• Mechanics of Binary Fission:
• The process begins with Karyokinesis, which is the division of the nucleus. In Amoeba, the DNA replicates, and the nucleus elongates and splits into two daughter nuclei.
• This is followed by Cytokinesis, where the cytoplasm of the cell divides, resulting in two separate organisms.
• In Amoeba, the division can take place in any plane, whereas in organisms like Leishmania, it occurs in a definite orientation relative to structures like the flagellum.
• Comparison with Sexual Reproduction:
• While asexual reproduction ensures rapid population growth, it lacks genetic diversity, making the population more vulnerable to environmental changes or diseases.
• Sexual reproduction, involving meiosis and gamete fusion, introduces variation which is a key driver for evolution and adaptation in species.

The correct answer is Cutting.

Key Points

• Vegetative propagation is a type of asexual reproduction in plants where new plants are produced from vegetative parts such as roots, stems, and leaves rather than from seeds.
• Cutting is an artificial vegetative propagation technique where a small part of a plant, like a stem, root, or leaf, is cut and placed in soil or water to develop into a new, independent plant. Hence, Option 3 is correct.
• Seed dispersal refers to the mechanism by which plant seeds are transported to new locations by wind, water, animals, or gravity to reduce competition with the parent plant. Hence, statement 1 is incorrect.
• Fertilisation is a process in sexual reproduction involving the fusion of male and female gametes to form a zygote, eventually leading to seed formation. Hence, statement 2 is incorrect.
• Pollination is the act of transferring pollen grains from the male anther of a flower to the female stigma, which is a prerequisite for sexual reproduction in flowering plants. Hence, statement 4 is incorrect.

Additional Information

• Artificial Methods of Vegetative Propagation:
• Layering: A branch of the plant is bent to the ground and covered with soil while still attached to the parent; it develops roots and is later separated.
• Grafting: This involves joining the Scion (a part of one plant) with the Stock (the rooted part of another plant) so they grow as a single plant.
• Tissue Culture: Also known as micropropagation, it involves growing plant cells or small pieces of plant tissue in a sterile nutrient medium under laboratory conditions.
• Natural Methods of Vegetative Propagation:
• Tubers: Thickened underground stems like Potato that have "eyes" or buds.
• Rhizomes: Horizontal underground stems found in plants like Ginger and Turmeric.
• Bulbs: Short underground stems surrounded by fleshy leaves, as seen in Onions and Lilies.
• Runners/Stolons: Creeping stems that grow along the ground, such as in Strawberries.

Important Points

• Advantages of Vegetative Propagation: It allows for the rapid multiplication of plants and ensures that the offspring are genetically identical (clones) to the parent, preserving desirable traits.
• Economic Uses: This method is vital for the commercial production of plants that do not produce seeds or have very low seed viability, such as Bananas, Sugarcane, Grapes, and Roses.
• Specific Examples: In Bryophyllum, vegetative propagation occurs through small adventitious buds located along the margins of the leaves, which fall off and grow into new plants.
• Stem Cuttings: Plants like Hibiscus, Bougainvillea, and Sugarcane are widely grown using stem cuttings because it is a simple and cost-effective method for farmers.

The correct answer is Brain, spinal cord and nerves.

Key Points

• Nervous tissue is a specialized primary tissue type that forms the nervous system, which is responsible for coordinating and controlling many body activities.
• The brain, spinal cord, and nerves are the fundamental structures composed almost entirely of this tissue. Hence, statement 4 is correct.
• This tissue is characterized by its ability to exhibit excitability and conductivity, allowing it to receive stimuli and transmit electrical impulses across long distances within the body.
• The primary cells of the nervous tissue are neurons, which are highly specialized cells that process and transmit information.
• Supporting these neurons are neuroglia (or glial cells), which provide nutrients, insulation, and structural support, although they do not conduct electrical impulses themselves.

Additional Information

• Structure of a Neuron:
• Every neuron consists of three main parts: the cell body (cyton), dendrites, and an axon.
• The cell body contains the nucleus and cytoplasm. Dendrites are short, branched projections that receive signals from other neurons.
• The axon is a long, single fiber that carries the nerve impulse away from the cell body toward a synapse or an effector organ.
• Classification of the Nervous System:
• The Central Nervous System (CNS) includes the brain and the spinal cord, acting as the integration and command center.
• The Peripheral Nervous System (PNS) consists of the nerves (cranial and spinal) that branch out from the CNS to connect it to the rest of the body.
• Other Tissue Types in the Human Body:
• Epithelial Tissue: Covers body surfaces and lines internal organs (e.g., skin, lining of the liver, and kidneys).
• Connective Tissue: Supports and binds other tissues (e.g., bones, cartilage, and blood).
• Muscular Tissue: Responsible for movement; includes skeletal, smooth, and cardiac muscle (found in the heart).

Important Points

• Kidneys and Liver: These organs are primarily composed of epithelial tissue (glandular epithelium) and connective tissue. While they are innervated by nerves, they are not "made up" of nervous tissue. Hence, statement 1 is incorrect.
• Lungs and Heart: The lungs consist of epithelial and connective tissues, while the heart is uniquely composed of cardiac muscular tissue. Hence, statement 2 is incorrect.
• Bones and Muscles: Bones are a form of mineralized connective tissue, and muscles are made of muscular tissue. Hence, statement 3 is incorrect.
• Synapse: The functional junction between two neurons where the neurotransmitter chemical is released to bridge the gap and continue the signal.
• The nerve impulse travels in one direction: from the dendrite to the cell body and then along the axon.

The correct answer is 81 u.

Key Points

• The formula unit mass of a substance is the sum of the atomic masses of all atoms in a formula unit of a compound.
• To calculate the formula unit mass of Zinc Oxide (ZnO), we need to consider the individual atomic masses of its constituent elements: Zinc (Zn) and Oxygen (O).
• The atomic mass of Zinc (Zn) is approximately 65 u.
• The atomic mass of Oxygen (O) is approximately 16 u.
• The formula ZnO indicates that one formula unit consists of one atom of Zinc and one atom of Oxygen.
• Therefore, the formula unit mass is calculated as: 1 x 65 u + 1 x 16 u = 81 u. Hence, the correct answer is 81 u.

Additional Information

• Definition of Formula Unit Mass:
• The term formula unit mass is used for those substances whose constituent particles are ions. While the calculation method is the same as molecular mass, the terminology differs for ionic compounds.
• For example, Sodium Chloride (NaCl) has a formula unit mass of 58.5 u (23 u for Na + 35.5 u for Cl).
• Understanding the Unit 'u':
• The symbol u stands for unified atomic mass unit. It was formerly known as amu (atomic mass unit).
• By international agreement, 1 u is defined as exactly 1/12th the mass of an atom of carbon-12.
• Properties and Uses of Zinc Oxide (ZnO):
• Zinc Oxide is an inorganic compound that usually appears as a white powder.
• It is insoluble in water and is widely used as an additive in numerous materials and products including rubbers, plastics, ceramics, glass, cement, lubricants, paints, ointments, adhesives, sealants, pigments, foods, batteries, ferrites, and fire retardants.
• In medicine, it is used to treat skin conditions like diaper rash and minor burns due to its antiseptic and protective properties.
• Molecular Mass vs. Formula Unit Mass:
• Molecular mass is used for compounds that exist as discrete molecules (like H2O or CO2).
• Formula unit mass is used for crystalline solids or ionic networks where atoms are arranged in a three-dimensional lattice.

Important Points

• Atomic Mass of Zinc: The exact value of Zinc on the periodic table is often given as 65.38 u, but for standard textbook calculations in exams, it is typically rounded to 65 u.
• Amphoteric Nature: ZnO is an amphoteric oxide, meaning it can react with both acids and bases to produce salts and water.
• Semiconductor Properties: Zinc Oxide is a wide-bandgap semiconductor of the II-VI semiconductor group, making it useful in electronics and optoelectronics.

The correct answer is tone and note.

Key Points

• A tone is defined as a sound that consists of a single frequency. It is often referred to as a "pure" sound because it lacks the complexity of multiple overlapping waves.
• A note is a sound produced by a mixture of several frequencies. When multiple tones are combined, they create a note, which is typically what we perceive when listening to musical instruments.
• In the context of music, a note is often more pleasant to the ear than a single tone because of its rich harmonic content and complexity.
• While a tone is characterized solely by its frequency and amplitude, a note also possesses a specific quality or timbre that allows listeners to distinguish between different sources, such as a guitar versus a flute, even if they are playing the same fundamental frequency.
• Noise, by contrast, is a collection of sounds that are unpleasant to the ear and consist of a random mixture of frequencies with no periodic structure.

Additional Information

• Frequency and Pitch:
• Frequency refers to the number of vibrations per second and is measured in Hertz (Hz).
• The pitch of a sound is how the human brain interprets the frequency; a high-frequency sound results in a high pitch, while a low-frequency sound results in a low pitch.
• Loudness and Amplitude:
• The loudness of a sound depends on the amplitude of the sound wave. Larger amplitudes produce louder sounds, measured in decibels (dB).
• Loudness is proportional to the square of the amplitude of the vibration producing the sound.
• Quality (Timbre):
• Quality or timbre is the characteristic that enables us to distinguish one sound from another having the same pitch and loudness.
• This is why a musical note played on a piano sounds different from the same note played on a violin.
• Audible Range:
• The human ear is typically sensitive to frequencies ranging from 20 Hz to 20,000 Hz (20 kHz).
• Sounds below 20 Hz are called infrasonic, and sounds above 20,000 Hz are called ultrasonic.

Important Points

• A pure tone can be produced by a tuning fork, which vibrates at one specific frequency.
• Most musical instruments do not produce pure tones; they produce a fundamental frequency along with several overtones or harmonics, which together form a note.
• Echo is the repetition of sound caused by the reflection of sound waves from a surface. To hear a distinct echo, the minimum distance between the source and the reflecting surface must be approximately 17.2 meters.
• The speed of sound varies depending on the medium; it travels fastest in solids, slower in liquids, and slowest in gases. In air at 0 degrees Celsius, the speed of sound is approximately 331 meters per second.

The correct answer is It acts as an electrolyte to help form a thicker oxide layer.

Key Points

• Anodising is an electrochemical process that converts the metal surface into a decorative, durable, and corrosion-resistant anodic oxide finish.
• In this process, Sulphuric Acid (usually at a concentration of 15% to 20%) serves as the electrolyte. An electrolyte is a medium that contains ions and conducts electricity through the movement of those ions.
• During the reaction, the aluminium object is made the anode (the positive electrode) in an electrolytic cell containing the acid. When an electric current is passed through the solution, water molecules are decomposed.
• Oxygen is released at the anode and reacts with the aluminium surface to form a controlled, uniform, and much thicker layer of aluminium oxide (Al2O3) than would occur naturally. Hence, the statement that it acts as an electrolyte to help form a thicker oxide layer is correct.
• While Sulphuric Acid does have the potential to dissolve aluminium, in the context of anodising, its primary role is to facilitate the oxidation process by allowing current to flow and creating micro-pores that help the oxide layer grow deeper into the metal.

Additional Information

• Structure of the Oxide Layer: The layer formed during Sulphuric Acid anodising (often called Type II Anodizing) is porous. These microscopic pores are essential because they allow the electrolyte to reach the unoxidized aluminium beneath, enabling the layer to continue growing to a thickness of 5 to 25 microns or more.
• Corrosion Resistance and Durability: The resulting aluminium oxide layer is extremely hard (second only to diamond in some forms) and provides excellent wear resistance and protection against atmospheric corrosion.
• Coloring and Aesthetics: Because the anodic coating is porous before it is sealed, it can easily absorb dyes or pigments. This is how various colored aluminium products, such as electronics, cookware, and architectural panels, are produced.
• Sealing Process: After the oxide layer reaches the desired thickness, the part is usually "sealed" by immersion in hot water or chemical solutions. This converts the oxide into a hydrated form, closing the pores and locking in any dyes while increasing corrosion resistance.

Important Points

• Comparison with Natural Oxidation: Aluminium naturally forms a very thin, passive oxide film (about 2 to 3 nanometers thick) when exposed to air. Anodising artificially increases this thickness by thousands of times to provide industrial-grade protection.
• Other Types of Anodising:
• Type I: Uses Chromic Acid, typically for aerospace parts where a thinner but highly corrosion-resistant layer is needed.
• Type III: Also known as Hard Anodizing, it uses Sulphuric Acid at lower temperatures and higher voltages to produce a very thick (up to 50-100 microns) and wear-resistant coating.
• Temperature Control: During the process, the temperature of the Sulphuric Acid bath is usually maintained around 20°C (68°F). If the temperature rises too high, the acid begins to dissolve the oxide layer faster than it forms, resulting in a powdery or "burnt" finish.
• Environmental Safety: Sulphuric Acid is preferred over Chromic Acid in many modern applications because it is generally less toxic and easier to dispose of in compliance with environmental regulations.

The correct answer is Because the amount of dissolved oxygen in water is very low.

Key Points

• Aquatic organisms, such as fish, prawns, and mollusks, obtain the oxygen necessary for respiration from the oxygen dissolved in water.
• The concentration of oxygen in the atmosphere is approximately 21%, whereas the amount of dissolved oxygen in water is significantly lower (roughly 0.7% or 4 to 10 milligrams per liter depending on conditions).
• Because the oxygen level is so low in aquatic environments, these organisms must take in water more frequently and pass it over their respiratory organs (like gills) to extract enough oxygen to meet their metabolic demands.
• This necessity results in a much faster breathing rate in aquatic animals compared to terrestrial animals, who have access to an abundant and concentrated supply of atmospheric oxygen.
• In fish, water is taken in through the mouth and forced past the gills, where the dissolved oxygen is absorbed into the bloodstream by diffusion across thin membranes.

Additional Information

• Oxygen Solubility: The solubility of oxygen in water is affected by temperature; cold water can hold more dissolved oxygen than warm water. Consequently, aquatic life may struggle more in higher temperatures due to hypoxia.
• Gills: These are specialized respiratory structures in aquatic organisms that provide a large surface area for gas exchange. They are highly vascularized to facilitate the efficient transfer of oxygen and carbon dioxide.
• Lungs vs. Gills: While terrestrial organisms use lungs to breathe air, gills are adapted specifically for aqueous environments. Gills would collapse and dry out in air, making them ineffective for terrestrial respiration.
• Diffusion Rate: The diffusion of oxygen in air is nearly 10,000 times faster than its diffusion in water. This physical limitation forces aquatic animals to evolve highly efficient mechanisms to extract oxygen.
• Viscosity of Water: Water is about 800 times denser and much more viscous than air. This means aquatic organisms expend significantly more energy to move the medium (water) over their respiratory surfaces than terrestrial animals do with air.

Important Points

• Counter-Current Exchange: Most fish utilize a counter-current flow mechanism where blood flows in the opposite direction to the water passing over the gills. This maintains a concentration gradient that allows them to extract up to 80-90% of the dissolved oxygen.
• Metabolic Rate: Despite the faster breathing rate, many aquatic organisms have a lower metabolic rate compared to endothermic (warm-blooded) terrestrial mammals because they do not need to generate internal body heat.
• Environmental Factors: Factors such as salinity, pollution, and depth also influence the dissolved oxygen levels. For instance, saltwater typically holds less oxygen than freshwater at the same temperature.
• Evolutionary Adaptation: The high breathing rate is an evolutionary adaptation to ensure survival in an oxygen-poor medium, highlighting the biological diversity in respiratory strategies across different habitats.

The correct answer is The Moon revolving around the Earth.

• Uniform Circular Motion is defined as the motion of an object traveling at a constant speed along a circular path. While the speed remains the same, the velocity changes continuously because the direction of motion changes at every point.
• The Moon revolving around the Earth is a classic example of this motion. The gravitational pull of the Earth provides the necessary centripetal force to keep the Moon in its nearly circular orbit at a relatively constant speed. Hence, statement 3 is correct.
• A car moving on a straight road represents Rectilinear Motion or linear motion. In this case, the direction of motion does not change, and if the speed is constant, it is called uniform linear motion. Hence, statement 1 is incorrect.
• A stone dropped from a height undergoes Free Fall, which is a type of Accelerated Linear Motion. As it falls, its speed increases due to the acceleration due to gravity (approximately 9.8 m/s²), and it moves in a vertical straight line. Hence, statement 2 is incorrect.
• A train moving on a straight track is another example of Linear Motion. Even if it moves at a constant speed, the path is not circular, so it cannot be classified under circular motion. Hence, statement 4 is incorrect.
• Characteristics of Uniform Circular Motion:
• In this motion, the speed of the particle remains constant, which means it covers equal distances along the circumference in equal intervals of time.
• The velocity is always tangential to the circular path. Since the tangent changes direction at every point on the circle, the velocity is variable.
• Because the velocity is changing, the object is said to be accelerating. This acceleration is called centripetal acceleration and is always directed toward the center of the circle.
• Centripetal Force:
• For any object to perform circular motion, an external force must act upon it. This center-seeking force is known as Centripetal Force.
• In the case of planets or satellites, this force is provided by Gravity. For a stone whirled in a string, the Tension in the string provides the force.
• Real-world Examples:
• The movement of the tips of the hands of a clock is a perfect example of uniform circular motion as they move with a constant angular speed.
• An artificial satellite orbiting the Earth at a constant altitude.
• The motion of electrons revolving around the nucleus in an atom (according to the Bohr model).
• Speed vs. Velocity: It is crucial to distinguish that while speed is a scalar quantity and remains unchanged in uniform circular motion, velocity is a vector quantity and is constantly changing due to the direction shift.
• Work Done: In uniform circular motion, the work done by the centripetal force is always zero. This is because the force is perpendicular to the instantaneous displacement (tangent) at every point.
• Formula: The speed in circular motion is calculated as the circumference divided by the time period, often expressed as v = (2 * pi * r) / T, where r is the radius and T is the time taken for one revolution.

The correct answer is Ovary → Fallopian tube → Uterus.

Key Points

• The human female reproductive system is designed to produce gametes (eggs/ova) and provide an environment for the development of a fetus. The correct physiological path of an egg starts at its site of production and ends at its site of implantation or exit.
• Statement 1: Ovary → Vagina → Uterus. This path is incorrect because the vagina is located below the uterus. An egg released from the ovary must pass through the fallopian tube to reach the uterus; it does not go from the ovary directly to the vagina and then back up to the uterus. Hence, statement 1 is incorrect.
• Statement 2: Fallopian tube → Ovary → Uterus. This sequence is incorrect because the ovary is the starting point where oogenesis (egg production) occurs. The fallopian tube receives the egg from the ovary, not the other way around. Hence, statement 2 is incorrect.
• Statement 3: Uterus → Ovary → Fallopian tube. This represents a reversed or scattered path. The egg is not produced in the uterus, and it does not travel from the uterus to the ovary. Hence, statement 3 is incorrect.
• Statement 4: Ovary → Fallopian tube → Uterus. This is the biologically correct path. The ovary releases a mature ovum during ovulation, which is then captured by the fimbriae of the fallopian tube. The egg travels through the tube toward the uterus, where it may implant if fertilized. Hence, statement 4 is correct.
• Hence, the correct answer is Ovary → Fallopian tube → Uterus.

Additional Information

• Ovaries:
• These are the primary female sex organs located on each side of the lower abdomen.
• Each ovary is about 2 to 4 cm in length and is responsible for producing the female gamete (ovum) and several steroid hormones like estrogen and progesterone.
• Fallopian Tubes (Oviducts):
• Each tube is about 10-12 cm long. It consists of three parts: the infundibulum (with finger-like fimbriae), the ampulla (the wider part), and the isthmus (which joins the uterus).
• The ampulla of the fallopian tube is the typical site where fertilization occurs when a sperm meets the egg.
• Uterus (Womb):
• The uterus is a large, muscular, pear-shaped organ where a fertilized egg (zygote) implants and develops into an embryo and then a fetus.
• The wall of the uterus has three layers: the outer perimetrium, the middle myometrium (smooth muscle), and the inner endometrium (glandular layer that sheds during menstruation).
• Vagina and Cervix:
• The uterus opens into the vagina through a narrow cervix.
• The cervical canal along with the vagina forms the birth canal during parturition (delivery of the baby).

Important Points

• Ovulation: This is the process where a mature follicle bursts to release an ovum from the ovary, usually occurring around day 14 of a 28-day menstrual cycle.
• Implantation: If the egg is fertilized in the fallopian tube, it moves to the uterus and attaches to the endometrial wall. This process is called implantation.
• Menstruation: If fertilization does not occur, the ovum degenerates, and the endometrium lining of the uterus is shed through the vagina, a process known as menstruation.
• Ciliary Movement: The inner lining of the fallopian tubes contains ciliated epithelium. The beating of these cilia helps push the non-motile egg from the ovary toward the uterus.

The correct answer is Peripheral nervous system.

Key Points

• The Peripheral Nervous System (PNS) consists of all the nerves that lie outside the brain and spinal cord. Its primary function is to connect the Central Nervous System (CNS) to the limbs and organs, essentially serving as a communication relay back and forth between the brain and the extremities.
• In the given scenario, if the brain scans show no damage, it indicates that the motor cortex and other integration centers in the CNS are functioning correctly. Therefore, the inability to move the left hand suggests an interruption in the signal path along the peripheral nerves that travel from the spinal cord to the muscles of the hand. Hence, the correct answer is the Peripheral nervous system.
• The PNS is divided into two main parts: the Somatic Nervous System and the Autonomic Nervous System. The Somatic Nervous System is specifically responsible for voluntary muscle movements and transmitting sensory information to the CNS.
• Damage to the Peripheral Nervous System (often called peripheral neuropathy) can result in weakness, numbness, and paralysis in specific areas like the hands or feet, even when the brain remains perfectly healthy.
• The PNS includes 12 pairs of cranial nerves and 31 pairs of spinal nerves. These nerves carry motor commands to the skeletal muscles to initiate movement.

Additional Information

• Central Nervous System (CNS):
• The CNS comprises the brain and the spinal cord. It acts as the "control center" of the body, processing sensory information and sending out instructions.
• Damage to the CNS (such as a stroke or traumatic brain injury) would typically show up on a brain scan (MRI or CT), which contradicts the conditions mentioned in the question.
• Autonomic Nervous System (ANS):
• This is a component of the PNS that regulates involuntary physiologic processes, including heart rate, blood pressure, respiration, and digestion.
• Since moving a hand is a voluntary action controlled by the Somatic Nervous System, the ANS is not the primary system responsible for this specific deficit.
• Parasympathetic Nervous System:
• This is a subdivision of the Autonomic Nervous System. It is often referred to as the "rest and digest" system.
• It helps conserve energy by slowing the heart rate and increasing intestinal and gland activity. It does not control the skeletal muscles used to move the hand.
• Sympathetic Nervous System:
• Another branch of the ANS, responsible for the "fight or flight" response. It prepares the body for physical activity by increasing heart rate and diverting blood flow to muscles, but it does not direct the specific voluntary motor signals for hand movement.

Important Points

• Nerve Conduction: For a muscle to contract, an electrical impulse must travel from the brain, down the spinal cord, through a peripheral motor nerve, and across the neuromuscular junction.
• Reflex Arc: Some movements are involuntary reflexes that involve only the spinal cord and PNS, bypassing the brain entirely. However, voluntary movement of the hand requires the full circuit.
• Location of Injury: Injuries to the brachial plexus (a network of nerves in the shoulder) or individual nerves like the radial, ulnar, or median nerves can cause localized hand paralysis despite a healthy brain.

The correct answer is To develop crops with better yield and useful traits.

Key Points

• Crop variety improvement is the primary method of enhancing agricultural productivity by selecting and breeding plant varieties that possess desirable characteristics.
• The main objective is to achieve a higher yield, which refers to increasing the total production of the crop per acre or unit of land. Hence, statement 2 is correct.
• It focuses on improved quality, where the criteria for quality vary from crop to crop; for example, protein quality in pulses, oil quality in oilseeds, and preserving quality in fruits and vegetables.
• Biotic and Abiotic resistance: Crops are developed to survive biotic stresses (diseases, insects, and nematodes) and abiotic stresses (drought, salinity, waterlogging, heat, cold, and frost), which can otherwise lead to significant crop loss.
• Change in maturity duration: By shortening the duration from sowing to harvesting, farmers can grow multiple rounds of crops in a single year, which also reduces the cost of production.
• Wider adaptability: Developing varieties that can grow under different climatic conditions helps in stabilizing the food supply across various geographical locations.

Additional Information

• Methods of Improvement:
• Hybridization: This involves crossing between genetically dissimilar plants. It can be intervarietal (between different varieties), interspecific (between two different species of the same genus), or intergeneric (between different genera).
• Genetic Modification: This involves introducing a gene that provides the desired characteristic, resulting in genetically modified crops (GM crops).
• Desirable Agronomic Characteristics: For cereal crops, dwarfness is a desired trait so that they consume fewer nutrients, whereas, for fodder crops, profuse branching and tallness are desired traits.
• Green Revolution: This period in the 1960s marked a significant leap in crop variety improvement in India, led by M.S. Swaminathan, known as the Father of the Green Revolution in India. Globally, Norman Borlaug is credited with this movement.
• Food Security: Continuous improvement in crop varieties is essential to meet the nutritional demands of the growing population and to ensure national food security.

Important Points

• Indian Council of Agricultural Research (ICAR): Established in 1929 and headquartered in New Delhi, it is the apex body for coordinating, guiding, and managing research and education in agriculture in India.
• Plant Breeding: It is the purposeful manipulation of plant species in order to create desired genotypes and phenotypes for specific uses.
• Selection: It is the oldest method of crop improvement where individual plants or groups of plants are picked from a mixed population based on their features.

The correct answer is m = -1.

Key Points

• In a concave mirror, when an object is placed at the centre of curvature (C), the image is formed exactly at the centre of curvature (C) itself.
• The nature of the image formed at this position is real and inverted. Hence, the statement regarding the nature is correct.
• The size of the image is equal to the size of the object. This is a unique position where the height of the image and the height of the object are identical.
• Magnification (m) is defined as the ratio of the height of the image (h') to the height of the object (h), expressed as m = h' / h.
• According to the Cartesian sign convention, the height of an inverted image is taken as negative, while the height of an erect object is taken as positive.
• Since the image is the same size but inverted, h' = -h. Therefore, the magnification m = -h / h = -1. Hence, the correct answer is m = -1.

Additional Information

• Linear Magnification (m): It is the ratio of the distance of the image from the mirror (v) to the distance of the object from the mirror (u), with a negative sign: m = -v / u. For an object at C, both u and v are equal to the radius of curvature (R), leading to m = -(-R) / (-R) = -1.
• Significance of Magnitude: If the absolute value of |m| = 1, the image is the same size as the object. If |m| > 1, the image is magnified (enlarged). If |m| < 1, the image is diminished.
• Significance of Sign: A negative sign in magnification always indicates a real image, which is consistently inverted. A positive sign indicates a virtual image, which is consistently erect.
• Concave Mirror Applications: Due to their ability to form magnified and real images, they are used in reflecting telescopes, shaving mirrors (when object is between P and F), and solar furnaces to concentrate sunlight.
• Convex Mirror Comparison: Unlike concave mirrors, a convex mirror always produces a magnification m < +1 because it always forms a virtual, erect, and diminished image, regardless of the object's position.

Important Points

• Position at Infinity: If the object is at infinity, the image forms at the focus (F), and the magnification is highly diminished (m much less than -1).
• Position between C and F: If the object is moved between the centre of curvature and the principal focus, the image is formed beyond C and is enlarged, meaning m < -1.
• Position at F: If the object is at the focus (F), the image is formed at infinity and is highly magnified.
• Position between P and F: This is the only case for a concave mirror where the magnification is positive (m > +1), as it forms a virtual, erect, and magnified image behind the mirror.

The correct answer is 56 J.

Key Points

• Work Done is defined as the product of the Force applied to an object and the Displacement of that object in the direction of the force.
• In this specific case, the Force (F) acting on the object is 7 N and the Displacement (s) is 8 m.
• The formula for work done is W = F × s × cos θ, where θ is the angle between the force and displacement vectors.
• Since the displacement is in the direction of the force, the angle θ is 0 degrees. As cos 0° = 1, the formula simplifies to W = F × s.
• By substituting the given values: W = 7 N × 8 m = 56 Nm. Since 1 Newton-meter (Nm) is equivalent to 1 Joule (J), the work done is 56 J. Hence, the correct answer is 56 J.

Additional Information

• Unit of Work: The SI unit of work is the Joule (J). It is named after the English physicist James Prescott Joule. One Joule is the amount of work done when a force of 1 Newton displaces an object through a distance of 1 meter.
• Scalar Quantity: Work is a scalar quantity, meaning it has only magnitude and no direction, even though both force and displacement are vector quantities.
• Dimensional Formula: The dimensional formula for work is [ML²T⁻²], which is the same as that for Energy.
• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its Kinetic Energy. This principle is fundamental in classical mechanics for solving problems involving motion.
• Other Options Analysis:
• 7 J is incorrect because it does not account for the 8 m displacement.
• 56 N is incorrect because Newton (N) is the unit of Force, not Work.
• 15 Nm is incorrect because it likely represents the sum (7 + 8) rather than the product required for work calculation.

Important Points

• Positive, Negative, and Zero Work:
• Positive Work: Occurs when the force and displacement are in the same direction (0° ≤ θ < 90°). An example is kicking a ball.
• Negative Work: Occurs when the force acts in the opposite direction to displacement (90° < θ ≤ 180°). An example is the force of friction acting against a sliding object.
• Zero Work: Occurs when the force is perpendicular to the displacement (θ = 90°) or if the displacement is zero. A coolie carrying a load on his head and walking on a horizontal platform does zero work against gravity because the force (upward) is perpendicular to the displacement (horizontal).
• CGS Unit: In the Centimetre-Gram-Second (CGS) system, the unit of work is the Erg. The relationship between Joule and Erg is 1 Joule = 10⁷ Ergs.
• Work against Gravity: When an object of mass (m) is lifted to a height (h), the work done is W = mgh, where g is the acceleration due to gravity (approximately 9.8 m/s²).

Shortcut Trick

Total Work (LCM of 65, 26) = 130 units.

Efficiency: A = 2, B = 5.

Pipe A works for the entire 30 minutes: 2 × 30 = 60 units.

Remaining work for Pipe B: 130 − 60 = 70 units.

Time for Pipe B: 70 ÷ 5 = 14 minutes.

∴ The correct answer is 14 min.

Alternate Method

Given:

Time taken by Pipe A = 65 minutes

Time taken by Pipe B = 26 minutes

Total time to fill the cistern = 30 minutes

Calculations:

⇒ Let the total capacity of the cistern be the LCM of 65 and 26 = 130 units.

⇒ Efficiency of Pipe A = 130 ÷ 65 = 2 units/min.

⇒ Efficiency of Pipe B = 130 ÷ 26 = 5 units/min.

⇒ Since the cistern is filled in 30 minutes, Pipe A worked for the full 30 minutes.

⇒ Work done by Pipe A = 30 × 2 = 60 units.

⇒ Remaining work to be done by Pipe B = 130 − 60 = 70 units.

⇒ Time for which Pipe B was open = 70 ÷ 5 = 14 minutes.

∴ The correct answer is 14 min.

Additional Information

Pipes & Cisterns Efficiency

Efficiency is inversely proportional to the time taken to fill a tank. Total Work = Efficiency × Time.

Inlet and Outlet Pipes

Inlet pipes have positive efficiency (filling), while outlet pipes (leaks) have negative efficiency (emptying).

Combined Work Formula

If two pipes fill a tank in x and y hours, together they fill it in (x × y) ÷ (x + y) hours.

Shortcut Trick

Given: 8 × (2 × Base Area) = 5 × CSA

⇒ 8 × (2πr²) = 5 × (2πrh) ⇒ 16r = 10h ⇒ h = 1.6r

Since r = 21/2 = 10.5 cm, then h = 1.6 × 10.5 = 16.8 cm.

Volume = πr²h = (22/7) × 10.5 × 10.5 × 16.8 = 346.5 × 16.8 = 5821.2 cm³.

∴ The correct answer is 5821.2 cm³.

Alternate Method

Given:

Base diameter of cylinder (d) = 21 cm ⇒ Radius (r) = 10.5 cm

8 × (Sum of 2 circular faces) = 5 × (Curved Surface Area)

Formula Used:

Area of circular face = πr²

Curved Surface Area (CSA) = 2πrh

Volume of Cylinder = πr²h

Calculations:

⇒ 8 × (2 × πr²) = 5 × (2πrh)

⇒ 16πr² = 10πrh

⇒ 16r = 10h

⇒ h = (16 × 10.5) ÷ 10

⇒ h = 168 ÷ 10 = 16.8 cm

⇒ Volume = (22/7) × (10.5)² × 16.8

⇒ Volume = (22/7) × 110.25 × 16.8

⇒ Volume = 22 × 15.75 × 16.8

⇒ Volume = 346.5 × 16.8

⇒ Volume = 5821.2 cm³

∴ The correct answer is 5821.2 cm³.

Additional Information

Total Surface Area of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Hollow Cylinder Volume

If R is the external radius and r is the internal radius, the volume of the material is V = π(R² − r²)h.

Relation between Volume and Area

The volume can also be expressed as Base Area × Height. For a cylinder, it is (πr²) × h.

Shortcut Trick

Net selling price = Initial CP × (100 + Profit%)/100 × (100 − Loss%)/100

Selling Price = 5,000 × (120/100) × (90/100)

Selling Price = 5,000 × 1.2 × 0.9

Selling Price = 6,000 × 0.9 = 5,400

∴ The correct answer is ₹5,400.

Alternate Method

Given: CP of Ramesh = ₹5,000 | Profit% = 20% | Loss% = 10% Formulas: SP = CP × (100 + P%)/100 | SP = CP × (100 − L%)/100

⇒ Selling price for Ramesh (which is Cost price for Suresh) = 5,000 × (120/100) = ₹6,000.

⇒ Selling price for Suresh = 6,000 × (90/100).

⇒ Suresh Selling Price = 60 × 90 = ₹5,400.

∴ The correct answer is ₹5,400.

Additional Information

Chain Rule in Profit and Loss

When an article is sold through multiple hands, the Selling Price of the previous person becomes the Cost Price for the next buyer.

Effective Percentage Change

Net effect = a + b + (ab/100). Here, 20 − 10 − (200/100) = 8% net profit on initial CP. 5,000 + 8% of 5,000 = 5,400.

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

⇒ (1/20 ÷ 11/60) × 100

⇒ (1/20 × 60/11) × 100

⇒ (3/11) × 100 ≈ 27.27%

∴ The correct answer is 27.27%.

Alternate Method

Given: First fraction = 1/20, Second fraction = 11/60

Formula Used: Percentage = (Value ÷ Base) × 100

⇒ Let 1/20 be x% of 11/60

⇒ 1/20 = (x/100) × (11/60)

⇒ x = (1/20) × (60/11) × 100

⇒ x = (3/11) × 100

⇒ x = 300 ÷ 11

⇒ x = 27.2727...

∴ The correct answer is 27.27%.

Additional Information

Fraction to Percentage Conversion

To convert any fraction a/b to a percentage, multiply the fraction by 100 and add the % symbol.

Reciprocal Percentages

Standard values like 1/11 ≈ 9.09% are helpful; thus 3/11 = 3 × 9.09% = 27.27%.

Shortcut Trick

Speed of train = 63 × (5/18) = 17.5 m/s.

Time taken to cross Platform = 60 seconds.

Time taken to cross Man (Train length only) = 40 seconds.

Time difference to cover Platform length = 60 − 40 = 20 seconds.

Platform Length = Speed × Time Difference = 17.5 × 20 = 350 m.

∴ The correct answer is 350 metres.

Alternate Method

Given: Speed = 63 km/hr, Time (Platform) = 60 s, Time (Man) = 40 s.

Formula Used: Distance = Speed × Time; 1 km/hr = 5/18 m/s.

⇒ Speed in m/s = 63 × (5/18) = 17.5 m/s

⇒ Length of Train (L_t) = Speed × Time to cross man

⇒ L_t = 17.5 × 40 = 700 m

⇒ Total distance to cross Platform = L_t + L_p = Speed × 60

⇒ 700 + L_p = 17.5 × 60 = 1050 m

⇒ L_p = 1050 − 700 = 350 m

∴ The correct answer is 350 metres.

Additional Information

Relative Speed

When two objects move in the same direction, relative speed is S₁ − S₂. 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 Objects

For a stationary point object (man/pole), distance = train length. For a stationary long object (bridge/platform), distance = train length + object length.

Shortcut Trick

The Mode is the observation with the highest repetition (frequency).

Scanning the data: 35 appears 3 times, while 24 and 20 appear only 2 times.

All other numbers in the set appear only once.

Because 35 occurs the most, it is the Mode.

∴ The correct answer is 35.

Alternate Method

Given: The set of observations is 21, 29, 35, 24, 25, 33, 24, 23, 26, 28, 35, 27, 20, 35, 20.

Formula Used:Mode = Value that occurs most frequently in the distribution.

⇒ Arrange observations in ascending order: 20, 20, 21, 23, 24, 24, 25, 26, 27, 28, 29, 33, 35, 35, 35.

⇒ Count of 20 = 2

⇒ Count of 24 = 2

⇒ Count of 35 = 3

⇒ Count of others = 1

⇒ The maximum frequency is 3, which corresponds to the value 35.

∴ The correct answer is 35.

Additional Information

Mean (Arithmetic Average)

The sum of all observations divided by the total number of observations. Mean = (Σx_i) ÷ n.

Median

The middle value of a data set when it is arranged in ascending or descending order. For n = 15, Median is the ((15+1)÷2)^th i.e., 8^th term.

Empirical Relationship

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

Range

The difference between the maximum and minimum values in a data set. Range = Max − Min.

Shortcut Trick

If every value in a set is decreased by a constant 'k', the new average is simply the old average minus 'k'.

Given Old Average = 23 and Decrease (k) = 7.

New Average = 23 − 7 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Number of observations (n) = 7

Initial Average = 23

Value decreased from each number = 7

Calculations:

⇒ Total Sum of 7 numbers = Average × n

⇒ Initial Sum = 23 × 7 = 161

⇒ Total decrease across all 7 numbers = 7 × 7 = 49

⇒ New Sum = 161 − 49 = 112

⇒ New Average = New Sum ÷ n

⇒ New Average = 112 ÷ 7 = 16

∴ The correct answer is 16.

Additional Information

Effect of Arithmetic Operations on Average

If every number is increased by 'k', the new average is Old Average + k. If every number is decreased by 'k', it is Old Average − k.

Multiplication and Division Rule

If every number is multiplied by 'k', the new average is Old Average × k. Similarly, if divided by 'k' (where k ≠ 0), it is Old Average ÷ k.

Sum of Observations

The Sum of Observations is always equal to the Average × Number of items. This is the fundamental basis for all average-related calculations.

Shortcut Trick

Let the Marked Price be MP.

Successive discounts of 35% and 20% mean the final Selling Price (SP) is 6,500.

Using the multiplier method: MP × (100 − 35) ÷ 100 × (100 − 20) ÷ 100 = SP

MP × 0.65 × 0.80 = 6,500

MP × 0.52 = 6,500 ⇒ MP = 6,500 ÷ 0.52 = 12,500

∴ The correct answer is ₹12,500.

Alternate Method

Given: Selling Price (SP) = ₹6,500, Discount₁ (d₁) = 35%, Discount₂ (d₂) = 20%. Formulas: SP = MP × (100 − d₁)% × (100 − d₂)% | Net Discount = a + b − (a × b) ÷ 100

⇒ Equivalent single discount = 35 + 20 − (35 × 20) ÷ 100

⇒ Net Discount = 55 − 7 = 48%

⇒ If MP is 100%, then SP = 100% − 48% = 52%

⇒ 52% of MP = 6,500

⇒ MP = (6,500 × 100) ÷ 52 = 12,500

∴ The correct answer is ₹12,500.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The second discount is calculated on the price remaining after the first discount.

Marked Price (MP)

It is the price displayed on the product label. Discount is always calculated on the Marked Price unless stated otherwise.

Shortcut Trick

Income ratio = 5 : 9, Expenditure ratio = 1 : 3. Difference in income parts = 9 − 5 = 4. Difference in expenditure parts = 3 − 1 = 2.

To make difference equal, multiply expenditure ratio by 2 → 2 : 6.

Savings (Income − Expenditure) = 5 − 2 = 3 units or 9 − 6 = 3 units.

3 units = ₹45,000 ⇒ 1 unit = ₹15,000.

Mukund monthly income = 5 units = 5 × ₹15,000 = ₹75,000.

∴ The correct answer is ₹75,000.

Alternate Method

Given:

Ratio of monthly income of Mukund and Sachin = 5 : 9

Ratio of their monthly expenditure = 1 : 3

Savings of each = ₹45,000

Formula Used:

Income − Savings = Expenditure

Calculations:

Let income of Mukund be 5x and Sachin be 9x.

⇒ (5x − 45000) / (9x − 45000) = 1 / 3

⇒ 3 × (5x − 45000) = 1 × (9x − 45000)

⇒ 15x − 1,35,000 = 9x − 45,000

⇒ 15x − 9x = 1,35,000 − 45,000

⇒ 6x = 90,000

⇒ x = 15,000

⇒ Mukund monthly income = 5x = 5 × 15,000 = 75,000

∴ The correct answer is ₹75,000.

Additional Information

Income, Expenditure and Savings Relationship

The fundamental relation used in these problems is: Income = Expenditure + Savings.

Ratio Balancing Method

When the absolute change (savings) is same for both, we equalize the difference between the ratio terms of Income and Expenditure to find 1 unit value.

Cross Multiplication Method

Used for two ratios with constant or different additions: (a × d − b × c) units = (Constant × d − Constant × c).

Shortcut Trick

Test small values for x because the total ₹935 is close to a multiple of 180.

If x = 5, then cost of pens = 180 × 5 = ₹900.

Remaining amount = 935 − 900 = ₹35.

Number of pencils y = 35 ÷ 3.5 = 10.

Check total items: x + y = 5 + 10 = 15 (Satisfied).

Calculation: x² + y² = 5² + 10² = 25 + 100 = 125.

∴ The correct answer is 125.

Alternate Method

Given:

Price of branded pen = ₹180

Price of pencil = ₹3.50

Total items (x + y) = 15

Total cost = ₹935

Formula Used:

Total Cost = (Price per Pen × Number of Pens) + (Price per Pencil × Number of Pencils)

Calculations:

⇒ 180x + 3.5y = 935 ... (i)

⇒ x + y = 15 → y = 15 − x ... (ii)

⇒ Substituting (ii) in (i):

⇒ 180x + 3.5(15 − x) = 935

⇒ 180x + 52.5 − 3.5x = 935

⇒ 176.5x = 935 − 52.5

⇒ 176.5x = 882.5

⇒ x = 882.5 ÷ 176.5 = 5

⇒ y = 15 − 5 = 10

⇒ x² + y² = 5² + 10²

⇒ x² + y² = 25 + 100 = 125

∴ The correct answer is 125.

Additional Information

Algebraic Identity

The sum of squares can be related to the sum of terms using: x² + y² = (x + y)² − 2xy.

Substitution Method

In a system of two linear equations, express one variable in terms of the other from the simpler equation and substitute it into the complex one.

Averages & Sums

If the average price is A, then Total Cost = A × (x + y). Here, Average ≈ 935 ÷ 15 ≈ 62.33.

Shortcut Trick

Use the Algebraic Identity: (a + b)² − (a − b)² = 4ab

Here, let a = 9x and b = 4y.

Numerator = 4 × (9x) × (4y) = 144xy.

Divide by 24x: 144xy ÷ 24x = 6y.

∴ The correct answer is 6y.

Alternate Method

Given: [(9x + 4y)² − (9x − 4y)²] ÷ 24x

Formula Used: Expansion of (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ Expansion 1: (9x + 4y)² = 81x² + 16y² + 72xy

⇒ Expansion 2: (9x − 4y)² = 81x² + 16y² − 72xy

⇒ Subtracting: (81x² + 16y² + 72xy) − (81x² + 16y² − 72xy)

⇒ Result of Numerator: 81x² − 81x² + 16y² − 16y² + 72xy + 72xy = 144xy

⇒ Final Division: 144xy ÷ 24x = (144/24) × (x/x) × y

⇒ Simplified Value: 6y

∴ The correct answer is 6y.

Additional Information

Standard Algebraic Identities

1. (a + b)² + (a − b)² = 2(a² + b²)
2. a² − b² = (a + b)(a − b)

Simplification Rules

Always apply BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) to solve multi-operational algebraic expressions efficiently.

Shortcut Trick

The angle between the bisectors of two adjacent angles of a quadrilateral is half the sum of the other two angles:

∠STR = (∠P + ∠Q) ÷ 2

⇒ 36° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 72°

Given the ratio of ∠P : ∠Q = 3 : 5, the total is 3 + 5 = 8 units.

⇒ 8 units = 72° ⇒ 1 unit = 9°

The difference between ∠Q and ∠P is 5 − 3 = 2 units ⇒ 2 × 9° = 18°.

∴ The correct answer is 18°.

Alternate Method

Given:

In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at T.

∠STR = 36°

Ratio of ∠P : ∠Q = 3 : 5

Formula Used:

Sum of angles in a triangle = 180°

Sum of angles in a quadrilateral = 360°

Calculations:

In triangle STR:

⇒ ∠STR + ∠TSR + ∠TRS = 180°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2 and ∠TRS = ∠R ÷ 2

⇒ 36° + (∠S ÷ 2) + (∠R ÷ 2) = 180°

⇒ (∠S + ∠R) ÷ 2 = 180° − 36°

⇒ (∠S + ∠R) ÷ 2 = 144°

⇒ ∠S + ∠R = 288°

In quadrilateral PQRS, the sum of all interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 288° = 360°

⇒ ∠P + ∠Q = 360° − 288° = 72°

Let the measures of ∠P and ∠Q be 3x and 5x respectively:

⇒ 3x + 5x = 72°

⇒ 8x = 72°

⇒ x = 9°

The difference between the measures of ∠Q and ∠P is:

⇒ ∠Q − ∠P = 5x − 3x = 2x

⇒ Difference = 2 × 9° = 18°

∴ The correct answer is 18°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles. For adjacent angles R and S, ∠STR = (∠P + ∠Q) ÷ 2.

Sum of Interior Angles of a Polygon

The sum of the interior angles of an n-sided polygon is (n − 2) × 180°. For a quadrilateral (n = 4), the sum is always 360°.

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral intersect, the angle between them is half the difference of the other two opposite angles.

Shortcut Trick

Substitute standard trigonometric values: sin 60° = √3/2, cos 30° = √3/2, and tan 45° = 1.

Expression = (√3/2 × √3/2) ÷ 1 = 3/4.

∴ The correct answer is 3/4.

Alternate Method

Given:

Expression: (sin 60° × cos 30°) ÷ tan 45°

Standard Trigonometric Values:

⇒ sin 60° = √3/2

⇒ cos 30° = √3/2

⇒ tan 45° = 1

Calculation:

⇒ Expression = [(√3/2) × (√3/2)] ÷ 1

⇒ Expression = [(√3 × √3) / (2 × 2)] ÷ 1

⇒ Expression = (3 / 4) / 1

⇒ Expression = 3/4

∴ The correct answer is 3/4.

Additional Information

Complementary Angles

For any angle θ, sin θ = cos(90° − θ). Therefore, sin 60° = cos 30° = √3/2.

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ.

Tangent of 45 Degrees

At 45°, the opposite and adjacent sides of a right-angled triangle are equal, so tan 45° = 1.

Shortcut Trick

Total Work = LCM of (2, 4, 4) = 4 units.

Individual efficiencies: A = 2, B = 1, C = 1.

Combined efficiency = 2 + 1 + 1 = 4 units/day.

Thrice the work = 3 × 4 = 12 units.

Time taken = 12 ÷ 4 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given: Time taken by A = 2 days, B = 4 days, C = 4 days.

Formula Used: Efficiency = Total Work ÷ Time Taken; Time = Total Work ÷ Combined Efficiency.

⇒ Total Efficiency of (A + B + C) = 2 + 1 + 1 = 4 units/day.

⇒ Work to be completed = 3 × Total Work = 3 × 4 = 12 units.

⇒ Required Time = 12 ÷ 4 = 3 days.

∴ The correct answer is 3 days.

Additional Information

Total Work Concept

Total work is usually assumed as the LCM (Least Common Multiple) of the number of days taken by each person to simplify calculations.

Work and Efficiency

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

Combined Work Rule

If multiple people work together, their individual efficiencies are added to find the total rate of work per day.

Shortcut Trick

Net discount for 6% and 5% = 6 + 5 − (6 × 5 ÷ 100) = 11 − 0.3 = 10.7%.

Net discount for 10.7% and 11% = 10.7 + 11 − (10.7 × 11 ÷ 100).

⇒ 21.7 − (117.7 ÷ 100) = 21.7 − 1.177 = 20.523%.

∴ The correct answer is 20.52%.

Alternate Method

Given: Three consecutive discounts d₁ = 6%, d₂ = 5%, and d₃ = 11%.

Formula: Net SP = MP × (100 − d₁)% × (100 − d₂)% × (100 − d₃)%

Let the Marked Price (MP) be 100.

⇒ SP = 100 × (94 ÷ 100) × (95 ÷ 100) × (89 ÷ 100)

⇒ SP = 100 × 0.94 × 0.95 × 0.89

⇒ SP = 89.3 × 0.89 = 79.477

⇒ Single Equivalent Discount = MP − SP = 100 − 79.477 = 20.523%

∴ The correct answer is 20.52%.

Additional Information

Successive Discount Formula

For two discounts a% and b%, the single equivalent discount is (a + b − ab ÷ 100)%.

Effective Price Calculation

When multiple discounts are applied, the order of application does not change the final Selling Price.

Shortcut Trick

Total deduction = 85 + 65 + 61 = ₹211

Remaining amount = 5575 − 211 = ₹5364

Divide ₹5364 in ratio 5 : 11 : 20 → Total parts = 36

Value of 1 part = 5364 ÷ 36 = ₹149

Difference (Y and Z) = (20 − 11) parts × 149 + (61 − 65) = 9 × 149 − 4 = 1341 − 4 = ₹1337

∴ The correct answer is ₹1337.

Alternate Method

Given: Total amount = ₹5,575. Deductions: X = ₹85, Y = ₹65, Z = ₹61. New ratio = 5 : 11 : 20.

⇒ Total Deductions = 85 + 65 + 61 = ₹211

⇒ Remaining Amount = 5575 − 211 = ₹5364

⇒ Sum of ratio parts = 5 + 11 + 20 = 36

⇒ 36 units = 5364 ⇒ 1 unit = 5364 ÷ 36 = 149

⇒ Reduced share of Y = 11 × 149 = ₹1639

⇒ Reduced share of Z = 20 × 149 = ₹2980

⇒ Original share of Y = 1639 + 65 = ₹1704

⇒ Original share of Z = 2980 + 61 = ₹3041

⇒ Difference = 3041 − 1704 = ₹1337

∴ The correct answer is ₹1337.

Additional Information

Ratio and Proportion Basics

If an amount A is divided in the ratio a:b:c, the shares are calculated as [a/(a+b+c)]×A, [b/(a+b+c)]×A, and [c/(a+b+c)]×A respectively.

Handling Deductions

When fixed amounts are subtracted from shares before applying a ratio, subtract the total deduction from the total sum before calculating unit values.

Difference Rule

To find the difference between original shares, calculate the difference between reduced shares and adjust by the difference in their respective deductions.

Shortcut Trick

Evaluate all the perfect square roots directly mentally:

⇒ √256 = 16, √169 = 13, √144 = 12, √25 = 5

⇒ Substitute these values directly: 16 − 13 + (12 − 5)²

⇒ 3 + (7)²

⇒ 3 + 49 = 52

∴ The correct answer is 52.

Alternate Method

Given:

Expression: √256 − √169 + (√144 − √25)²

Method Used:

Apply the BODMAS rule: evaluate Order (square roots), then Brackets, followed by Exponents, and finally Addition/Subtraction.

Calculations:

⇒ Find the values of individual square roots:

⇒ √256 = 16

⇒ √169 = 13

⇒ √144 = 12

⇒ √25 = 5

⇒ Substitute the values back into the main expression:

⇒ 16 − 13 + (12 − 5)²

⇒ Simplify the bracket first (12 − 5):

⇒ 16 − 13 + (7)²

⇒ Evaluate the square of 7:

⇒ 16 − 13 + 49

⇒ Perform subtraction and addition from left to right:

⇒ 3 + 49

⇒ 52

∴ The correct answer is 52.

Additional Information

BODMAS Rule

It defines the correct sequence for operations: Brackets, Order (Powers and Roots), Division × Multiplication (left-to-right), and Addition − Subtraction (left-to-right).

Perfect Squares up to 20

Memorising squares up to 30 significantly improves calculation speed in competitive exams. For example: 14² = 196, 15² = 225, 16² = 256, 17² = 289.

Basic Algebraic Identities

Although arithmetic simplification is best here, knowing (a − b)² = a² − 2ab + b² is essential for cases involving unknown variables.

Shortcut Trick

Fourth Proportional of 4, 8, 19 → (8 × 19) ÷ 4 = 38

Mean Proportional of 9, 4 → √(9 × 4) = 6

Required Ratio → 38 : 6 = 19 : 3

∴ The correct answer is 19 : 3.

Alternate Method

Given: Numbers for 4th proportional (a=4, b=8, c=19) and mean proportional (m=9, n=4).

Formula Used: Fourth proportional x = (b × c) / a; Mean proportional y = √(m × n).

⇒ Fourth proportional (x) = (8 × 19) / 4

⇒ x = 2 × 19 = 38

⇒ Mean proportional (y) = √(9 × 4) = √36

⇒ y = 6

⇒ Ratio = x : y = 38 : 6

⇒ Ratio = 19 : 3

∴ The correct answer is 19 : 3.

Additional Information

Third Proportional

For two numbers a and b, the third proportional c is such that a : b :: b : c, giving c = b² / a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b), also known as the geometric mean.

Duplicate and Sub-duplicate Ratio

Duplicate ratio of a:b is a²:b² and sub-duplicate ratio is √a:√b.

Shortcut Trick

Identify the LCM of the given divisors: 12, 15, 18, and 20.

LCM (12, 15, 18, 20) = 180.

Required number = LCM − Increase Value

Required number = 180 − 10 = 170.

∴ The correct answer is 170.

Alternate Method

Given: Divisors are 12, 15, 18, and 20; Number is increased by 10.

Formula Used: Required Number = LCM(divisors) − k (where k is the increment).

⇒ Prime factors of 12 = 2² × 3

⇒ Prime factors of 15 = 3 × 5

⇒ Prime factors of 18 = 2 × 3²

⇒ Prime factors of 20 = 2² × 5

⇒ LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180

⇒ According to the condition: x + 10 = 180

⇒ x = 180 − 10 = 170

∴ The correct answer is 170.

Additional Information

LCM (Least Common Multiple)

It is the smallest positive integer that is evenly divisible by a given set of numbers.

Remainder and Increment Rules

If a number is increased by 'k', subtract 'k' from the LCM. If it leaves a remainder 'r', add 'r' to the LCM.

Common Remainder Case

If a number when divided by a, b, c leaves remainders x, y, z respectively such that (a−x) = (b−y) = (c−z) = K, then the number is LCM(a, b, c) − K.

Shortcut Trick

Find factors of Area (40) whose squares sum to 89.

Pairs: (1,40), (2,20), (4,10), (5,8).

⇒ 5² + 8² = 25 + 64 = 89 (Matches diagonal squared).

⇒ Perimeter = 2 × (8 + 5) = 2 × 13 = 26 cm.

∴ The correct answer is 26 cm.

Alternate Method

Given:

Diagonal (d) = √89 cm

Area (A) = l × b = 40 cm²

Formula Used:

⇒ d² = l² + b²

⇒ (l + b)² = l² + b² + 2lb

Calculations:

⇒ l² + b² = (√89)² = 89

⇒ lb = 40

⇒ (l + b)² = 89 + 2 × 40

⇒ (l + b)² = 89 + 80 = 169

⇒ l + b = √169 = 13 cm

⇒ Perimeter = 2(l + b) = 2 × 13 = 26 cm

∴ The correct answer is 26 cm.

Additional Information

Pythagorean Property of Rectangles

The diagonal of a rectangle forms a right-angled triangle with the length and breadth, satisfying l² + b² = d².

Algebraic Identity for Perimeter

Perimeter is related to diagonal and area by the identity: (Perimeter/2)² = d² + 2 × Area.

Area and Perimeter Relationship

For a fixed area, the perimeter of a rectangle is minimized when the rectangle is a square (l = b).

Shortcut Trick

Using the Man-Days relation: M₁ × D₁ = M₂ × D₂

20 × 12 = 80 × D₂

D₂ = 240 ÷ 80 = 3 days

∴ The correct answer is 3 days.

Alternate Method

Given:

Initial strength (M₁) = 20 pupils

Initial days (D₁) = 12 days

New strength (M₂) = 80 pupils

Formula Used:

M₁ × D₁ = M₂ × D₂

Calculations:

⇒ Total provisions = M₁ × D₁

⇒ Total provisions = 20 × 12 = 240 pupil-days

⇒ New days (D₂) = Total provisions ÷ M₂

⇒ D₂ = 240 ÷ 80

⇒ D₂ = 3 days

∴ The correct answer is 3 days.

Additional Information

Inverse Proportion

When the number of people increases, the food provisions will last for fewer days. This is a case of inverse variation, represented as x × y = k (constant).

Chain Rule Formula

For work and provisions, the general formula is (M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂, where M is men, D is days, H is hours, and W is work.

Shortcut Trick

Use the Successive Percentage method for 2 years: Net Rate = x + y + (xy/100)

Net Interest Rate = 15 + 15 + (15 × 15)/100 = 30 + 2.25 = 32.25%

Total Amount = 132.25% of Principal = 4800 × 1.3225

Amount = 48 × 132.25 = ₹6,348

∴ The correct answer is ₹6,348.

Alternate Method

Given:

Principal (P) = ₹4,800

Rate (R) = 15% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4800 × (1 + 15/100)²

⇒ A = 4800 × (1 + 3/20)²

⇒ A = 4800 × (23/20) × (23/20)

⇒ A = 4800 × (529/400)

⇒ A = (4800 ÷ 400) × 529

⇒ A = 12 × 529

⇒ A = 6348

∴ The correct answer is ₹6,348.

Additional Information

Difference between CI and SI

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

Compound Interest Formula

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

Compounding Half-Yearly

If interest is compounded half-yearly, the Rate becomes R/2 and the Time (n) becomes 2n.

Shortcut Trick

When the distance for both journeys is the same (from A to B and back to A), use the harmonic mean formula.

Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

Average Speed = (2 × 70 × 111) ÷ (70 + 111) = 15540 ÷ 181

Average Speed ≈ 85.8563...

∴ The correct answer is 85.86 km/hr.

Alternate Method

Given:

Distance (A to B) = 933 km

Speed (A to B), S₁ = 70 km/hr

Speed (B to A), S₂ = 111 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 933 + 933 = 1866 km

⇒ Time taken from A to B (T₁) = 933 ÷ 70 = 13.3285 hr

⇒ Time taken from B to A (T₂) = 933 ÷ 111 = 8.4054 hr

⇒ Total Time = 13.3285 + 8.4054 = 21.7339 hr

⇒ Average Speed = 1866 ÷ 21.7339 ≈ 85.856 km/hr

∴ The correct answer is 85.86 km/hr.

Additional Information

Average Speed (Equal Distances)

When an object covers equal distances at different speeds, the average speed is independent of the actual distance and is calculated as the harmonic mean of speeds.

Relative Speed

Relative speed in same direction is (S₁ − S₂) and in opposite direction 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.

Shortcut Trick

Ratio of ages of Ram and Lalith = 6 : 1

Product of ratio parts = 6 × 1 = 6 units²

Given product = 216 ⇒ 1 unit² = 216 ÷ 6 = 36 ⇒ 1 unit = √36 = 6 years

Difference in ratio = 6 − 1 = 5 units

Difference in ages = 5 × 6 = 30 years

∴ The correct answer is 30 years.

Alternate Method

Given:

Product of Ram and Lalith ages = 216 years

Ratio of their ages = 6 : 1

Let the age of Ram be 6x and Lalith be x.

⇒ 6x × x = 216

⇒ 6x² = 216

⇒ x² = 216 ÷ 6

⇒ x² = 36

⇒ x = √36 = 6

⇒ Age of Ram = 6 × 6 = 36 years

⇒ Age of Lalith = 1 × 6 = 6 years

⇒ Difference = 36 − 6 = 30 years

∴ The correct answer is 30 years.

Additional Information

Ratio and Product

When the product of two quantities in ratio a : b is given as P, always use a constant x such that (ax) × (bx) = P, leading to x = √(P / ab).

Age Difference Constant

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

Ratio Difference

To find the difference directly from ratio a : b, calculate (a − b) × x, where x is the value of one ratio unit.

Shortcut Trick

Assume savings = 100. Then expenditure = 100 + 200 = 300.

Income = 300 + 100 = 400.

New expenditure = 300 − 10% of 300 = 270. New savings = 100 + 34% of 100 = 134.

New income = 270 + 134 = 404.

% Increase in income = (4 / 400) × 100 = 1%.

∴ The correct answer is 1%.

Alternate Method

Given that Expenditure is 200% more than Savings.

⇒ Let Savings = S and Expenditure = E.

⇒ E = S + 2S = 3S.

⇒ Income (I) = E + S = 3S + S = 4S.

⇒ If S = 100, then E = 300 and I = 400.

⇒ New E = 300 × (90/100) = 270.

⇒ New S = 100 × (134/100) = 134.

⇒ New I = 270 + 134 = 404.

⇒ Increase in Income = 404 − 400 = 4.

⇒ Percentage increase = (4 / 400) × 100 = 1%.

∴ The correct answer is 1%.

Additional Information

Income-Expenditure Relationship

The fundamental formula is Income = Expenditure + Savings. Any change in two variables directly impacts the third.

Percentage Change in Income

It can also be calculated using weighted averages: %ΔIncome = (%ΔE × E + %ΔS × S) / (E + S).

X% More Than Concept

If A is x% more than B, then A = B × (1 + x/100). For 200% more, the multiplier is (1 + 2) = 3.

Given:

→ Seven boxes L, M, N, O, Q, R and S are kept one above the other.

→ Only L is kept above Q.

→ Only three boxes are kept between L and R.

→ Only O is kept below N.

→ S is kept immediately above M.

Step 1: Arrange N and O

Only O is below N.

So, N must be at the bottom second position and O at the bottom.

 

Step 2: Place S and M

S is immediately above M.

Possible suitable positions:

 

Step 3: Place L and R

There are exactly three boxes between L and R.

So, possible positions are:

→ L at 1 and R at 5

 

Step 4: Place Q

Only L is above Q.

So Q must be immediately below L.

 

Step 5: Find the box second above M

M is at position 4.

First above M = S

Second above M = Q

Hence, the correct answer is "Option 1 - Q".

Given Statements:

All pilots are captains.

All captains are drivers.

Some drivers are teachers.

Venn Diagram Logic:

Conclusions:

(I) All pilots are drivers: Follows (Because all pilots are captains and all captains are drivers. Therefore, all pilots are drivers.)

(II) Some teachers are pilots: Does not follows (Because only “some drivers are teachers” is given. There is no definite relation between teachers and pilots.)

∴ Only conclusion (I) follows.

Hence, the correct answer is "Only conclusion (I) follows".

Given:

→ Seven friends B, C, D, T, U, V and W are sitting in a straight line facing north.

→ Only two people sit between V and D.

→ Only U sits to the right of T.

→ Only one person sits between D and T.

→ C sits to the right of B but to the left of W.

B is at position 2.

W is at position 5.

People between them = C and D = 2 people

Hence, the correct answer is "Option 4 - Two".

Given:

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

We need to find the odd digits which are:

→ Immediately preceded by an odd digit

→ Immediately followed by an odd digit

Required pattern: Odd digit → Odd digit → Odd digit

Checking the series:

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

No odd digit is both preceded and followed by an odd digit.

Therefore, there are no such odd digits.

Hence, the correct answer is "Option 1".

Given series: DER UVI LMZ CDQ ?

The pattern followed here is: 

∴ According to the pattern followed TUH will replace the (?) mark.

Hence, the correct answer is "Option 3".

Given:

Mr. Akte ranked 54^th from the top and 19^th from the bottom.

Formula:

Total students = Rank from top + Rank from bottom − 1

Calculation:

= 54 + 19 − 1

= 73 − 1

= 72

Hence, the correct answer is "Option 2".

Shortcut Trick

Interchange the signs as given: (− → +), (÷ → ×), (+ → −), (× → ÷).

Rewrite the expression: 33 + 15 × 6 − 120 ÷ 8.

Calculate division and multiplication: 120 ÷ 8 = 15 and 15 × 6 = 90.

Solve: 33 + 90 − 15 = 123 − 15 = 108.

∴ The correct answer is 108.

Alternate Method

Given:

Original equation: 33 − 15 ÷ 6 + 120 × 8

Interchanges: (+) ↔ (−) and (×) ↔ (÷)

Method:

Follow the BODMAS rule: Bracket, Of, Division, Multiplication, Addition, Subtraction.

Calculations:

⇒ New Equation: 33 + 15 × 6 − 120 ÷ 8

⇒ Step 1 (Division): 120 ÷ 8 = 15

⇒ Step 2 (Multiplication): 15 × 6 = 90

⇒ Step 3 (Addition): 33 + 90 = 123

⇒ Step 4 (Subtraction): 123 − 15

⇒ ? = 108

∴ The correct answer is 108.

Additional Information

BODMAS Rule

It defines the order of operations in arithmetic: Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction.

Operator Interchange Logic

In these problems, replace every occurrence of the specified symbols simultaneously before starting any arithmetic calculation.

Priority of Division and Multiplication

Division and Multiplication hold the same rank and should be performed from left to right as they appear in the modified expression.

Logic followed here is:

The difference between the positional values of consecutive letters in the letter-cluster follows a specific pattern.

Option 1: LGJ

Option 2: RMQ

 

Option 3: PKN

 

Option 4: MHK

 

The pattern observed is: (1st letter - 5 = 2nd letter) and (2nd letter + 3 = 3rd letter). Letter-cluster 'RMQ' does not follow this pattern.

Therefore, the odd letter-cluster is 'RMQ'.

Hence, the correct answer is "Option 2".