RRB Group D Official Paper (Held On: 05 Feb, 2026 Shift 3) (Previous Year Paper)

The correct answer is 9 A.

Key Points

• The relationship between voltage (V), current (I), and resistance (R) in an electrical circuit is governed by Ohm's Law, which states that V = IR.
• In the first scenario, the electric heater is connected to a 70 V source and draws a current of 5 A. Using these values, we can calculate the resistance (R) of the heater's heating element: R = V / I = 70 / 5 = 14 Ohm.
• The resistance of the heater remains constant because it is a physical property of the material used in the heating coil (assuming the temperature change doesn't significantly alter the resistance for this calculation).
• In the second scenario, the potential difference is increased to 126 V. To find the new current, we apply Ohm's Law again using the previously calculated resistance: I = V / R.
• By substituting the new values, we get I = 126 / 14 = 9 A. Thus, the new current drawn by the heater is 9 A. Hence, the calculation confirms the correct answer is 9 A.

Additional Information

• Ohm's Law:
• Formulated by German physicist Georg Simon Ohm in 1827, it states that the current through a conductor between two points is directly proportional to the voltage across the two points, provided physical conditions like temperature remain constant.
• The SI unit of resistance is the Ohm (Ω), the unit of potential difference is the Volt (V), and the unit of current is the Ampere (A).
• Resistance Factors:
• The resistance of a conductor depends on its length, cross-sectional area, nature of the material, and temperature.
• In heating appliances like electric heaters, irons, and toasters, the coils are typically made of alloys like Nichrome because they have high resistance and do not oxidize (burn) easily at high temperatures.
• Linear Relationship:
• In an Ohmic conductor, a graph plotted between Voltage (V) and Current (I) results in a straight line passing through the origin, where the slope of the line represents the resistance.

Important Points

• Power Calculation: The power consumed by the heater also changes with voltage. Using the formula P = VI, the initial power was 70 * 5 = 350 Watts, while the new power is 126 * 9 = 1134 Watts.
• Safety Considerations: When voltage is nearly doubled (from 70 V to 126 V), the current also increases proportionally. In real-world applications, this could lead to overheating or tripping a circuit breaker if the wire's current rating is exceeded.
• Application: This principle is crucial for the design of voltage stabilizers and transformers, which ensure that appliances receive the correct potential difference to operate at their rated current.

The correct answer is vertically upward.

Key Points

• Buoyancy is the upward force exerted by a fluid (liquid or gas) that opposes the weight of a partially or fully immersed object. In any fluid, the pressure increases with depth; therefore, the fluid exerts more pressure on the bottom of an object than on the top, resulting in a net force acting vertically upward. Hence, option 1 is correct.
• The direction of the buoyant force does not depend on the shape of the object but rather on the gravity field, which dictates that the net pressure difference will always push the object away from the center of gravity (upwards). Hence, statement 2 is incorrect.
• The weight of the object acts vertically downward due to gravity. The buoyant force is the opposing force that pushes against this weight. Hence, statement 3 is incorrect.
• The buoyant force acts independent of the direction of motion of the object. Whether an object is moving horizontally or vertically, the buoyant force remains directed vertically upward relative to the fluid's surface. Hence, statement 4 is incorrect.
• The magnitude of this force is determined by Archimedes' Principle, which states that the upward buoyant force is equal to the weight of the fluid that the object displaces.

Additional Information

• Archimedes' Principle: Discovered by the Greek mathematician Archimedes in the 3rd century BC, this principle is fundamental to fluid mechanics. It explains why heavy ships made of steel can float while a small iron nail sinks.
• Mathematical Formula: The buoyant force (Fb) can be expressed as Fb = ρ × V × g, where ρ (rho) is the density of the fluid, V is the volume of the displaced fluid, and g is the acceleration due to gravity (approximately 9.8 m/s²).
• Factors Affecting Buoyancy:
• Density of the Fluid: A higher fluid density results in a greater buoyant force. This is why it is easier to float in saltwater (like the Dead Sea) than in freshwater.
• Volume of Displaced Fluid: The more fluid an object displaces, the larger the upward force. Large hulls of ships are designed to displace massive amounts of water to support their weight.
• Floating and Sinking:
• If the buoyant force is greater than the weight of the object, the object will rise to the surface and float.
• If the weight is greater than the buoyant force, the object will sink.
• If the weight equals the buoyant force, the object remains neutrally buoyant and stays suspended at its current depth.

Important Points

• Center of Buoyancy: This is the point where the buoyant force is considered to act. It is the geometric center (centroid) of the displaced volume of fluid. For a floating object to be stable, the relationship between the Center of Gravity and the Center of Buoyancy is critical.
• Apparent Weight: When an object is submerged in water, it feels lighter. This "loss" in weight is called the apparent weight, which is the actual weight minus the buoyant force.
• Applications: The principle of buoyancy is used in the design of submarines (using ballast tanks to change density), hot air balloons (using heated air which is less dense than cool air), and hydrometers (used to measure the specific gravity of liquids).

The correct answer is Always negative.

Key Points

• According to the New Cartesian Sign Convention, the pole (P) of the spherical mirror is taken as the origin, and the principal axis is taken as the X-axis of the coordinate system.
• A concave mirror is a converging mirror where the reflecting surface is curved inwards, towards the center of curvature.
• The principal focus (F) of a concave mirror is the point on the principal axis where rays of light parallel to the axis actually meet after reflection. This focus is situated in front of the mirror.
• Since the object is always placed on the left side of the mirror, the incident light travels from left to right. Distances measured in the direction of incident light are positive, while those measured opposite to the direction of incident light are negative.
• The focal length (f) is the distance between the pole (P) and the principal focus (F). Because the focus of a concave mirror lies to the left of the pole, this distance is measured against the direction of incident light. Hence, the focal length of a concave mirror is always negative.

Additional Information

• New Cartesian Sign Convention Rules:
• The object is always placed to the left of the mirror. This implies that the object distance (u) is always negative.
• All distances parallel to the principal axis are measured from the pole of the mirror.
• Distances measured to the right of the origin (+x-axis) are positive, and those to the left (-x-axis) are negative.
• Distances measured perpendicularly above the principal axis (+y-axis) are positive, and those below (-y-axis) are negative.
• Convex Mirror:
• A convex mirror is a diverging mirror with its reflecting surface curved outwards.
• Its principal focus is a virtual point located behind the mirror (on the right side).
• Therefore, the focal length (f) of a convex mirror is always positive.
• Mirror Formula and Magnification:
• The mirror formula represents the relationship between focal length (f), object distance (u), and image distance (v): 1/f = 1/v + 1/u.
• Magnification (m) is the ratio of the height of the image (h') to the height of the object (h), also expressed as m = -v/u.
• For a concave mirror, the radius of curvature (R) is also negative, and the relationship is R = 2f.
• Applications of Concave Mirrors:
• They are used in shaving mirrors and makeup mirrors to produce an enlarged and erect image when the object is held close.
• They are used as reflectors in car headlights, searchlights, and torches to produce a powerful parallel beam of light.
• Solar furnaces use large concave mirrors to concentrate sunlight to produce high temperatures for melting metals.
• Dentists use them to see larger images of the teeth of patients.

Important Points

• The focal length of a plane mirror is considered to be infinity because the radius of curvature of a flat surface is infinite.
• The sign of the image distance (v) in a concave mirror can be positive or negative depending on the position of the object. If the image is real (formed in front), v is negative; if the image is virtual (formed behind), v is positive.
• The focal length of a spherical mirror depends only on its radius of curvature and is independent of the medium in which it is placed (unlike lenses).

The correct answer is It is the sum of the atomic masses of all atoms present in a formula unit of an ionic compound.

Key Points

• The formula unit mass of a substance is defined as the sum of the atomic masses of all atoms in a formula unit of a compound. Hence, statement 1 is correct.
• It is calculated in the same manner as molecular mass; however, the term "formula unit" is specifically used for substances whose constituent particles are ions (ionic compounds) rather than discrete molecules.
• Statement 2 suggests it is the product of atomic masses, which is scientifically incorrect as masses are always additive in this context to determine the total mass of a unit. Hence, statement 2 is incorrect.
• Statement 3 refers to the total number of atoms in a molecule, which is actually the definition of atomicity (e.g., O2 has an atomicity of 2). Hence, statement 3 is incorrect.
• Statement 4 describes the mass of one mole of a compound, which is known as the molar mass, typically expressed in grams per mole (g/mol). Hence, statement 4 is incorrect.
• The unit used for expressing formula unit mass is the atomic mass unit, represented by the symbol ‘u’ (unified mass).

Additional Information

• Ionic Compounds vs. Molecular Compounds:
• In molecular compounds (like water, H2O), atoms are bonded covalently to form distinct molecules. We use the term molecular mass here.
• In ionic compounds (like Sodium Chloride, NaCl), the structure consists of a three-dimensional crystal lattice of alternating cations and anions. Since there are no individual "NaCl molecules," we use the formula unit (the simplest whole-number ratio of ions) to calculate mass.
• Calculation Example (Sodium Chloride):
• The formula unit of sodium chloride is NaCl.
• Atomic mass of Sodium (Na) = 23 u.
• Atomic mass of Chlorine (Cl) = 35.5 u.
• Formula unit mass = 1 × 23 + 1 × 35.5 = 58.5 u.
• Calculation Example (Calcium Chloride):
• The formula unit is CaCl2.
• Atomic mass of Calcium (Ca) = 40 u.
• Atomic mass of Chlorine (Cl) = 35.5 u.
• Formula unit mass = 40 + (2 × 35.5) = 40 + 71 = 111 u.

Important Points

• Relative Atomic Mass: All these calculations are based on the relative atomic mass of elements, which is determined relative to the Carbon-12 (12C) isotope, where 1 u is 1/12th the mass of one atom of carbon-12.
• Significance in Stoichiometry: Understanding formula unit mass is crucial for quantitative chemistry and stoichiometry, allowing scientists to calculate the exact amount of reactants and products in a chemical reaction involving ionic substances.
• Molar Mass Connection: If the formula unit mass of NaCl is 58.5 u, then its molar mass is numerically the same but expressed in grams, i.e., 58.5 g, which contains 6.022 × 10^23 (Avogadro's number) formula units.

The correct answer is Pushing a wall.

Key Points

• In physics, Work is defined as the product of the Force applied to an object and the Displacement caused by that force in the direction of the force.
• The mathematical formula for work done is W = F × s × cos(θ), where W is work, F is the magnitude of force, s is the displacement, and θ is the angle between the force and displacement vectors.
• For work to be considered "done" in a scientific sense, two conditions must be met: a force must act on the object, and the object must possess a non-zero displacement.
• Pulling a trolley along a flat surface: In this case, a force is applied and the trolley moves a certain distance. Since there is displacement, work is done. Hence, statement 1 is an example of work.
• Lifting a brick from the floor: A force is applied upwards to overcome gravity, and the brick moves from the floor to a height. Since displacement occurs in the direction of the force, work is done. Hence, statement 2 is an example of work.
• Pushing a wall: When a person pushes a rigid wall, they apply a significant amount of force. However, because the wall does not move, the displacement (s) is 0. According to the formula, if s = 0, then W = 0. Hence, statement 3 is NOT an example of work done.
• Sliding luggage on the floor: Force is applied to overcome friction, and the luggage moves across the floor. Since displacement occurs, work is performed. Hence, statement 4 is an example of work.

Additional Information

• Nature of Work:
• Positive Work: When the force and displacement are in the same direction (angle θ = 0°), the work done is positive.
• Negative Work: When the force and displacement are in opposite directions (angle θ = 180°), such as the force of friction acting on a moving body, the work done is negative.
• Zero Work: This occurs when displacement is zero or when the force is perpendicular to the displacement (angle θ = 90°). A classic example is a coolie carrying a load on his head while walking horizontally; the force of gravity is downward, but displacement is horizontal.
• Units and Measurements:
• The S.I. unit of work is the Joule (J). One Joule is defined as the work done when a force of 1 Newton displaces an object by 1 meter.
• In the CGS system, the unit of work is the erg. 1 Joule is equal to 10^7 ergs.
• Work is a scalar quantity, meaning it has only magnitude and no direction.

Important Points

• Work-Energy Theorem: It states that the net work done on an object is equal to the change in its Kinetic Energy. This establishes a direct link between motion and the effort exerted.
• Dimensional Formula: The dimensional representation of work is [M1 L2 T-2], which is the same as that of energy and torque.
• Power: While work represents the total energy transferred, Power is the rate at which work is done. It is measured in Watts (W), where 1 Watt = 1 Joule per second.
• Conservative Forces: Work done by gravitational or electrostatic forces depends only on the initial and final positions, not on the path taken. These are known as conservative forces.

The correct answer is At any temperature.

Key Points

• Evaporation is a fundamental physical process where a liquid changes into its vapor state at the surface. This process occurs at any temperature below the boiling point of the liquid.
• It is classified as a surface phenomenon because it only involves the molecules located at the top layer of the liquid, unlike boiling which affects the entire volume.
• Molecules in a liquid are in constant motion and possess different levels of kinetic energy. At any temperature, a small fraction of molecules at the surface has enough energy to overcome the intermolecular forces of attraction and escape into the air as gas.
• Since the molecules with the highest energy escape, the average kinetic energy of the remaining liquid decreases, which leads to a cooling effect.
• The process of evaporation is continuous and happens even at very low temperatures, though the rate may be significantly slower compared to higher temperatures. Hence, the correct answer is At any temperature.

Additional Information

• Factors Affecting the Rate of Evaporation:
• Temperature: As the temperature increases, more molecules gain enough kinetic energy to convert into vapor, thereby increasing the rate of evaporation.
• Surface Area: Since evaporation is a surface phenomenon, increasing the exposed surface area (e.g., spreading out wet clothes) allows more molecules to escape simultaneously.
• Humidity: The rate of evaporation is inversely proportional to humidity. If the air is already saturated with water vapor, the liquid evaporates more slowly.
• Wind Speed: An increase in wind speed helps move the vapor particles away from the surface, making room for more molecules to evaporate.
• Evaporation vs. Boiling:
• Boiling occurs only at a specific temperature called the boiling point (which is 100 degrees Celsius for pure water at sea level).
• Boiling is a bulk phenomenon where bubbles form throughout the liquid, whereas evaporation is limited to the surface.
• While evaporation happens at all temperatures, boiling requires a constant supply of external heat to maintain the temperature at the boiling point.

Important Points

• Latent Heat of Vaporization: This is the amount of heat energy required to change 1 kilogram of a liquid into gas at its boiling point without a change in temperature. During evaporation, the liquid absorbs this "latent heat" from the surroundings, resulting in the characteristic cooling effect.
• Real-world Applications:
• Earthen Pots (Matkas): Water stays cool in earthen pots because water evaporates through the pores of the pot, taking heat from the water inside.
• Sweating: The human body uses evaporation to regulate internal temperature. As sweat evaporates from the skin, it absorbs body heat, providing a cooling sensation.
• Acetone/Perfume: When you put acetone or perfume on your palm, it evaporates rapidly because of its low boiling point, making your palm feel very cold.
• Vapor Pressure: Evaporation continues until the vapor pressure above the liquid reaches a state of equilibrium with the liquid itself in a closed container. In an open system, it can continue until the liquid is completely gone.

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

Key Points

• Carboxylic acids are organic compounds containing the carboxyl functional group (-COOH). Common examples include methanoic acid and ethanoic acid.
• Statement A claims that carboxylic acids are strong acids that completely ionize in water. This is scientifically inaccurate because most carboxylic acids are weak acids. In an aqueous solution, they only partially dissociate into hydrogen ions (H+) and carboxylate ions (RCOO-). Hence, statement A is incorrect.
• The degree of ionization for these acids is typically very low, often less than 1% to 5% depending on the concentration, meaning the majority of the acid remains in its molecular form in the solution.
• Statement B posits that ethanoic acid is a weak acid compared to hydrochloric acid. This is correct because hydrochloric acid (HCl) is a strong mineral acid that undergoes 100% dissociation in water, whereas ethanoic acid (CH3COOH) only partially ionizes. Hence, statement B is correct.
• The strength of an acid is determined by its acid dissociation constant (Ka). Strong acids have very high Ka values, while weak acids like those in the carboxylic family have relatively low Ka values.

Additional Information

• Ethanoic Acid (Acetic Acid):
• It has the chemical formula CH3COOH and is a colorless liquid with a distinctive sour taste and pungent smell.
• A 5% to 8% solution of ethanoic acid in water is commonly known as vinegar, which is widely used as a food preservative and in cooking.
• When pure ethanoic acid is frozen, it forms ice-like crystals; because its melting point is 290 K (16.6 degrees Celsius), it is often referred to as glacial acetic acid.
• Ionization and pH:
• The pH scale measures the concentration of hydrogen ions. For the same molar concentration, a strong acid like HCl will have a much lower pH (around 1.0) compared to a weak acid like ethanoic acid (around 3.0).
• The stability of the carboxylate ion via resonance is what allows carboxylic acids to act as acids, even though they are generally weaker than mineral acids.

Important Points

• Mineral Acids vs. Organic Acids: Mineral acids (like Sulfuric acid, Nitric acid, and Hydrochloric acid) are generally strong because they lack carbon-hydrogen bonds in the same structural context as organic acids and donate protons easily. Organic acids (like Citric acid, Tartaric acid, and Ethanoic acid) are usually weak.
• Physical Properties: Carboxylic acids have higher boiling points than alkanes, ethers, or aldehydes of comparable molecular weight because they form intermolecular hydrogen bonds, often existing as dimers even in the vapor phase.
• Solubility: Simple carboxylic acids (with 1 to 4 carbon atoms) are highly miscible in water due to their ability to form hydrogen bonds with water molecules.

The correct answer is Sodium.

Key Points

• Sodium is a member of the alkali metal group, located in Group 1 of the periodic table with the symbol Na and Atomic Number 11.
• One of the most distinctive physical properties of Sodium is its extreme softness; it has a consistency similar to soft wax or clay at room temperature.
• Unlike most common metals such as Iron or Copper, Sodium is so soft that it can be easily sliced with a stainless steel knife.
• Because of this low structural hardness and tensile strength, Sodium can be flattened or shaped into thin leaves or sheets simply by applying the pressure of the fingers.
• This property is a result of the metallic bonding in alkali metals, which involves only one valence electron, leading to a relatively weak lattice structure compared to transition metals.

Additional Information

• Reactivity and Storage:
• Sodium is highly reactive, especially with water and oxygen. To prevent it from reacting with moisture in the atmosphere, it is strictly stored under kerosene oil or liquid paraffin.
• When it reacts with water, it produces sodium hydroxide and hydrogen gas, often resulting in a vigorous or explosive reaction.
• Comparison with Other Options:
• Gold (Au): It is the most malleable and ductile metal known. While it can be beaten into incredibly thin gold leaves (foils), this process requires mechanical hammering or rolling; it cannot be shaped from a solid block into a leaf by mere finger pressure due to its higher density and atomic weight.
• Silver (Ag): Similar to gold, silver is highly malleable and used for making fine foils (like Chandi ka Varq used in sweets), but it possesses a crystalline structure that is much firmer than Sodium.
• Iron (Fe): A transition metal known for its strength and hardness. It has a high melting point of 1538°C and requires heavy machinery or high temperatures (forging) to be shaped.
• Physical Constants of Sodium:
• The melting point of Sodium is relatively low at approximately 97.8°C.
• Its density is 0.97 g/cm³, which is less than that of water, meaning it will float while reacting.

Important Points

• Electronic Configuration: Sodium has an electronic configuration of [Ne] 3s¹. The presence of a single outer shell electron makes it highly likely to lose that electron to achieve a stable octet, forming the Na+ cation.
• Flame Test: When Sodium or its salts are introduced to a Bunsen burner flame, they impart a characteristic persistent golden-yellow color, which is a standard qualitative analysis test.
• Natural Occurrence: Due to its extreme reactivity, Sodium is never found in its elemental state in nature. It is primarily obtained through the electrolysis of molten sodium chloride (Downs process).

The correct answer is The principal axis is normal (perpendicular) to the mirror at its pole.

Key Points

• The Principal Axis of a spherical mirror is an imaginary straight line that passes through the Center of Curvature (C) and the Pole (P) of the mirror.
• A spherical mirror (whether Concave or Convex) is a segment of a Hollow Sphere. According to the laws of geometry, any line drawn from the center of a sphere to its surface is a Radius.
• A Radius is always Normal (Perpendicular) to the surface of the sphere at the point where it intersects. Since the Principal Axis passes through the Center of Curvature, it acts as a normal to the mirror’s surface at the Pole (P). Hence, the statement regarding its perpendicularity is correct.
• This geometric relationship ensures that a ray of light traveling along the Principal Axis strikes the mirror at an Angle of Incidence of 0 degrees relative to the normal, causing it to reflect back along the same path.
• At the Pole, the Principal Axis is at an angle of 90 degrees to the Tangent drawn at the pole on the mirror's reflecting surface.

Additional Information

• Components of a Spherical Mirror:
• Pole (P): This is the Geometric Center of the reflecting surface of a spherical mirror. It is located exactly on the surface of the mirror.
• Center of Curvature (C): It is the center of the Hollow Glass Sphere of which the mirror is a part. For a Concave Mirror, it lies in front of the reflecting surface, whereas for a Convex Mirror, it lies behind it.
• Radius of Curvature (R): This is the linear distance between the Pole and the Center of Curvature (PC). It represents the radius of the sphere from which the mirror was cut.
• Principal Focus (F): It is a point on the Principal Axis where light rays that are parallel to the axis converge (in Concave Mirrors) or appear to diverge from (in Convex Mirrors) after reflection.
• Focal Length (f): The distance between the Pole and the Principal Focus is known as the Focal Length. For mirrors with a small Aperture, the relationship is R = 2f.
• Types of Spherical Mirrors:
• Concave Mirror: Also known as a Converging Mirror, its reflecting surface is curved inwards. It is used in Headlights, Solar Furnaces, and by Dentists to see enlarged images.
• Convex Mirror: Also known as a Diverging Mirror, its reflecting surface is curved outwards. It always forms a Virtual, Erect, and Diminished image and is primarily used as Rear-view Mirrors in vehicles because of its wide field of view.

Important Points

• Sign Convention (New Cartesian Sign Convention):
• The Pole (P) is taken as the Origin.
• Objects are always placed to the Left of the mirror.
• Distances measured in the direction of Incident Light (to the right) are taken as Positive, while those measured against it (to the left) are Negative.
• The Focal Length of a Concave Mirror is always Negative, and for a Convex Mirror, it is always Positive.
• Aperture: The diameter of the reflecting surface of a spherical mirror is called its Aperture. In most educational problems, we assume the Aperture is much smaller than the Radius of Curvature.

p-The correct answer is in moist air.

 

Key Points

• Corrosion is a natural process that converts a refined metal into a more chemically stable form such as oxide, hydroxide, or sulfide. It is an electrochemical process that gradually destroys materials through chemical reactions with their environment.
• The presence of moist air, which contains both oxygen and water vapor, is the essential condition for the rusting of iron and the corrosion of many other metals. Water acts as an electrolyte, allowing the flow of ions between the anodic and cathodic regions on the metal surface. Hence, the statement "in moist air" is correct.
• In dry air, the lack of moisture prevents the formation of an electrolytic medium. Without water, the movement of ions is restricted, and the oxidation of the metal happens extremely slowly or not at all at room temperature. Hence, the statement "in dry air" is incorrect.
• Galvanisation is actually a protective measure against corrosion. It involves coating iron or steel with a thin layer of zinc to prevent oxygen and moisture from reaching the underlying metal. Hence, the statement "by galvanisation" is incorrect as it prevents rather than causes damage.
• The presence of oil acts as a barrier that seals the metal surface from the atmosphere. By excluding both oxygen and moisture, oil prevents the chemical reaction required for corrosion to occur. Hence, the statement "in both dry air and oil" is incorrect.

Additional Information

• Rusting of Iron: This is the most common form of corrosion. The chemical formula for rust is Fe2O3 · xH2O (hydrated iron(III) oxide). It occurs when iron reacts with oxygen in the presence of water. Unlike some other metal oxides, rust is porous and flakes off, exposing fresh metal to further corrosion.
• Corrosion of Copper: When exposed to moist air, copper develops a green coating. This green substance is a mixture of copper hydroxide and copper carbonate, often referred to as patina.
• Tarnishing of Silver: Silver objects turn black when exposed to air containing hydrogen sulfide gas. The black coating is silver sulfide (Ag2S), which is a result of a chemical reaction with sulfur compounds in the atmosphere.
• Factors Accelerating Corrosion: The rate of corrosion increases in the presence of impurities in the metal, high temperature, and electrolytes like saltwater. This is why ships and structures in coastal areas corrode much faster than those in inland regions.

Important Points

• Prevention Methods: To save metals from damage, several techniques are used, including painting, oiling, greasing, electroplating, and alloying.
• Alloying: Creating an alloy such as stainless steel (a mixture of iron, carbon, chromium, and nickel) makes the metal highly resistant to corrosion because the chromium forms a thin, invisible layer of chromium oxide on the surface.
• Economic Impact: Every year, corrosion causes significant damage to car bodies, bridges, iron railings, and ships, leading to massive financial losses in the global industrial sector.
• Sacrificial Protection: In this method, a more reactive metal (like magnesium or zinc) is attached to the metal to be protected. The more reactive metal corrodes first, "sacrificing" itself to protect the primary structure.

The correct answer is Daughter cells have the same number of chromosomes as the parent cell.

Key Points

• Mitosis is a fundamental process of cell division where a single parent cell divides to produce two genetically identical daughter cells.
• The primary function of mitosis is the growth of the organism and the repair or replacement of damaged tissues.
• During the S phase of Interphase (the phase preceding mitosis), the cell replicates its DNA. This ensures that there is a duplicate copy of every chromosome available for the division process.
• Because the genetic material is duplicated and then partitioned equally, the daughter cells maintain the exact same chromosomal number as the original parent cell.
• In biological terms, if the parent cell is diploid (2n), the resulting daughter cells will also be diploid (2n). Hence, the statement that daughter cells have the same number of chromosomes is correct.

Additional Information

• The Stages of Mitosis: Mitosis is a continuous process but is generally divided into four main stages:
• Prophase: The chromatin condenses into visible chromosomes, and the nuclear envelope begins to disappear.
• Metaphase: The chromosomes align along the equatorial plane (metaphase plate) of the cell, ensuring proper separation.
• Anaphase: The sister chromatids are pulled apart by spindle fibers toward opposite poles of the cell.
• Telophase: New nuclear membranes form around the two sets of separated chromosomes, which begin to de-condense back into chromatin.
• Equational Division: Mitosis is often referred to as equational division because the number of chromosomes remains constant from the parent to the daughter cells.
• Comparison with Meiosis: Unlike mitosis, meiosis is a "reduction division" occurring in germ cells to produce gametes (sperm and eggs). In meiosis, the daughter cells end up with half (haploid, n) the number of chromosomes compared to the parent cell.
• Cytokinesis: This is the final step of the cell cycle where the cytoplasm of the parent cell physically divides into two, resulting in two distinct daughter cells.

Important Points

• Chromosome Count in Humans: In a typical human somatic cell, there are 46 chromosomes (23 pairs). After undergoing mitosis, each of the two daughter cells will also contain 46 chromosomes.
• Genetic Stability: Mitosis is crucial for maintaining genomic integrity. By producing clones of the parent cell, it ensures that every cell in a multicellular organism (except gametes) carries the same genetic information.
• Occurrence: In humans and many animals, mitosis occurs almost everywhere in the body, such as the skin, bone marrow, and lining of the gut, where constant cell turnover is required.
• Discovery: The process of mitosis was first observed and described by the German biologist Walther Flemming in the year 1882.

The correct answer is Cytokinesis.

Key Points

• Cytokinesis is the physical process of cell division, which divides the cytoplasm of a parental cell into two daughter cells.
• It typically begins during the late stages of nuclear division (Anaphase or Telophase) and ensures that each daughter cell receives a nucleus and a share of cytoplasmic organelles.
• In animal cells, the process occurs through the formation of a cleavage furrow. A contractile ring composed of actin and myosin filaments constricts the cell membrane inward until the cell is pinched into two.
• In plant cells, due to the presence of a rigid cell wall, a cleavage furrow cannot form. Instead, a cell plate is synthesized in the center of the cell by the fusion of Golgi-derived vesicles, which then grows outward to join the existing lateral walls.
• This stage marks the completion of the M phase (Mitotic phase) of the cell cycle, resulting in two distinct cells that can then enter the G1 phase of Interphase.

Additional Information

• Karyokinesis:
• This refers to the division of the cell nucleus during mitosis or meiosis. It is divided into four main stages: Prophase, Metaphase, Anaphase, and Telophase.
• While cytokinesis deals with the cytoplasm, karyokinesis ensures the equal distribution of replicated DNA (chromosomes) into the daughter nuclei.
• Interphase:
• Known as the preparatory phase, it accounts for approximately 95% of the duration of the cell cycle.
• It is divided into three sub-phases: G1 (cell growth), S (DNA replication), and G2 (further growth and protein synthesis). During the S phase, the amount of DNA per cell doubles, but the chromosome number remains the same.
• Chromatin Condensation:
• This process occurs primarily during Prophase. The diffuse network of DNA and proteins (chromatin) undergoes coiling and folding to form compact, visible chromosomes.
• Condensation is crucial for the safe transport of genetic material to the opposite poles of the cell without entanglement or breakage.
• Syncytium:
• In some organisms, karyokinesis is not followed by cytokinesis, leading to a multinucleate condition.
• A classic biological example is the liquid endosperm in coconut, where many nuclei exist within a common cytoplasm before cellularization occurs.

Important Points

• The term mitosis is often used interchangeably with the M phase, but technically, mitosis refers to karyokinesis (nuclear division), while the total cell splitting is only complete after cytokinesis.
• The centrosomes and spindle fibers play a vital role in karyokinesis by pulling sister chromatids apart during Anaphase.
• Errors in cytokinesis or karyokinesis can lead to aneuploidy (abnormal number of chromosomes) or polyploidy, which are often linked to cancer development or specific evolutionary traits in plants.
• The regulatory proteins called cyclins and cyclin-dependent kinases (CDKs) control the transition between these various stages of the cell cycle to ensure genomic stability.

The correct answer is the Western Ghats and Eastern Himalayas.

Key Points

• The term Biodiversity Hotspot refers to a biogeographic region with significant levels of biodiversity that is under threat from human habitation. The concept was first introduced by British ecologist Norman Myers in 1988.
• In India, there are four major biodiversity hotspots: the Himalayas, the Western Ghats, the Indo-Burma region, and Sundaland (which includes the Nicobar Islands).
• The Western Ghats run parallel to the western coast of the Indian peninsula, spanning states like Kerala, Karnataka, Tamil Nadu, Maharashtra, and Goa. It is one of the eight "hottest hotspots" of biological diversity in the world. Hence, the inclusion of Western Ghats in the option is correct.
• The Eastern Himalayas region includes parts of North-East India, Bhutan, and Nepal. It is home to a vast variety of flora and fauna, including many endemic species like the Red Panda and the Snow Leopard. Hence, the inclusion of Eastern Himalayas in the option is correct.
• Option 1 mentions Central India and the Deccan Plateau; while these regions are ecologically significant, they do not meet the strict international criteria to be classified as global biodiversity hotspots. Hence, statement 1 is incorrect.
• Option 3 mentions the Cold desert of Ladakh; while it is a unique ecosystem with specific biodiversity, it is not categorized as a primary biodiversity hotspot under the global classification. Hence, statement 3 is incorrect.
• Option 4 refers to the Mediterranean, Sardinia, and Sicily; these locations are part of the Mediterranean Basin hotspot, which is entirely outside the geographical boundaries of India. Hence, statement 4 is incorrect.

Additional Information

• Criteria for Biodiversity Hotspots: To qualify as a hotspot, a region must meet two strict criteria:
• It must contain at least 1,500 species of vascular plants (more than 0.5% of the world’s total) as endemics.
• It has to have lost at least 70% of its original native habitat.
• Global Context: Currently, there are 36 biodiversity hotspots identified worldwide. These areas represent just 2.4% of Earth’s land surface but support more than half of the world’s plant species as endemics.
• Indo-Burma Region: This hotspot encompasses a vast area including the North-Eastern states of India (specifically south of the Brahmaputra River), Myanmar, Thailand, and Vietnam. It is known for its high degree of endemism in freshwater turtle species.
• Sundaland: This region covers the Nicobar group of Islands in India, along with Indonesia, Malaysia, and Singapore. It is characterized by its tropical rainforests and iconic species like the Orangutan (though not found in the Indian portion).

Important Points

• The Western Ghats was declared a UNESCO World Heritage Site in 2012 due to its immense influence on the Indian monsoon weather pattern.
• India is recognized as one of the 17 mega-diverse countries in the world, harboring about 7-8% of all recorded species despite having only 2.4% of the global land area.
• The Ministry of Environment, Forest and Climate Change (MoEFCC) is the nodal agency in India responsible for the conservation of these hotspots through various National Parks, Wildlife Sanctuaries, and Biosphere Reserves.
• Major threats to these hotspots include habitat fragmentation, invasive species, over-exploitation, and climate change. For instance, the Shola grasslands in the Western Ghats are highly sensitive to temperature fluctuations.

The correct answer is The embryo fails to implant properly and is expelled.

Key Points

• The Uterus (womb) is a major female hormone-responsive secondary sex organ. Each month, the inner lining of the uterus, known as the Endometrium, prepares itself to receive a Fertilised Egg.
• For a successful pregnancy, the Endometrium must become Thick and Spongy, enriched with Blood Vessels and nutrients to support the Embryo.
• Implantation is the process where the Blastocyst (the early-stage embryo) attaches to the uterine wall. If the lining is Thin and not spongy, the embryo cannot find a secure place to anchor or receive nourishment. Hence, the statement that the embryo fails to implant properly and is expelled is correct.
• If the Fertilised Egg (embryo) fails to implant, the body does not receive the hormonal signal (HCG) to maintain the thickened lining. Consequently, the lining breaks down, and the embryo is shed along with the blood and tissues during Menstruation.
• The idea that the egg survives longer than usual is biologically incorrect, as a non-implanted embryo has a very limited window of viability without maternal support. Hence, statement 1 is incorrect.
• The claim that the egg gets fertilised again is impossible because Fertilisation occurs in the Fallopian Tube, and once an egg is fertilised, the Zona Pellucida hardens to prevent Polyspermy. Hence, statement 4 is incorrect.

Additional Information

• Role of Progesterone: This hormone is primarily responsible for making the Endometrium "spongy" and Secretory. It is produced by the Corpus Luteum after Ovulation. If Progesterone levels are low, the lining remains thin, leading to Infertility.
• The Menstrual Cycle: It consists of several phases. During the Proliferative Phase, Estrogen causes the lining to thicken. During the Secretory Phase, the lining becomes prepared for Implantation. If no implantation occurs, the Menstrual Phase begins.
• Fertilisation vs. Implantation: Fertilisation is the union of Sperm and Ovum, typically occurring in the Ampulla of the Fallopian Tube. Implantation occurs about 6 to 10 days later in the Uterus.
• Endometrial Thickness: In clinical medicine, a thickness of at least 7 mm to 8 mm is usually considered necessary for a high chance of successful Embryo Implantation during IVF or natural cycles.

Important Points

• Human Chorionic Gonadotropin (HCG): This is the "pregnancy hormone." It is produced by the cells surrounding a growing embryo, which eventually form the Placenta. Its primary role is to keep the Corpus Luteum producing Progesterone, preventing the Menstrual Cycle from starting.
• Luteal Phase Deficiency: This is a condition where the Uterine Lining does not develop sufficiently because of inadequate Progesterone levels or the lining's failure to respond to the hormone, often leading to Early Miscarriage.
• Expulsion Process: When implantation fails, the embryo, which is microscopic at this stage, is expelled naturally. This often goes unnoticed as it occurs around the same time as a regular Menstrual Period.

The correct answer is Pushing a stationary car and making it move.

Key Points

• Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This means that forces always exist in pairs, and the two forces act on different bodies simultaneously.
• A rocket moving upward due to gases expelled downward: This is a classic example of the third law. The rocket engine expels hot exhaust gases downward (action), and these gases exert an equal force on the rocket upward (reaction). Hence, this is an example of the third law.
• Foot pushes the ground backward and ground pushes the foot forward: When we walk, our feet exert a backward force on the ground (action). According to the law, the ground exerts an equal and forward force on our feet (reaction), allowing us to move. Hence, this is an example of the third law.
• The swimmer pushes the water backward with her hands: A swimmer exerts a force on the water in the backward direction (action). In response, the water exerts an equal force on the swimmer in the forward direction (reaction), pushing her through the water. Hence, this is an example of the third law.
• Pushing a stationary car and making it move: While there is a reaction force on the person's hands, the act of changing the state of a stationary object involves overcoming inertia and applying an unbalanced force to create acceleration. This is primarily described by Newton's Second Law (F = ma) and the First Law (Law of Inertia). Hence, this is NOT a primary example of the third law in this context.

Additional Information

• Newton's First Law of Motion (Law of Inertia): It states that an object will remain at rest or move in a straight line at a constant velocity unless acted upon by an external unbalanced force. The tendency of an object to resist changes in its state of motion is called Inertia, which depends on the mass of the object.
• Newton's Second Law of Motion: This law defines force quantitatively. It states that the rate of change of momentum of an object is directly proportional to the applied force and takes place in the direction of the force. The mathematical formula is Force (F) = mass (m) × acceleration (a).
• Sir Isaac Newton: He was an English mathematician and physicist who formulated the three laws of motion in his work "Philosophiae Naturalis Principia Mathematica", published in 1687. These laws form the foundation of classical mechanics.
• Conservation of Momentum: Newton's Third Law leads to the principle of conservation of linear momentum. In an isolated system where no external forces act, the total momentum remains constant over time, regardless of internal changes or collisions.

Important Points

• Force Pairs: It is important to note that the action and reaction forces never act on the same body. If they did, the net force on every object would always be zero, and motion would be impossible.
• Simultaneity: There is no time lag between action and reaction; they occur at the exact same instant.
• Magnitude and Direction: The two forces in a third-law pair always have the same magnitude but act in exactly opposite directions.
• Applications in Technology: The third law is essential for the design of jet engines, propellers, recoil mechanisms in guns, and even the simple act of rowing a boat.

The correct answer is Electric motor.

Key Points

• The Electric motor is a rotating device that converts electrical energy into mechanical energy. It is based on the principle that a current-carrying conductor placed in a magnetic field experiences a mechanical force.
• When a rectangular coil carrying current is placed in a magnetic field, the magnetic force acts on the arms of the coil, causing it to rotate continuously. Hence, the correct answer is Electric motor.
• Electric fuse: It is a safety device used in electrical circuits that works on the heating effect of electric current (Joule heating). It consists of a wire with a low melting point that melts and breaks the circuit if the current exceeds a safe limit. Hence, statement 1 is incorrect.
• Electric heater: This device converts electrical energy into heat energy using a high-resistance heating element, typically made of Nichrome. It operates purely on the thermal effect of current rather than mechanical force. Hence, statement 2 is incorrect.
• Electric bell: While it uses electromagnetism, its operation relies on the magnetic effect of electric current to attract an armature (striker) periodically. It does not use the continuous force on a conductor in the same way a motor does to produce rotation. Hence, statement 4 is incorrect.

Additional Information

• Fleming’s Left-Hand Rule: This rule is used to find the direction of the force acting on a current-carrying conductor in a magnetic field. If the forefinger represents the magnetic field and the middle finger represents the current, then the thumb points in the direction of the motion or force.
• Lorentz Force: The fundamental physics behind this is the Lorentz force law, which states that a point charge moving in a magnetic field experiences a force. For a wire of length L carrying current I in a magnetic field B, the force F is calculated as F = BIL sin(theta).
• Applications of Electric Motors: These are used in a wide variety of daily appliances including water pumps, electric fans, washing machines, mixers, refrigerators, and electric cars.
• Commutator: In a DC motor, a device called a split-ring commutator is used to reverse the direction of current every half rotation, ensuring the torque remains in the same direction and the motor continues to spin.

Important Points

• Armature: The soft iron core on which the coil is wound is called an armature. This enhances the power of the motor by increasing the magnetic flux.
• Magnetic Field Sources: Commercial motors use an electromagnet instead of a permanent magnet to create a much stronger magnetic field, allowing the motor to perform heavy-duty work.
• Energy Transformation: While a motor converts electrical to mechanical energy, an electric generator works on the reverse principle (Electromagnetic Induction) to convert mechanical to electrical energy.
• Efficiency: Modern electric motors can be highly efficient, often converting over 90 percent of electrical energy into useful mechanical work, far surpassing internal combustion engines.

The correct answer is 25,000 Hz.

Key Points

• The human ear is sensitive to sound waves within a specific frequency range, typically defined as 20 Hz to 20,000 Hz (or 20 kHz).
• Any sound frequency that falls outside this range is generally considered inaudible to the average human being.
• 25,000 Hz is greater than the upper limit of 20,000 Hz, making it inaudible to humans. Hence, option 2 is correct.
• 5000 Hz falls comfortably within the audible range. Hence, option 1 is audible and incorrect.
• 50 Hz is at the lower end of the audible spectrum but is still perceivable by the human ear. Hence, option 3 is audible and incorrect.
• 2000 Hz is within the range where human hearing is most sensitive, particularly for speech. Hence, option 4 is audible and incorrect.
• The unit of frequency is the Hertz (Hz), named after the German physicist Heinrich Rudolf Hertz, which represents one cycle per second.

Additional Information

• Classification of Sound based on Frequency:
• Infrasonic Sound: These are sounds with frequencies lower than 20 Hz. They are produced by large sources like earthquakes, volcanic eruptions, and ocean waves. Animals like elephants and whales can communicate using infrasound over long distances.
• Audible Sound: The standard range for humans is 20 Hz to 20,000 Hz. Sensitivity varies with age and exposure to loud noises; children can often hear slightly higher frequencies than adults.
• Ultrasonic Sound (Ultrasound): These are sounds with frequencies higher than 20,000 Hz. While humans cannot hear them, animals such as bats, dogs, and dolphins can perceive and utilize these frequencies.
• Applications of Ultrasound:
• Medical Imaging: Used in ultrasonography (sonogram) to visualize internal organs and monitor fetal development during pregnancy.
• SONAR (Sound Navigation and Ranging): Used by ships and submarines to detect underwater objects or measure the depth of the sea.
• Industrial Cleaning: High-frequency waves are used to clean parts located in hard-to-reach places, such as spiral tubes and electronic components.
• Physics of Sound:
• Sound is a mechanical longitudinal wave that requires a medium (solid, liquid, or gas) to travel. It cannot travel through a vacuum.
• The pitch of a sound is directly related to its frequency; higher frequency results in a higher pitch (shrill sound), while lower frequency results in a lower pitch (bass sound).

Important Points

• Hearing Loss and Age: As humans age, a condition known as presbycusis often occurs, where the ability to hear high-frequency sounds (near 15,000 Hz - 20,000 Hz) diminishes first.
• Dogs and Whistles: Special whistles known as Galton's whistles produce sounds in the ultrasonic range (above 20,000 Hz). Dogs can hear these signals, but they remain silent to human observers.
• Speed of Sound: The speed of sound is approximately 343 meters per second in dry air at 20°C. It travels fastest in solids and slowest in gases.
• Intensity vs. Frequency: It is important to distinguish between frequency (pitch, measured in Hz) and intensity/loudness (measured in Decibels or dB). A frequency of 25,000 Hz is inaudible regardless of how "loud" (intense) it is played.

The correct answer is It breaks down and is discharged as blood and mucus.

Key Points

• The uterus (womb) prepares itself every month to receive a fertilized egg. To provide a nourishing environment for a potential embryo, the inner lining of the uterus, known as the endometrium, becomes thick and spongy, rich with blood vessels.
• If fertilization by a sperm does not occur, the ovum (egg) survives for about one day. Since the egg is not fertilized, the thick lining of the uterus is no longer required for implantation or pregnancy.
• As the levels of hormones like progesterone and estrogen drop, the uterine lining begins to break down slowly.
• This broken-down lining, along with the unfertilized egg and blood, is discharged from the body through the vagina. This process is called menstruation. Hence, the correct answer is option 2.
• The flow of blood and mucus usually lasts for about 2 to 8 days, and the entire menstrual cycle typically repeats approximately every 28 days.

Additional Information

• Phases of the Menstrual Cycle:
• Menstrual Phase: This is the bleeding phase where the endometrium is shed. It marks the start of a new cycle (Day 1).
• Follicular Phase: Under the influence of Follicle Stimulating Hormone (FSH), new follicles develop in the ovary, and the estrogen levels rise to rebuild the uterine lining.
• Ovulation: Around Day 14 of a 28-day cycle, a mature egg is released from the ovary into the fallopian tube.
• Luteal Phase: After ovulation, the ruptured follicle turns into the corpus luteum, which secretes progesterone to maintain the thickness of the endometrium.
• Key Biological Terms:
• Menarche: The very first menstruation that occurs at puberty, usually between the ages of 10 and 15 years.
• Menopause: The permanent cessation of the menstrual cycle, which typically occurs in females between the ages of 45 and 55 years.
• Hormonal Regulation: The cycle is controlled by the pituitary gland (FSH and LH) and the ovaries (Estrogen and Progesterone).

Important Points

• Effect of Fertilization: If the egg is fertilized, the corpus luteum does not degenerate. It continues to produce progesterone, which keeps the uterine lining intact to support the developing fetus, and menstruation does not occur during pregnancy.
• Endometrium Function: Its primary role is to provide nutrition and oxygen to the embryo via the placenta once implantation occurs.
• Variation: While 28 days is the average, the cycle length can vary between 21 and 35 days depending on health, stress, and dietary factors.
• Excretion: The discharge consists of blood, tissue debris from the uterine wall, and unfertilized oocytes.

The correct answer is Mother always contributes X, father contributes X or Y.

Key Points

• In humans, the sex of a child is determined by the types of chromosomes inherited from the parents. Human cells contain 23 pairs of chromosomes, out of which 22 pairs are autosomes and one pair is the sex chromosome.
• Females have a homogametic sex chromosome makeup, meaning they possess two identical X chromosomes (XX). During the process of meiosis and gamete formation, every egg produced by a female will carry an X chromosome. Hence, the statement that the mother always contributes an X chromosome is correct.
• Males have a heterogametic sex chromosome makeup, consisting of one X chromosome and one Y chromosome (XY). Consequently, 50% of the sperms produced by a male carry the X chromosome, while the other 50% carry the Y chromosome. Hence, the statement that the father contributes X or Y is correct.
• If a sperm carrying an X chromosome fertilizes an egg (which always has an X), the resulting zygote will be XX, leading to a female child. Hence, statement 1 is incorrect as the father can also contribute a Y chromosome.
• If a sperm carrying a Y chromosome fertilizes an egg, the resulting zygote will be XY, leading to a male child. Hence, statement 4 is incorrect because the mother cannot contribute a Y chromosome.
• The genetic probability for the birth of a boy or a girl is 50:50, and it depends entirely on the chromosome contributed by the father's sperm. Hence, statement 3 is incorrect because the mother cannot contribute "either" X or Y.

Additional Information

• Sex Determination Systems:
• While humans use the XY system, other organisms have different mechanisms. In birds, the sex determination follows a ZW system where the female is heterogametic (ZW) and the male is homogametic (ZZ).
• In certain insects like grasshoppers, an XO system is present, where females have XX and males have only one X chromosome (XO).
• The Y Chromosome and SRY Gene:
• The Y chromosome is significantly smaller than the X chromosome. It contains the SRY gene (Sex-determining Region Y), which is responsible for triggering the development of male characteristics in an embryo.
• In the absence of a functional SRY gene, the embryo will typically develop female reproductive structures.
• Chromosomal Abnormalities:
• Errors during cell division can lead to conditions like Turner Syndrome (45, XO), where an individual has only one X chromosome, or Klinefelter Syndrome (47, XXY), where a male has an extra X chromosome.

Important Points

• History: The X chromosome was first identified by Hermann Henking in 1891 during his study of insects, though he initially called it the "X-body."
• Inheritance: Since a male (XY) always receives his Y chromosome from his father, the Y chromosome can be used to trace patrilineal ancestry. Similarly, because mitochondrial DNA is inherited only from the mother, it is used for matrilineal tracing.
• Function of X Chromosome: Unlike the Y chromosome, the X chromosome contains over 900 genes, many of which are essential for biological functions unrelated to sex, such as blood clotting and color vision.

The correct answer is Non-biodegradable.

Key Points

• Non-biodegradable materials are substances that cannot be decomposed, dissolved, or broken down by natural biological agents such as bacteria, fungi, or saprophytes.
• While these materials resist biological decay, they are subject to physical processes such as photodegradation (breakdown by UV radiation from the sun), thermal degradation, and mechanical weathering caused by wind or water.
• These substances are typically synthetic or man-made, featuring strong chemical bonds that natural enzymes cannot penetrate or metabolize.
• Because they are not part of the natural nutrient cycle, they persist in the environment for extremely long periods, ranging from decades to thousands of years.
• Common examples of such materials include plastics, polythene bags, glass, metal scraps, synthetic fibers like nylon, and certain pesticides like DDT.

Additional Information

• Biodegradable Materials:
• These are organic substances that can be easily broken down into simpler compounds like carbon dioxide, water, and methane by the action of microorganisms.
• Examples include food waste, paper, wood, leaves, and animal waste. These materials are eco-friendly as they return nutrients to the soil.
• Saprophytes:
• These are organisms, including certain bacteria and fungi, that obtain their nutrition from dead and decaying organic matter.
• They play a critical role as decomposers in the ecosystem, but they lack the capacity to process non-biodegradable waste due to the absence of specific catalytic enzymes.
• Physical Degradation:
• This refers to the fragmentation of materials into smaller pieces without changing their chemical nature significantly.
• In the case of plastics, physical processes lead to the formation of microplastics (particles smaller than 5mm), which pose a severe threat to marine life and enter the human food chain.

Important Points

• Biomagnification: Non-biodegradable pollutants like mercury and DDT do not decompose and instead accumulate in the bodies of organisms. Their concentration increases at each successive trophic level of the food chain.
• Environmental Persistence: A simple plastic bottle can take up to 450 years to fragment through physical processes, while glass may take over 1 million years to break down, highlighting the permanence of non-biodegradable waste.
• Waste Management: Since biological methods fail, the management of these materials relies on the 3Rs: Reduce, Reuse, and Recycle. Specialized industrial processes like pyrolysis or incineration are sometimes used to manage synthetic polymers.
• Impact on Soil: The accumulation of non-biodegradable waste reduces soil porosity and interferes with the nitrogen-fixing capability of soil bacteria, eventually leading to land degradation.

The correct answer is Only statements 1 and 3 are correct.

Key Points

• Statement 1: When two or more elements undergo a chemical reaction, they combine to form a new substance known as a compound. This process involves the breaking of old bonds and the formation of new chemical bonds, resulting in a product that has entirely different physical and chemical properties from its constituents. Hence, statement 1 is correct.
• Statement 2: A compound is characterized by a fixed composition of its constituent elements by mass. Unlike mixtures, which can have variable composition (meaning the components can be present in any ratio), the elements in a compound are always present in definite proportions. For instance, pure water will always contain Hydrogen and Oxygen in a mass ratio of 1:8. Hence, statement 2 is incorrect.
• Statement 3: The constituents of a compound are chemically unified and cannot be separated by simple physical methods such as filtration, evaporation, distillation, or magnetic separation. Separation of these elements requires the application of energy through chemical reactions or electrochemical reactions, such as the process of electrolysis to separate water into hydrogen and oxygen. Hence, statement 3 is correct.
• Consequently, out of the three given statements, only the first and the third accurately describe the characteristics of a chemical compound.

Additional Information

• Definition of a Compound: A compound is a pure substance made up of two or more elements chemically combined in a fixed ratio. Examples include Sodium Chloride (NaCl), Methane (CH4), and Carbon Dioxide (CO2). A compound is always homogeneous in nature, meaning its properties are uniform throughout the sample.
• Properties of Compounds: The properties of a compound are distinctly different from the properties of the elements that form it. For example, Sodium (Na) is a highly reactive metal and Chlorine (Cl) is a poisonous gas, but their compound Sodium Chloride (Table Salt) is an edible solid essential for life. Furthermore, compounds have sharp and fixed melting points and boiling points.
• Comparison with Mixtures: Mixtures are formed by the physical blending of two or more substances where no chemical reaction takes place. In a mixture, the components retain their original properties and can be separated by physical means. Mixtures do not have a fixed chemical formula or a fixed melting point.
• The Law of Definite Proportions: Also known as Proust's Law, it was formulated by the French chemist Joseph Proust in 1799. It states that a given chemical compound always contains its component elements in a fixed ratio (by mass) and does not depend on its source or method of preparation. This is a fundamental law in stoichiometry.
• Energy Changes: The formation of a compound is usually accompanied by a significant energy change in the form of heat, light, or sound (either exothermic or endothermic reactions), whereas the formation of a mixture involves little to no energy change.

Important Points

• Types of Compounds: Compounds can be classified into organic compounds (containing carbon-hydrogen bonds) and inorganic compounds. They can also be classified based on the type of bonding, such as ionic compounds (held by electrostatic forces) and covalent compounds (formed by sharing electrons).
• Electrolysis: This is a common electrochemical reaction used to decompose compounds. For example, during the electrolysis of water, electric current is passed through acidified water to yield Hydrogen gas at the cathode and Oxygen gas at the anode.
• Valency: The formation of a compound depends on the valency or combining capacity of the elements involved. The fixed composition of a compound is a direct result of the specific number of atoms of each element required to satisfy their valence shells and reach a stable electronic configuration.

The correct answer is Vegetable oil.

• Vegetable oil is primarily composed of triglycerides, which are esters derived from glycerol and three fatty acids. These fatty acids consist of long carbon chains typically ranging from 12 to 22 carbon atoms in length.
• The "unstable" nature mentioned in the question refers to the unsaturated characteristic of most vegetable oils. Unsaturated fatty acids contain one or more double bonds within their carbon chains, which makes them chemically more reactive than saturated fats.
• These double bonds are sites where oxidation can occur, leading to the oil becoming rancid when exposed to air and light. This chemical reactivity is why they are considered "unstable" compared to shorter or fully saturated carbon structures.
• In contrast, Glucose has a relatively short and highly stable six-carbon chain (often forming a ring structure), while Water and Table salt are inorganic compounds that do not contain carbon chains at all.

• Structure of Vegetable Oils:
• Vegetable oils like sunflower oil, mustard oil, and groundnut oil are liquids at room temperature because the "kinks" created by double bonds in their unsaturated carbon chains prevent the molecules from packing closely together.
• The process of hydrogenation is often used in the food industry to convert these unstable liquid oils into saturated solid fats (like vanaspati ghee) by adding hydrogen across the double bonds in the presence of a Nickel catalyst.
• Glucose (C6H12O6):
• It is a monosaccharide and serves as the primary energy source for living organisms.
• It contains only 6 carbon atoms, which is much shorter than the chains found in oils. Its structure is very stable under normal physiological conditions.
• Water (H2O):
• Known as the universal solvent, water is an inorganic polar covalent compound.
• It contains zero carbon atoms and therefore does not possess carbon chains.
• Table Salt (NaCl):
• Scientifically known as Sodium Chloride, it is an ionic compound formed by the attraction between sodium cations and chloride anions.
• It forms a crystalline lattice structure and, like water, contains no carbon.

• Health Implications: Unsaturated vegetable oils are generally considered healthier for the human heart compared to saturated fats, as they help maintain better cholesterol levels, despite being chemically less stable.
• Carbon Chain Length: In organic chemistry, the stability of a molecule is often dictated by the types of chemical bonds (single vs. double vs. triple) rather than just the length. However, in the context of nutrition and lipids, long unsaturated chains are the hallmark of liquid oils.
• Reactivity: The presence of multiple double bonds makes an oil polyunsaturated. These are the most "unstable" and prone to polymerization and oxidation when heated to high temperatures (the smoke point).

The correct answer is act on different bodies.

Key Points

• Newton's Third Law of Motion states that whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first. This is often summarized as "to every action, there is always an equal and opposite reaction."
• The action and reaction forces are always equal in magnitude but opposite in direction. They represent a mutual interaction between two entities.
• A fundamental characteristic of these forces is that they act on different bodies. For instance, if Body A exerts a force on Body B (action), Body B exerts a force on Body A (reaction). Because they are applied to different objects, they do not cancel each other out. Hence, the statement in option 4 is correct.
• The forces do not produce the same effect on both bodies if their masses are different. According to Newton's Second Law (F = ma), the body with smaller mass will experience a greater acceleration compared to the body with larger mass. Hence, statement 1 is incorrect.
• Action and reaction forces never act in the same direction; they are always oriented 180 degrees apart. Hence, statement 2 is incorrect.
• If action and reaction were to act on the same body, the net force would always be zero, making acceleration or motion impossible for any object under interaction. Hence, statement 3 is incorrect.

Additional Information

• Newton's Laws of Motion: These three physical laws were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, published in 1687. They form the foundation of classical mechanics.
• First Law (Law of Inertia): An object remains at rest or in uniform motion in a straight line unless acted upon by an external force.
• Second Law: The rate of change of momentum of an object is proportional to the applied force and takes place in the direction of the force. It is mathematically expressed as F = ma.
• Examples of the Third Law:
• Walking: When a person walks, they push the ground backward with their feet (action), and the ground pushes the person forward with an equal force (reaction).
• Recoil of a Gun: When a bullet is fired from a gun, the gun exerts a forward force on the bullet. The bullet exerts an equal and opposite backward force on the gun, causing it to recoil.
• Rocket Propulsion: A rocket accelerates by expelling gas backward at high velocity. The reacting force of the gas pushes the rocket forward in the vacuum of space.

Important Points

• Simultaneity: There is no time lag between action and reaction. They occur simultaneously. One is not the "cause" of the other in a chronological sense; they are two sides of a single interaction.
• Nature of Forces: Both the action and reaction forces are of the same nature. For example, if the action is a gravitational force, the reaction is also a gravitational force. If the action is a contact force (like friction or normal force), the reaction is also a contact force.
• Vector Quantity: Force is a vector quantity, meaning it has both magnitude and direction. The sum of the action-reaction vector pair across the system is zero, which is a basis for the Law of Conservation of Momentum.

The correct answer is Muscles.

Key Points

• Animal tissues are broadly classified into four types: Epithelial, Connective, Muscular, and Neural tissues.
• Muscles are made up of muscular tissue, which consists of elongated cells called muscle fibers. This tissue is responsible for the movement and locomotion of the body through contraction and relaxation. Hence, Muscles are NOT made of connective tissue.
• Connective tissues are the most abundant and widely distributed in the body of complex animals. They serve the function of linking and supporting other tissues/organs of the body.
• Blood is considered a fluid connective tissue because it contains plasma, red blood cells (RBC), white blood cells (WBC), and platelets, and it circulates throughout the body to transport nutrients and oxygen.
• Cartilage and Bone are types of specialized connective tissues. They provide the structural framework for the body and protect internal organs.

Additional Information

• Muscular Tissue:
• It is categorized into three types: Skeletal muscle (attached to bones and voluntary), Smooth muscle (found in walls of internal organs like the stomach and involuntary), and Cardiac muscle (found only in the heart).
• These tissues play a vital role in maintaining posture and generating heat within the body.
• Bone:
• Bones have a hard and non-pliable ground substance rich in calcium salts and collagen fibers which give bone its strength.
• The bone cells, known as osteocytes, are located in spaces called lacunae. They provide a structural frame to the body and interact with skeletal muscles to bring about movement.
• Cartilage:
• The intercellular material of cartilage is solid and pliable and resists compression.
• Cells of this tissue, called chondrocytes, are enclosed in small cavities within the matrix secreted by them. It is found in the tip of the nose, outer ear joints, and between adjacent bones of the vertebral column.
• Blood:
• It is the only connective tissue that is liquid. Unlike other connective tissues, the fibers are absent in blood unless it starts to clot.
• It serves as the main circulating fluid that helps in the transport of various substances like hormones, waste products, and gases.

Important Points

• Connective Tissue Proper: Includes Areolar tissue (serves as a support framework for epithelium) and Adipose tissue (located mainly beneath the skin and stores fats).
• Dense Connective Tissue: Includes Tendons, which attach skeletal muscles to bones, and Ligaments, which attach one bone to another.
• The primary difference between connective and muscular tissue is that connective tissue contains a large amount of extracellular matrix, whereas muscular tissue is primarily composed of cellular components designed for contractility.

The correct answer is 2n².

Key Points

• The maximum number of electrons that can be accommodated in a particular shell is determined by the formula 2n², where 'n' is the orbit number or principal quantum number.
• This formula was proposed as part of the Bohr-Bury scheme to explain the distribution of electrons in various energy levels surrounding the atomic nucleus.
• For the first shell (K shell), the value of n is 1; therefore, the maximum capacity is 2(1)² = 2 electrons. Hence, the formula 2n² is correct.
• For the second shell (L shell), the value of n is 2; therefore, the maximum capacity is 2(2)² = 8 electrons. Hence, the formula 2n² is correct.
• For the third shell (M shell), the value of n is 3; therefore, the maximum capacity is 2(3)² = 18 electrons. Hence, the formula 2n² is correct.
• For the fourth shell (N shell), the value of n is 4; therefore, the maximum capacity is 2(4)² = 32 electrons. Hence, the formula 2n² is correct.

Additional Information

• Bohr's Model of the Atom: Proposed by Niels Bohr in 1913, this model states that electrons move in discrete circular orbits called energy levels or shells. Electrons do not radiate energy while revolving in these stable orbits.
• Quantum Numbers: The principal quantum number (n) defines the main energy level of an electron. As the numerical value of n increases, the electron's distance from the nucleus and its energy level also increase.
• Shell Designations: The shells are represented by the letters K, L, M, N, O, P, etc., or by the numbers n = 1, 2, 3, 4, 5, 6. The K shell is closest to the nucleus and has the lowest energy.
• Subshells and Orbitals: Within each main shell, there are subshells labeled s, p, d, and f. Each subshell contains a specific number of orbitals, and each orbital can hold a maximum of 2 electrons with opposite spins.
• Octet Rule: Although the 2n² rule defines the theoretical capacity, the outermost shell (valence shell) of a stable atom cannot accommodate more than 8 electrons, regardless of the shell's total capacity (except for the K shell, which is stable with 2 electrons).

Important Points

• Pauli Exclusion Principle: This principle, introduced by Wolfgang Pauli in 1925, states that no two electrons in an atom can have the identical set of four quantum numbers, ensuring that only two electrons occupy a single orbital.
• Electronic Configuration: The distribution of electrons across different shells and subshells is called electronic configuration. For example, Chlorine has an atomic number of 17, and its configuration is 2, 8, 7.
• Filling Sequence: According to the Aufbau Principle, electrons occupy the lowest energy shells first before filling higher energy levels. This ensures the stability of the atom in its ground state.
• Valency: The number of electrons in the outermost shell determines the valency of an element, which governs its chemical reactivity and bonding properties with other atoms.

Shortcut Trick

When n identical cubes with edge a are joined in a row, the number of exposed square faces is (4n + 2).

Here, number of cubes n = 6 and edge a = 10 cm.

Total exposed faces = (4 × 6 + 2) = 26 faces.

Surface Area = Number of faces × a² ⇒ 26 × 10² = 2600 cm².

∴ The correct answer is 2600 cm².

Alternate Method

Given: Edge of each cube = 10 cm | Number of cubes = 6.

Formula Used: Total Surface Area (TSA) of Cuboid = 2(lb + bh + hl).

When cubes are joined end to end, the length of the resulting cuboid increases while breadth and height remain equal to the cube's edge.

⇒ Length (l) = 6 × 10 = 60 cm

⇒ Breadth (b) = 10 cm

⇒ Height (h) = 10 cm

⇒ TSA = 2(lb + bh + hl)

⇒ TSA = 2(60 × 10 + 10 × 10 + 10 × 60)

⇒ TSA = 2(600 + 100 + 600)

⇒ TSA = 2 × 1300

⇒ TSA = 2600 cm²

∴ The correct answer is 2600 cm².

Additional Information

Surface Area of a Cube

A cube has 6 identical square faces; hence the total surface area is 6a² and the lateral surface area is 4a².

Volume of the Resulting Solid

The volume remains the sum of individual volumes: Volume = n × a³. For 6 cubes, V = 6 × 10³ = 6000 cm³.

Diagonal of a Cuboid

The longest rod that can be placed in a cuboid is its diagonal, given by d = √(l² + b² + h²).

Shortcut Trick

Use the ratio relationship between Marked Price (MP) and Cost Price (CP): MP ÷ CP = (100 + Gain%) ÷ (100 − Discount%).

MP ÷ 651 = (100 + 68) ÷ (100 − 86) = 168 ÷ 14.

MP ÷ 651 = 12.

MP = 651 × 12 = 7,812.

∴ The correct answer is ₹7,812.

Alternate Method

Given: CP = ₹651, Gain% = 68%, Discount% = 86%. Formulas: SP = CP × (100 + Gain%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%).

⇒ Step 1: Find Selling Price (SP) = 651 × (168 ÷ 100) = 1,093.68.

⇒ Step 2: Since Discount is 86%, SP is 14% of Marked Price (MP).

⇒ Step 3: MP = 1,093.68 × 100 ÷ 14.

⇒ Step 4: MP = 109,368 ÷ 14 = 7,812.

∴ The correct answer is ₹7,812.

Additional Information

Relation between CP and MP

CP ÷ MP = (100 − D%) ÷ (100 + G%). This ratio is the fastest way to solve problems involving both discount and gain percentage.

Marked Price

The price listed on the label of an article is the Marked Price (List Price). Discounts are always calculated on this value.

Shortcut Trick

Use the Successive Discount formula: a + b − (a × b) ÷ 100.

Here, a = 16 and b = 22.

Equivalent Discount = 16 + 22 − (16 × 22) ÷ 100.

Equivalent Discount = 38 − (352 ÷ 100) = 38 − 3.52.

Equivalent Discount = 34.48%.

∴ The correct answer is 34.48%.

Alternate Method

Given: First discount (d₁) = 16%, Second discount (d₂) = 22%.

Formula: Net Discount = [1 − (1 − d₁/100) × (1 − d₂/100)] × 100

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 16% discount = 100 − 16 = 84.

⇒ Price after second discount (22% of 84) = 84 − (22/100 × 84) = 84 − 18.48.

⇒ Final Selling Price (SP) = 65.52.

⇒ Single equivalent discount = MP − SP = 100 − 65.52 = 34.48%.

∴ The correct answer is 34.48%.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The total effective discount is always less than the sum of individual discounts.

Marked Price & Discount

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

Shortcut Trick

Let Emma's age be 1 unit.

Sarah's age = 2 units + 5 years.

Total units = 1 + 2 + 5 = 29 ⇒ 3 units = 24.

1 unit = 8.

∴ The correct answer is 8 years.

Alternate Method

Given:

Let Emma's age = x years.

Sarah's age = 2x + 5 years.

Sum of their ages = 29 years.

Calculations:

⇒ x + (2x + 5) = 29

⇒ 3x + 5 = 29

⇒ 3x = 29 − 5

⇒ 3x = 24

⇒ x = 24 ÷ 3

⇒ x = 8

∴ The correct answer is 8 years.

Additional Information

Linear Equations in One Variable

An equation of the form ax + b = c, where x is the unknown variable and a, b, c are constants.

Age Relation Problems

Always define the base person's age as 'x' to simplify the relationship expressions for others.

Verification Method

Substitute the answer back: If Emma is 8, Sarah is (2 × 8) + 5 = 21. Sum: 8 + 21 = 29. Correct.

Shortcut Trick

Fourth Proportional of 10, 15, 18 = (15 × 18) ÷ 10 = 270 ÷ 10 = 27

Mean Proportional of 9 and 4 = √(9 × 4) = √36 = 6

Required Ratio = 27 : 6 = 9 : 2

∴ The correct answer is 9:2.

Alternate Method

Given: Numbers for Fourth Proportional (a=10, b=15, c=18) and Mean Proportional (x=9, y=4).

Formula Used: Fourth Proportional (d) = (b × c) ÷ a; Mean Proportional = √(x × y)

⇒ Fourth Proportional (d) = (15 × 18) ÷ 10

⇒ d = 270 ÷ 10 = 27

⇒ Mean Proportional (m) = √(9 × 4)

⇒ m = √36 = 6

⇒ Ratio = 27 : 6 = 9 : 2

∴ The correct answer is 9:2.

Additional Information

Third Proportional

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

Mean Proportional Definition

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

Continued Proportion

Three quantities a, b, c are in continued proportion if a/b = b/c, where b is the mean proportional.

Shortcut Trick

Let the number be N.

Translate the percentage and fraction: 80% = 4/5 and 10% = 1/10.

The equation becomes: N × (4/5) × (1/10) × (2/3) = 748.

Simplify: N × (8/150) = 748 ⇒ N = (748 × 150) ÷ 8.

N = 93.5 × 150 = 14025.

∴ The correct answer is 14025.

Alternate Method

Given:

Product result = 748

Percentage 1 = 80%

Percentage 2 = 10%

Fraction = 2/3

Calculations:

⇒ (80/100) × (10/100) × (2/3) × N = 748

⇒ (4/5) × (1/10) × (2/3) × N = 748

⇒ (8/150) × N = 748

⇒ N = (748 × 150) ÷ 8

⇒ N = 112200 ÷ 8

⇒ N = 14025

∴ The correct answer is 14025.

Additional Information

Percentage to Fraction Conversion

To convert a percentage to a fraction, divide the value by 100. For example, x% = x/100.

Successive Percentage Application

When multiple percentages or fractions are applied to a number, they are multiplied together as factors.

Reciprocal Multiplication

To find the original value from a final result, multiply the result by the reciprocal of each applied fraction.

Shortcut Trick

Total Cost Price (CP) of 9 articles = ₹16.

Total Selling Price (SP) of 9 articles = 9 × 2 = ₹18.

Profit = SP − CP = 18 − 16 = ₹2.

Profit % = (Profit ÷ CP) × 100 = (2 ÷ 16) × 100.

Profit % = (1 ÷ 8) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Alternate Method

Given → CP of 9 articles = ₹16, SP of 1 article = ₹2. Formulas: Profit = SP − CP | Profit % = (Profit ÷ CP) × 100.

⇒ Cost Price (CP) for 9 articles = ₹16. ⇒ Selling Price (SP) for 1 article = ₹2. ⇒ SP for 9 articles = 9 × 2 = ₹18. ⇒ Profit = 18 − 16 = ₹2. ⇒ Profit Percentage = (2 ÷ 16) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Additional Information

Profit and Loss Calculation

Profit occurs when SP > CP. Profit % is always calculated on the CP unless stated otherwise.

Unitary Method

You can also find CP of 1 article (16 ÷ 9) and compare it with SP of 1 article (2) to get the same result.

Shortcut Trick

Mode is defined as the value that appears most frequently in a data set.

By observing the given data, the value 80 appears 5 times.

All other values such as 81 (3 times), 66 (3 times), and others appear less than 5 times.

Therefore, the maximum frequency corresponds to the number 80.

∴ The correct answer is 80.

Alternate Method

Given: Data set = {75, 70, 80, 70, 80, 71, 80, 66, 87, 82, 68, 73, 80, 75, 66, 64, 66, 87, 90, 68, 89, 82, 81, 81, 65, 81, 80}

To find the Mode, we construct a Frequency Table for the observed values:

⇒ Value 80: Observed at positions 3, 5, 7, 13, 27 → Frequency = 5

⇒ Value 81: Observed at positions 23, 24, 26 → Frequency = 3

⇒ Value 66: Observed at positions 8, 15, 17 → Frequency = 3

⇒ Value 75: Observed at positions 1, 14 → Frequency = 2

⇒ Value 70: Observed at positions 2, 4 → Frequency = 2

⇒ Value 87: Observed at positions 9, 18 → Frequency = 2

⇒ Value 82: Observed at positions 10, 22 → Frequency = 2

⇒ Value 68: Observed at positions 11, 20 → Frequency = 2

⇒ Other values (64, 65, 71, 73, 89, 90) each have a frequency of 1.

⇒ Comparing frequencies: 5 > 3 > 2 > 1.

⇒ Since 80 has the highest frequency (5), it is the Mode.

∴ The correct answer is 80.

Additional Information

Definition of Mode

The mode is the value that appears most often in a set of data. A data set can be unimodal (one mode), bimodal (two modes), or multimodal.

Empirical Relationship

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Median and Mean

Mean is the average of all observations, while Median is the middle value when data is arranged in ascending or descending order.

Shortcut Trick

To find the greatest number leaving the same remainder, calculate the HCF of the differences between the numbers.

Differences: (59 − 47) = 12, (74 − 59) = 15, and (74 − 47) = 27.

HCF of 12, 15, and 27 is 3.

∴ The correct answer is 3.

Alternate Method

Given: Numbers are 47, 59, and 74.

Formula: For the same remainder, the number is HCF(|x−y|, |y−z|, |z−x|).

⇒ Difference 1 = 59 − 47 = 12

⇒ Difference 2 = 74 − 59 = 15

⇒ Difference 3 = 74 − 47 = 27

⇒ Prime factors of 12 = 2² × 3

⇒ Prime factors of 15 = 3 × 5

⇒ Prime factors of 27 = 3³

⇒ HCF(12, 15, 27) = 3

∴ The correct answer is 3.

Additional Information

HCF with Same Remainder

The greatest number which divides x, y, and z leaving the same remainder in each case is HCF of (x−y), (y−z), and (z−x).

HCF with Different Remainders

To find the greatest number that divides x, y, z to leave remainders a, b, c, find HCF of (x−a), (y−b), and (z−c).

LCM Applications

LCM is used for finding the smallest number divisible by given values or finding when events (like bells ringing) occur together.

Shortcut Trick

The ratio of Total Surface Area (TSA) depends on the factor r(r + h) since 2π is constant.

For Original Cylinder: Factor₁ = 5 × (5 + 12) = 5 × 17 = 85

For Small Cylinder: Factor₂ = 2 × (2 + 3) = 2 × 5 = 10

Required Ratio (Small : Original) = 10 : 85 = 2 : 17

∴ The correct answer is 2 : 17.

Alternate Method

Given:

Original Cylinder: Radius (R) = 5 cm, Height (H) = 12 cm

Small Cylinder: Radius (r) = 2 cm, Height (h) = 3 cm

Formula Used:

Total Surface Area (TSA) of a Cylinder = 2πr(r + h)

Calculations:

⇒ TSA of Original Cylinder = 2 × π × R × (R + H)

⇒ TSA_Original = 2 × π × 5 × (5 + 12) = 10π × 17 = 170π cm²

⇒ TSA of one Small Cylinder = 2 × π × r × (r + h)

⇒ TSA_Small = 2 × π × 2 × (2 + 3) = 4π × 5 = 20π cm²

⇒ Required Ratio = TSA_Small : TSA_Original

⇒ Required Ratio = 20π : 170π = 20 : 170 = 2 : 17

∴ The correct answer is 2 : 17.

Additional Information

Volume of a Cylinder

The amount of space occupied by a cylinder is given by the formula V = πr²h.

Curved Surface Area (CSA)

The area of the curved part of the cylinder is 2πrh. Total surface area includes the two circular bases: TSA = CSA + 2πr².

Melting and Recasting Rule

When a solid is melted and recast into other shapes, its total Volume remains constant. Volume_Original = n × Volume_Small.

Shortcut Trick

If sinA = 1/2, then the angle A = 30°.

Substituting A in sin3A → sin(3 × 30°) = sin90°.

The value of sin90° is 1.

∴ The correct answer is 1.

Alternate Method

Given: sinA = 1/2

Formula Used: sin3A = 3sinA − 4sin³A

⇒ sin3A = 3 × (1/2) − 4 × (1/2)³

⇒ sin3A = 3/2 − 4 × (1/8)

⇒ sin3A = 3/2 − 1/2

⇒ sin3A = 2/2 = 1

∴ The correct answer is 1.

Additional Information

Trigonometric Triple Angle Formulas

The formula for sin3A is 3sinA − 4sin³A and for cos3A is 4cos³A − 3cosA.

Standard Values of Sine

sin0° = 0, sin30° = 1/2, sin45° = 1/√2, sin60° = √3/2, sin90° = 1.

Range of Sine Function

The value of sinθ for any real θ always lies in the interval [−1, 1].

Shortcut Trick

In ΔBDC, ∠C = 90° − ∠DBC = 90° − 76° = 14°.

In ΔAEC, ∠AEB is the exterior angle at vertex E.

By the Exterior Angle Theorem: ∠AEB = ∠CAE + ∠C.

∠AEB = 53° + 14° = 67°.

∴ The correct answer is 67°.

Alternate Method

Given: In ΔABC, BD ⊥ AC at D, ∠DBC = 76°, and ∠CAE = 53°.

Formula Used: Angle Sum Property of a triangle and Exterior Angle Theorem.

In ΔBDC:

⇒ ∠BDC = 90° (Given BD ⊥ AC)

⇒ ∠DBC = 76° (Given)

⇒ ∠C = 180° − (∠BDC + ∠DBC)

⇒ ∠C = 180° − (90° + 76°) = 14°.

In ΔAEC:

⇒ ∠CAE = 53° (Given)

⇒ ∠ACE = 14° (Calculated above)

⇒ ∠AEB is the exterior angle to ΔAEC at vertex E.

⇒ ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 53° + 14° = 67°.

∴ The correct answer is 67°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of all interior angles of any triangle is always 180°.

Complementary Angles

In a right-angled triangle, the two acute angles are complementary, meaning their sum is exactly 90°.

Shortcut Trick

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

Ratio of cost WM : TV = 3 : 4. For 8 WM and 4 TV, cost ratio = (8 × 3) : (4 × 4) = 24 : 16 = 3 : 2.

Using Weighted Average: % Change = [(3 × 30%) + (2 × −19%)] ÷ (3 + 2).

% Change = (90 − 38) ÷ 5 = 52 ÷ 5 = 10.4%.

Since the result is positive, it is an increase.

∴ The correct answer is Increase by 10.4%.

Alternate Method

Given → TV cost = 100, WM cost = 75 (25% less). Formulas: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂) | % Change = (Change ÷ Original) × 100.

⇒ Initial Total Cost = (8 × 75) + (4 × 100) = 600 + 400 = 1000. ⇒ New WM Cost = 75 + 30% of 75 = 75 + 22.5 = 97.5. ⇒ New TV Cost = 100 − 19% of 100 = 81. ⇒ New Total Cost = (8 × 97.5) + (4 × 81) = 780 + 324 = 1104. ⇒ Percentage Change = [(1104 − 1000) ÷ 1000] × 100 = (104 ÷ 1000) × 100 = 10.4%.

∴ The correct answer is Increase by 10.4%.

Additional Information

Weighted Average Percentage

When different components of a total change by different percentages, the net change is the sum of (weight × individual change) divided by the total weight.

Percentage Increase/Decrease

Percentage change = [(Final Value − Initial Value) ÷ Initial Value] × 100. A positive value indicates an increase, while a negative value indicates a decrease.

Shortcut Trick

Let Total Work = LCM(30, 45, 24) = 360 units.

Efficiencies: A = 12, B = 8, C = −15.

Work done by A + B in first 6 minutes = (12 + 8) × 6 = 120 units.

Remaining work = 360 − 120 = 240 units. Combined efficiency (A+B+C) = 12 + 8 − 15 = 5.

Time for remaining part = 240 ÷ 5 = 48 minutes. Total time = 6 + 48 = 54 minutes.

∴ The correct answer is 54 min.

Alternate Method

Given: Pipe A fills in 30 min, Pipe B fills in 45 min, Pipe C empties in 24 min.

⇒ Work done in first 6 min = (Eff_A + Eff_B) × 6

⇒ Work done = (12 + 8) × 6 = 120 units

⇒ Remaining work = 360 − 120 = 240 units

⇒ Combined efficiency of A, B, and C = 12 + 8 − 15 = 5 units/min

⇒ Time taken to complete remaining work = 240 ÷ 5 = 48 min

⇒ Total time required = Initial time + Remaining time = 6 + 48 = 54 min

∴ The correct answer is 54 min.

Additional Information

Pipes and Cisterns Efficiency

Filling pipes have positive efficiency, while emptying (outlet) pipes have negative efficiency in calculations.

Total Work Calculation

Total capacity of the tank is usually taken as the LCM of the individual times to simplify division.

Combined Rate

When pipes operate together, the net rate is the algebraic sum of their individual rates: Net Rate = R₁ + R₂ − R_outlet.

Shortcut Trick

Given a + b = 4 and ab = 3.

Use the direct identity: a³ + b³ = (a + b)³ − 3ab(a + b).

Substitute the values: 4³ − 3 × 3 × 4

Value = 64 − 36 = 28.

∴ The correct answer is 28.

Alternate Method

Given:

Sum of terms (a + b) = 4

Product of terms (ab) = 3

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

⇒ (4)³ = a³ + b³ + 3 × 3 × 4

⇒ 64 = a³ + b³ + 36

⇒ a³ + b³ = 64 − 36

⇒ a³ + b³ = 28

∴ The correct answer is 28.

Additional Information

Sum of Cubes Formula

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

Difference of Cubes Formula

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

Standard Binomial Expansion

The cube of a sum is expanded as (a + b)³ = a³ + 3a²b + 3ab² + b³, which simplifies to a³ + b³ + 3ab(a + b).

Shortcut Trick

Ratio of Income : Expenditure = 8 : 5.

Saving = Income − Expenditure = 8 − 5 = 3 units.

Saving % = (Saving ÷ Income) × 100 = (3 ÷ 8) × 100.

Since 1/8 = 12.5%, then 3/8 = 3 × 12.5% = 37.5%.

∴ The correct answer is 37.5%.

Alternate Method

Given: Ratio of Income (I) to Expenditure (E) is 8 : 5.

Formula Used: Saving (S) = Income − Expenditure; Saving % = (S/I) × 100

⇒ Let Income = 8x and Expenditure = 5x

⇒ Saving = 8x − 5x = 3x

⇒ Saving Percentage = (3x ÷ 8x) × 100

⇒ Saving Percentage = 0.375 × 100

⇒ Saving Percentage = 37.5%

∴ The correct answer is 37.5%.

Additional Information

Income-Expenditure Identity

Total Income is always the sum of Expenditure and Savings (I = E + S).

Percentage Conversion

To find the percentage of Savings, always use Income as the base (denominator) unless specified otherwise.

Common Fraction to Percentage

1/8 = 12.5%, 2/8 (1/4) = 25%, 3/8 = 37.5%, 4/8 (1/2) = 50%.

Shortcut Trick

Combined efficiency of A, B, and C = (1/3 + 1/3 + 1/12) = 9/12 = 3/4 units/day.

Days taken for same work = 1 ÷ (3/4) = 4/3 days.

Days taken for thrice the work = 3 × (4/3) = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Days taken by A = 3, B = 3, C = 12.

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

⇒ Total Efficiency of A + B + C = 4 + 4 + 1 = 9 units/day.

⇒ Work to be completed = 3 × Total Work = 3 × 12 = 36 units.

⇒ Time taken = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Work-Efficiency Relation

Efficiency is inversely proportional to the time taken to complete the work. More efficiency means less time.

Work and Wages

Wages are distributed in the ratio of work done or the ratio of efficiencies when all persons work for the same duration.

Pipes and Cisterns

Similar logic applies where filling pipes have positive efficiency and leakage/outlet pipes have negative efficiency.

Shortcut Trick

Let the price of a short notebook be x. Then the long notebook price is x + 20.

According to the question: 4(x + 20) + 5x = 305

9x + 80 = 305 ⇒ 9x = 225 ⇒ x = 25.

Price of long notebook = 25 + 20 = 45.

∴ The correct answer is 45, 25.

Alternate Method

Given:

Total cost of 4 long and 5 short notebooks = ₹305.

Difference in price = ₹20.

Formula Used:

Formation of Linear Equations: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂).

Calculations:

Let L be the price of a long notebook and S be the price of a short notebook.

⇒ 4L + 5S = 305 --- (Eq. 1)

⇒ L − S = 20 ⇒ L = S + 20 --- (Eq. 2)

Substituting the value of L from (Eq. 2) into (Eq. 1):

⇒ 4(S + 20) + 5S = 305

⇒ 4S + 80 + 5S = 305

⇒ 9S = 305 − 80

⇒ 9S = 225

⇒ S = 25

Now, finding the value of L:

⇒ L = 25 + 20 = 45

∴ The correct answer is 45, 25.

Additional Information

Substitution Method

A process of solving a system of equations by expressing one variable in terms of the other and substituting it into the remaining equation.

Elimination Method

A technique where equations are added or subtracted to eliminate one variable, making it easier to solve for the other.

Linear Equation in Two Variables

An algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line, typically written as ax + by = c.

Shortcut Trick

Convert all to base 2:

32 = 2⁵, 64 = 2⁶, 16 = 2⁴, 4 = 2²

Expression inside root:

= 2⁵ × (2⁶)⁻² × 2 × (2⁴)⁻³ × (2²)^m+3

⇒ = 2^{5 − 12 + 1 − 12 + 2m + 6}

⇒ = 2^{2m − 12}

Square root ⇒ power becomes half:

⇒ = 2^{(2m − 12)/2} = 2^{m − 6}

Given value = 2 ⇒ 2^m−6 = 2¹

⇒ m − 6 = 1 ⇒ m = 7

∴ Answer = 7

Alternate Method

Given:

√(32 × 64⁻² × 2 / 16³ × 4^m+3) = 2

Formula used:

a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn

Calculation:

⇒ Convert to base 2 and combine powers

⇒ Inside root = 2^{2m − 12}

⇒ After square root = 2^{m − 6}

⇒ 2^{m − 6} = 2

⇒ m − 6 = 1

⇒ m = 7

∴ The correct answer is 7.

Additional Information

Exponent Rule

Convert all numbers to same base for easy simplification.

Square Root Rule

√(aⁿ) = a^n/2

Power Matching

If bases are same, equate exponents.

Shortcut Trick

Speed is constant → Distance is directly proportional to Time.

Ratio of Time = 6 : 4.5 = 4 : 3

4 units = 240 km

1 unit = 60 km

3 units = 3 × 60 = 180 km

∴ The correct answer is 180 km.

Alternate Method

Given:

Distance₁ = 240 km

Time₁ = 6 hours

Time₂ = 4.5 hours

Formula Used:

Speed = Distance ÷ Time

Distance = Speed × Time

Calculations:

⇒ Speed of the bus = 240 ÷ 6

⇒ Speed = 40 kmph

⇒ New Distance = Speed × New Time

⇒ New Distance = 40 × 4.5

⇒ New Distance = 180 km

∴ The correct answer is 180 km.

Additional Information

Direct Proportionality

When speed is constant, distance (D) is directly proportional to time (T), expressed as D₁/T₁ = D₂/T₂.

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds u and v, Average Speed = 2uv/(u+v).

Unit Conversion

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

Shortcut Trick

Speed ratio = 5 : 6 → Time ratio = 6 : 5 (since distance is constant).

Difference in time ratio parts = 6 − 5 = 1 unit.

Given 1 unit = 25 minutes.

Usual time = 5 units = 5 × 25 = 125 minutes.

∴ The correct answer is 125 min.

Alternate Method

Given: New speed = 5/6 of normal speed, Time delay = 25 minutes.

Formula Used: Distance = Speed × Time. If distance is constant, Speed ∝ 1/Time.

⇒ Let normal speed be S and usual time be T.

⇒ Normal Distance = S × T

⇒ New Distance = (5/6)S × (T + 25)

⇒ S × T = (5/6)S × (T + 25)

⇒ T = 5/6(T + 25)

⇒ 6T = 5T + 125

⇒ T = 125 minutes.

∴ The correct answer is 125 min.

Additional Information

Inverse Proportion of Speed and Time

When the distance is constant, the ratio of speeds is inversely proportional to the ratio of time taken: S₁/S₂ = T₂/T₁.

Late/Early Concept Shortcut

Usual Time = (Late Time × Numerator) ÷ (Difference between Numerator and Denominator) = (25 × 5) ÷ (6 − 5) = 125 min.

Average Speed

If a person covers two equal distances at speeds x and y, the average speed is 2xy/(x+y).

Shortcut Trick

संख्या 6336 के लिए 11 की विभाज्यता की जाँच करें।

विषम स्थानों के अंकों का योग = 6 + 3 = 9

सम स्थानों के अंकों का योग = 3 + 6 = 9

अंतर = 9 − 9 = 0

चूँकि अंतर 0 है, इसलिए संख्या पहले से ही 11 से विभाज्य है।

∴ The correct answer is 0.

Alternate Method

दिया गया है: संख्या = 6336

प्रयुक्त सूत्र: यदि (विषम स्थानों के अंकों का योग) − (सम स्थानों के अंकों का योग) = 0 या 11 का गुणज हो, तो संख्या 11 से विभाज्य होती है।

⇒ विषम स्थानों का योग (1st + 3rd) = 6 + 3 = 9

⇒ सम स्थानों का योग (2nd + 4th) = 3 + 6 = 9

⇒ अंतर = 9 − 9 = 0

⇒ चूँकि अंतर 0 है, इसलिए संख्या 11 से पूर्णतः विभाज्य है।

⇒ अतः, संख्या में 0 जोड़ने या घटाने पर वह 11 से विभाज्य बनी रहेगी।

∴ The correct answer is 0.

Additional Information

9 की विभाज्यता का नियम

यदि किसी संख्या के सभी अंकों का योग 9 से विभाज्य है, तो वह संख्या 9 से विभाज्य होती है (यहाँ 6+3+3+6 = 18, जो 9 से विभाज्य है)।

11 की विभाज्यता का नियम

एकांतर अंकों के योग का अंतर 0 या 11 का गुणज होना चाहिए।

99 की विभाज्यता

यदि कोई संख्या 9 और 11 दोनों से विभाज्य है, तो वह उनके गुणनफल 99 से भी विभाज्य होगी।

Shortcut Trick

Rate = 16% = 4/25 ⇒ Ratio of Principal to Amount for 1 year = 25 : 29.

For 2 years, the ratio becomes 25² : 29² = 625 : 841.

Given 625 units = ₹7500 ⇒ 1 unit = ₹12.

Total Amount = 841 units × 12 = ₹10,092.

∴ The correct answer is ₹10,092.

Alternate Method

Given:

Principal (P) = ₹7,500

Rate of Interest (R) = 16% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 7500 × (1 + 16/100)²

⇒ A = 7500 × (116/100) × (116/100)

⇒ A = 7500 × (29/25) × (29/25)

⇒ A = (7500 ÷ 625) × (29 × 29)

⇒ A = 12 × 841

⇒ A = 10092

∴ The correct answer is ₹10,092.

Additional Information

Effective Rate of Interest

For 2 years at R%, the effective compound interest rate is R + R + (R×R)/100. For 16%, it is 16 + 16 + 2.56 = 34.56%.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is given by P(R/100)².

Successive Percentage Change

Compound interest is essentially a successive percentage increase on the initial principal value over the given time periods.

Shortcut Trick

Use the direct empirical relationship formula: Mean = (3 × Median − Mode) ÷ 2.

Substitute the given values: (3 × 31 − 72.8) ÷ 2.

⇒ (93 − 72.8) ÷ 2 = 20.2 ÷ 2 = 10.1.

∴ The correct answer is 10.1.

Alternate Method

Given:

Mode of the data = 72.8

Median of the data = 31

Formula Used:

The Empirical Relationship between measures of central tendency is:

Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 72.8 = 3 × 31 − 2 × Mean

⇒ 72.8 = 93 − 2 × Mean

⇒ 2 × Mean = 93 − 72.8

⇒ 2 × Mean = 20.2

⇒ Mean = 20.2 ÷ 2

⇒ Mean = 10.1

∴ The correct answer is 10.1.

Additional Information

Empirical Formula for Skewed Distributions

For moderately asymmetrical distributions, the relationship is Mean − Mode = 3(Mean − Median), which simplifies to Mode = 3 Median − 2 Mean.

Types of Skewness

In a Symmetric Distribution, Mean = Median = Mode. In a Positively Skewed distribution, Mean > Median > Mode. In a Negatively Skewed distribution, Mode > Median > Mean.

Measures of Central Tendency

Mean is the average, Median is the middle value, and Mode is the most frequent value in a data set.

Shortcut Trick

Total amount to be divided = ₹10,896.

Total deductions = 91 + 4 + 57 = ₹152.

Remaining amount = 10,896 − 152 = ₹10,744.

Divide ₹10,744 in ratio 3 : 8 : 6 (Total units = 17).

Difference between B and C shares (after deduction) = 8 − 6 = 2 units.

Difference between original shares = (Value of 2 units) + (Difference in deductions) = (10,744 ÷ 17 × 2) + (4 − 57) = 1,264 − 53 = ₹1,211.

∴ The correct answer is ₹1,211.

Alternate Method

Given: Total sum = ₹10,896. Deductions from A, B, C are ₹91, ₹4, ₹57. Ratio after deduction = 3 : 8 : 6.

Calculations:

⇒ Sum of ratio parts = 3 + 8 + 6 = 17 units

⇒ Total deduction = 91 + 4 + 57 = ₹152

⇒ Remaining amount = 10,896 − 152 = ₹10,744

⇒ Value of 1 unit = 10,744 ÷ 17 = ₹632

⇒ B's share after deduction = 8 × 632 = ₹5,056

⇒ B's original share = 5,056 + 4 = ₹5,060

⇒ C's share after deduction = 6 × 632 = ₹3,792

⇒ C's original share = 3,792 + 57 = ₹3,849

⇒ Required Difference = 5,060 − 3,849 = ₹1,211

∴ The correct answer is ₹1,211.

Additional Information

Ratio and Proportion

If a sum S is divided in ratio a:b:c, the shares are [a/(a+b+c)]×S, [b/(a+b+c)]×S, and [c/(a+b+c)]×S respectively.

Deduction problems

Always subtract the specific deductions from the total sum before applying the ratio units to find the individual shares.

BODMAS Table: 

Decoding the given information:

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

⇒ 608 ÷ 76 − 6 × 37 + 259 = ?

⇒ 8 − 222 + 259 = ?

⇒ 267 − 222 = ?

⇒ 45

Hence, the correct option is "Option 4".

Given:

Mr. Hepa is 9th from the left end and Mr. Annu sits 56 places to the right of Mr. Hepa in a row of 76 students.

Here,

Mr. Annu’s position from the left end = 9 + 56 = 65th

Now, Mr. Annu’s position from the right end = 76 − 65 + 1 = 12th

So, Mr. Annu is 12th from the right end.

Hence, the correct answer is "Option 4".

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

Now,

Even numbers, which are immediately preceded by an odd number and also immediately followed by an odd number.

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

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

So, total valid even numbers = 4

Hence, the correct answer is Option 3.

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

Now,

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

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

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

So, total valid odd numbers = 5

Hence, the correct answer is Option 4.

BODMAS Table: 

Decoding the given information:

 

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

⇒ 9 × 2 − 24 ÷ 6 + 3 = ? − 5

⇒ 18 − 4 + 3 = ? − 5

⇒ 14 + 3 = ? − 5

⇒ 17 = ? − 5

⇒ ? = 17 + 5

⇒ ? = 22

Hence, the correct option is "Option 4".

The logic followed here is:

thus, "4" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

Logic: First Number = Second Number × Third Number.

For, (15 - 3 - 5) →

⇒ 3 × 5 = 15

For, (24 - 4 - 6) →

⇒ 4 × 6 = 24

Now, checking the options:

Option 1) (20 - 2 - 10)

⇒ 2 × 10 = 20 

Option 2) (35 - 5 - 8)

⇒ 5 × 8 = 40 ≠ 35

Option 3) (63 - 7 - 11)

⇒ 7 × 11 = 77 ≠ 63

Option 4) (50 - 3 - 19)

⇒ 3 × 19 = 57 ≠ 50

Hence, the correct answer is Option 1.

Given Statements:

All rings are kettles.

All dogs are bottles.

No rings are bottles.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some kettles are not dogs – Follows (Rings are kettles and no rings are bottles. Since all dogs are bottles, rings cannot be dogs. Therefore, some kettles are not dogs).

II. No rings are dogs – Follows (All dogs are bottles and no rings are bottles, so rings cannot be dogs).

So, both conclusion I and conclusion II follow.

Hence, the correct answer is "Option 1".

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

A sits third to the left of E.

E sits second to the left of F.

D sits to the immediate right of E.

B sits to the immediate left of C.

Now,

People between C and F → E and D = 2

Hence, the correct answer is "Option 3".

Given: 745, 768, 729, 716, 721

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

Descending order: 768 > 745 > 729 > 721 > 716

Second highest number = 745

Second digit of second highest number = 4

Lowest number = 716

Third digit of lowest number = 6

So, 4 + 6 = 10

Hence, the correct answer is Option 1.

The logic followed here is:

Option 1) EF – HL

Option 2) VW – ZA

Option 3) OP – ST

Option 4) MN – QR

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

Hence, the correct answer is "Option 1".

Given Statements:

Some easels are brushes.

Some canvases are brushes.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some brushes are canvases – Follows (From “Some canvases are brushes”, it follows that some brushes are canvases).

II. All easels are canvases – Does not follow (No direct relation is given between easels and canvases).

So, only conclusion I follows.

Hence, the correct answer is "Option 1".

The logic followed here is:

Given:

Karthik starts from Point Y and drives 86 km towards the east.

He then takes a right turn, drives 41 km, turns right and drives 61 km.

He then takes a right turn and drives 92 km.

He takes a left turn, drives 25 km.

He then turns right, drives 31 km and stops at Point Z

 

​

So,To reach back to Point Y, he must move 82 km towards the South

Hence, the correct answer is Option 4.

The logic followed here is:

thus, "224" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

The logic followed here is:

Given:

‘dirt is there’ → mp rc gp

‘clean the dirt’ → hk mp bo

So, the code for dirt = mp.

Hence, the correct answer is Option 1.

The logic followed here is:

thus, "60" will replace the question mark in the given series.

Hence, the correct option is "Option 4".

Family Chart:

Decoding the Symbols:

Given: K & T − R × Z + V

It means: K is the father of T, T is the brother of R, R is the wife of Z, and Z is the son of V.

Therefore, K is related to V as son’s wife’s father.

Hence, the correct answer is "Option 1".

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

Now,

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

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

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

So, total valid even numbers = 3

Hence, the correct answer is Option 3.

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

D sits at the extreme left end of the row.

E sits third to the right of A.

B sits to the immediate right of A.

Now, People between D and E → A, B, C = 3

Hence, the correct answer is "Option 1".

The logic followed here is:

Given:

‘move the rocks’ → ld an ur

‘rocks block roads’ → mp cq an

So, the code for rocks = an.

Hence, the correct answer is Option 1.

The logic followed here is:

So, the missing term = ECY 116

Hence, the correct answer is Option 2.

The logic followed here is:

For MLPO → NMQP

Now, for GFJI → HGKJ

Now, applying the same logic for ONRQ:

So, ONRQ → POSR

Hence, the correct answer is "Option 1".

BODMAS Table: 

Decoding the given information:

 

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

⇒ 97 × 11 + 67 − 864 ÷ 16 = ?

⇒ 1067 + 67 − 54 = ?

⇒ 1134 − 54 = ?

⇒ 1080

Hence, the correct option is "Option 4".

The logic followed here is:

For IS69 → LO61

Now, verify with WE43 → ZA35

Now, applying the same logic for GI91:

So, GI91 → JE82

Hence, the correct answer is "Option 3".

Given:

Kiran is 10th from the left end and Sandeep is 20th from the right end in a row of 60 people.

Here, Number of people between them = 60 − (10 + 20)

= 60 − 30

= 30

So, there are 30 people between Kiran and Sandeep.

Hence, the correct answer is "Option 2".

The logic followed here is:

So, the missing term = UAE 41

Hence, the correct answer is Option 2.

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

Only three people sit to the left of C.

Only D sits to the right of B.

Only three people sit between B and F.

A sits at some place to the left of G but at some place to the right of E.

Now, People to the right of A : 4

Hence, the correct answer is "Option 3".

The logic followed here is:

Given:

Town X is to the north of Town P

Town X is to the east of Town Y

Town W is to the south of Town Y

Town Z is to the east of Town W

Town Z is to the north of Town P

Therefore, Town X is to the north of Town Z.

Hence, the correct answer is Option 3.

The logic followed here is:

For BDH → EGK

Now, for KMQ → NPT

Now, applying the same logic to the given options:

Option 1: ITD → YDS

Option 2: COD → NCX

Option 3: BTD → FGY

Option 4: HQW → KTZ

Hence, HQW → KTZ follows the same logic.

Hence, the correct answer is "Option 4".

The logic followed here is:

Option 1) ZWY

Option 2) QOP

Option 3) JGI

Option 4) IFH

Here, QOP is odd from the given options.

Hence, the correct answer is "Option 2".

The correct answer is Prabowo Subianto.

Key Points

• The Government of India invited Prabowo Subianto, the President of Indonesia, to be the Chief Guest for the 76th Republic Day parade held on January 26, 2025.
• This invitation marks a significant milestone in the bilateral relations between India and Indonesia, particularly as both nations are celebrating 75 years of established diplomatic relations in 2024-2025.
• Prabowo Subianto, who took office as the 8th President of Indonesia in October 2024, made his first official state visit to India during this event, reinforcing the Comprehensive Strategic Partnership signed in 2018.
• The Republic Day parade, held at Kartavya Path in New Delhi, serves as a platform to showcase India's military strength, cultural heritage, and technological advancements to the global community.
• The choice of an Indonesian leader is historically symbolic, as President Sukarno was the first-ever Chief Guest at India's very first Republic Day celebrations in 1950.

Additional Information

• India-Indonesia Strategic Ties:
• India and Indonesia share deep maritime boundaries and are key partners in the Indo-Pacific region.
• The "Act East Policy" of India considers Indonesia a vital pillar for regional stability and maritime security.
• Economic cooperation is robust, with Indonesia being India's largest trading partner in the ASEAN region, particularly in the trade of palm oil, coal, and crude petroleum.
• Recent Chief Guests:
• In 2024 (the 75th Republic Day), the Chief Guest was Emmanuel Macron, the President of France.
• In 2023, the Chief Guest was Abdel Fattah el-Sisi, the President of Egypt.
• During the years 2021 and 2022, there were no foreign Chief Guests due to the COVID-19 pandemic.
• Republic Day Significance:
• Republic Day commemorates the date on which the Constitution of India came into effect on January 26, 1950, replacing the Government of India Act (1935).
• The date January 26 was chosen because it was on this day in 1930 that the Indian National Congress proclaimed Purna Swaraj (Complete Independence).

Important Points

• Defense Cooperation: India and Indonesia regularly conduct joint military exercises, including Garuda Shakti (Army) and Samudra Shakti (Navy), to enhance interoperability.
• Geopolitical Context: The invitation to Prabowo Subianto highlights India's focus on the Global South and its commitment to strengthening ties with Southeast Asian neighbors amid changing geopolitical dynamics in the Indo-Pacific.
• Cultural Connection: The two nations share a "shared heritage," with influences of Ramayana and Mahabharata deeply embedded in Indonesian culture and art forms.

The correct answer is Punjab and Haryana.

Key Points

• In September 2025, the Commission for Air Quality Management (CAQM) conducted high-level review meetings with the state governments of Punjab and Haryana to finalize their action plans for preventing stubble burning during the upcoming Kharif harvest season.
• The primary objective of these meetings was to monitor the ground-level preparedness of district administrations in these two states to ensure a significant reduction in fire counts compared to previous years.
• The CAQM emphasized the optimal utilization of Crop Residue Management (CRM) machinery, highlighting that thousands of machines have been subsidized to help farmers manage paddy straw without resorting to burning.
• Both Punjab and Haryana presented detailed strategies involving In-situ (mixing straw back into the soil) and Ex-situ (using straw for industrial purposes) management to mitigate air pollution in the National Capital Region (NCR).
• The commission directed the states to intensify Information, Education, and Communication (IEC) activities to sensitize farmers about the harmful effects of Parali burning and the available government incentives for alternative methods.

Additional Information

• Commission for Air Quality Management (CAQM):
• It is a statutory body established under the Commission for Air Quality Management in National Capital Region and Adjoining Areas Act, 2021.
• The commission replaced the Environment Pollution (Prevention and Control) Authority (EPCA) and has the power to issue directions and entertain complaints for protecting air quality in the NCR and adjoining areas like Punjab, Haryana, Rajasthan, and Uttar Pradesh.
• Stubble Burning (Parali Burning):
• This is the practice of setting fire to paddy straw to quickly clear the fields for the next crop cycle, specifically Wheat (Rabi crop).
• It is a major source of Particulate Matter (PM2.5 and PM10), Carbon Monoxide (CO), and Methane (CH4), which causes severe smog and health issues during the winter months in North India.
• Management Techniques:
• In-situ Management: Involves using machines like the Happy Seeder, Super Seder, and Smart Seder to sow seeds directly into the field containing residue. The use of Pusa Bio-decomposers (developed by ICAR-IARI) is also promoted to speed up the natural decomposition of straw.
• Ex-situ Management: Involves the collection of straw for use in Biomass Power Plants, Bio-CNG production, Paper and Pulp industries, and as cattle fodder.
• Government Initiatives:
• The Central Government provides funds under the Central Sector Scheme for Crop Residue Management to provide 50% subsidy to individual farmers and 80% subsidy to Custom Hiring Centres (CHCs) for purchasing machinery.
• The Graded Response Action Plan (GRAP) is also implemented by the CAQM to enforce emergency measures based on the Air Quality Index (AQI) levels in Delhi-NCR.

Important Points

• The harvesting window between Paddy and Wheat is very short (about 15 to 20 days), which is why farmers often resort to burning as the fastest clearing method.
• Punjab and Haryana are the largest contributors to smoke plumes seen via satellite monitoring by ISRO during October and November.
• The CAQM has the authority to impose Environmental Compensation (EC) on farmers caught burning stubble, as per National Green Tribunal (NGT) guidelines.