RRB Group D Official Paper (Held On: 10 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is particles are continuously moving.

Key Points

• The primary reason the smell of hot food reaches a distance is that the particles of matter are continuously moving. This is a fundamental property of matter described by the Kinetic Molecular Theory. Hence, the correct statement is that particles are continuously moving.
• In the case of hot food, the temperature is high, which significantly increases the kinetic energy of the aroma particles (volatile organic compounds) released from the food.
• When the kinetic energy increases, the velocity of these particles also increases, allowing them to mix rapidly with air particles and move across large distances in a short amount of time.
• This process of intermixing of particles of two different types of matter on their own is known as diffusion. The rate of diffusion is much faster in gases compared to liquids or solids.
• Regarding other options: Sound waves are mechanical waves that carry energy through vibrations, not chemical aromas. Hence, statement 1 is incorrect.
• Light energy does not carry physical matter like aroma particles; it is an electromagnetic wave. Hence, statement 2 is incorrect.
• While temperature affects the state of food, it does not increase the weight of the food in a way that makes smell travel; in fact, evaporation might slightly decrease the mass. Hence, statement 3 is incorrect.

Additional Information

• Diffusion: It is the movement of particles from a region of higher concentration to a region of lower concentration. In gases, diffusion is extremely rapid due to high particle speed and large intermolecular spaces.
• Kinetic Energy and Temperature: The average kinetic energy of particles is directly proportional to the absolute temperature (measured in Kelvin). This explains why the smell of cold food does not reach us as quickly as the smell of hot sizzling food.
• Brownian Motion: This is the random, zig-zag motion of microscopic particles in a fluid (liquid or gas) resulting from their collision with the fast-moving atoms or molecules in the gas or liquid. This was first observed by Robert Brown in 1827.
• States of Matter: The rate of diffusion follows the order: Gases > Liquids > Solids. Because gas particles have the least intermolecular force of attraction, they move most freely.

Important Points

• Volatility: Only volatile substances produce a smell. When food is heated, more molecules transition into the vapour phase, increasing the concentration of aroma particles in the air.
• Practical Examples:
• The smell of a perfume spreading in a room is due to the diffusion of perfume vapours in the air.
• The smell of an incense stick (agarbatti) when lit reaches us even in another room because the heat increases the movement of its particles.
• Dissolving potassium permanganate in water shows diffusion in liquids, where the purple color spreads throughout the beaker.
• Atmospheric Factors: The speed at which a smell travels can also be influenced by wind currents and convection, but the underlying molecular mechanism remains the continuous motion and diffusion of particles.

The correct answer is a, b and c.

Key Points

• A suspension is defined as a heterogeneous mixture in which the solute particles do not dissolve but remain suspended throughout the bulk of the medium. Unlike a solution, the components can be clearly distinguished. Hence, statement a is correct.
• The particles of a suspension are large enough to be seen by the naked eye. This is due to their size being typically greater than 1000 nm (or 1 micrometer) in diameter, which makes them physically distinct from the solvent. Hence, statement c is correct.
• Suspensions exhibit the Tyndall effect, which is the scattering of a beam of light as it passes through the mixture. Because the particles are large, they collide with light waves and deflect them, making the path of light visible. Hence, statement b is correct.
• If a suspension is left undisturbed, the particles will eventually settle down at the bottom of the container due to the force of gravity. This indicates that a suspension is unstable in nature.
• When the particles settle down, the mixture loses its ability to scatter light. Therefore, the Tyndall effect is only observed while the particles are still actively suspended in the medium.

Additional Information

• Separation Methods: Since the particles in a suspension are relatively large and do not dissolve, they can be separated from the mixture using the process of filtration. The residue remains on the filter paper while the liquid passes through.
• Common Examples: Familiar examples of suspensions include muddy water, chalk powder in water, sand in water, and flour mixed in water. In these cases, the solid phase is clearly visible and eventually precipitates.
• Comparison with Solutions: In contrast to suspensions, a true solution is a homogeneous mixture where particle sizes are less than 1 nm. These particles cannot be seen by the naked eye and do not scatter light.
• Comparison with Colloids: Colloids sit between solutions and suspensions. Their particle size ranges from 1 nm to 1000 nm. While they appear homogeneous, they are actually heterogeneous and do scatter light, but their particles do not settle under gravity.

Important Points

• Particle Size: The distinguishing factor for a suspension is a particle size exceeding 1000 nm. This size threshold is what allows the particles to be filtered and seen without a microscope.
• Optical Properties: The visibility of the light path is a key diagnostic tool in chemistry to differentiate between a true solution and a suspension or colloid.
• Stability: A suspension is chemically unstable because the solute-solvent interaction is not strong enough to overcome gravitational pull over time, leading to sedimentation.
• State of Matter: Suspensions can exist in various forms, such as solids in liquids (mud in water) or solids in gases (dust particles in the air).

The correct answer is Meristematic cells are dead at maturity..

Key Points

• Meristematic tissues consist of a group of actively dividing cells that help in the increase of length and girth of the plant. These cells continue to divide throughout the life cycle of the plant. Hence, the statement 'Meristematic tissues consist of actively dividing cells' is correct.
• The cells of these tissues are very active, have dense cytoplasm, thin cellulose walls, and prominent nuclei. The thin walls facilitate rapid cell division. Hence, the statement 'Cells are small, thin-walled, and densely cytoplasmic with large nuclei' is correct.
• Unlike animals, plants have localized regions of growth. Meristematic tissues are present in these growth regions, allowing plants to grow indefinitely or exhibit lifelong growth. Hence, the statement 'Enable lifelong growth in plants' is correct.
• Meristematic cells are living cells because they need to be metabolically active to undergo mitosis. Cells that are dead at maturity, such as sclerenchyma or xylem vessels, serve structural or transport functions and cannot divide. Hence, the statement 'Meristematic cells are dead at maturity' is incorrect.
• The vacuoles are usually absent in meristematic cells because these cells do not need to store food or waste materials, as they are constantly dividing.

Additional Information

• Classification by Position:
• Apical Meristem: Located at the growing tips of stems and roots. Its primary function is to increase the length of the plant, known as primary growth.
• Intercalary Meristem: Found at the base of leaves or internodes (e.g., in grasses). It helps in the regeneration of parts removed by grazing herbivores.
• Lateral Meristem: Occurs on the sides of the stem and root. It is responsible for increasing the thickness or girth of the plant, known as secondary growth. Examples include vascular cambium and cork cambium.
• Differentiation:
• When cells produced by meristematic tissue take up a permanent shape, size, and function, they lose the ability to divide. This process is called differentiation, leading to the formation of permanent tissues.
• Types of Permanent Tissues:
• Simple Permanent Tissue: Includes Parenchyma (storage), Collenchyma (flexibility), and Sclerenchyma (strength/rigidity).
• Complex Permanent Tissue: Includes Xylem (water transport) and Phloem (food transport).

Important Points

• The term "Meristem" was coined by C. Nageli in 1858.
• Meristematic cells lack intercellular spaces; they are compactly arranged to ensure structural continuity during rapid division.
• The metabolic rate in meristematic cells is extremely high because of the energy demands of DNA replication and cytokinesis.
• In woody plants, the lateral meristem adds layers of secondary xylem (wood) and secondary phloem each year, creating annual rings used to determine the age of the tree.

The correct answer is Light.

Key Points

• Cross-pollination, also known as allogamy, involves the transfer of pollen grains from the anther of a flower of one plant to the stigma of a flower of another plant of the same species.
• To facilitate this transfer over a distance, plants rely on various pollinating agents, which are broadly classified into abiotic (non-living) and biotic (living) agents.
• Wind is a common abiotic agent of cross-pollination. Plants that use wind for pollination, such as grasses, corn, and many conifers, produce large quantities of light, non-sticky pollen. Hence, statement 4 is a correct agent.
• Water is another abiotic agent, primarily used by aquatic plants like Vallisneria and Hydrilla. In these cases, pollen grains are often long and needle-like to float and reach the stigma. Hence, statement 2 is a correct agent.
• Animals serve as the primary biotic agents. This includes insects (bees, butterflies), birds (sunbirds, hummingbirds), and mammals (bats). These animals are attracted to the flower's nectar, color, or scent. Hence, statement 3 is a correct agent.
• Light is an essential environmental factor for photosynthesis and photoperiodism (the physiological reaction of organisms to the length of night or day). While light influences flowering time, it does not act as a physical carrier to transport pollen grains from one plant to another. Hence, statement 1 is NOT an agent of cross-pollination.

Additional Information

• Anemophily (Wind Pollination):
• Flowers are generally small, inconspicuous, and lack nectar or scent.
• The pollen grains are light, dry, and smooth-walled so they can be easily carried by the wind.
• The stigma is often feathery or branched to catch the airborne pollen.
• Hydrophily (Water Pollination):
• This is relatively rare, occurring in only about 30 genera of monocotyledons.
• Pollen grains are protected from wetting by a mucilaginous covering.
• Zoophily (Animal Pollination):
• Entomophily refers specifically to pollination by insects. These flowers are usually brightly colored and fragrant.
• Ornithophily is pollination by birds, while Chiropterophily is pollination by bats.
• Pollinators receive a "reward" in the form of pollen or nectar.
• Self-Pollination vs. Cross-Pollination:
• Self-pollination (Autogamy) occurs within the same flower or between flowers of the same plant.
• Cross-pollination increases genetic diversity and leads to the production of stronger, more adaptable offspring.

Important Points

• Light's Role: It is important to distinguish that Light triggers the biological clock of a plant to start the reproductive phase, but the actual transport of genetic material (pollen) requires a physical medium like air, water, or a living creature.
• Evolutionary Adaptation: Many plants have evolved specific morphological features to favor cross-pollination over self-pollination, such as Dichogamy (different maturation times of anther and stigma) or Self-incompatibility.

The correct answer is Right-hand rule (curling fingers along current).

Key Points

• The Right-hand rule (curling fingers along current) is specifically used to find the direction of the magnetic field at the centre of a circular loop. According to this rule, if we curl the fingers of our right hand in the direction of the current flowing through the circular wire, the thumb will point in the direction of the magnetic field at the center. Hence, statement 3 is correct.
• When the current flows in an anti-clockwise direction, the magnetic field lines emerge outward from the center of the loop, representing a North Pole.
• When the current flows in a clockwise direction, the magnetic field lines point inward (into the plane), representing a South Pole.
• The magnetic field produced by a circular loop is strongest at its centre because the field contributions from every small segment of the loop add up vectorially in the same direction at that point.
• This rule is often considered a variation of Maxwell’s Right-Hand Grip Rule, adapted for curved geometries to simplify the visualization of electromagnetic effects.

Additional Information

• Fleming’s Left-Hand Rule:
• This rule is primarily used to determine the direction of the magnetic force (motion) acting on a conductor.
• It states that if the forefinger points in the direction of the magnetic field and the middle finger in the direction of the current, then the thumb represents the direction of the force or motion. Hence, statement 2 is incorrect for finding field direction.
• Right-Hand Palm Rule:
• This is a simplified method to find the direction of magnetic force.
• The fingers point in the direction of the magnetic field, the thumb points in the direction of the velocity of a positive charge (or current), and the palm faces the direction of the magnetic force. Hence, statement 1 is incorrect.
• Maxwell’s Corkscrew Rule:
• Also known as the Right-Hand Screw Rule, it states that if a right-handed corkscrew is rotated such that it advances in the direction of the current, the direction of rotation of the handle gives the direction of the magnetic field.
• This is typically used for straight conductors to visualize the concentric field lines.

Important Points

• Magnetic Field Strength: The magnitude of the magnetic field (B) at the centre of a circular loop of radius r carrying current I is given by the formula B = (μ₀I) / (2r), where μ₀ is the permeability of free space (approximately 4π × 10⁻⁷ T·m/A).
• Number of Turns: If the circular coil has n turns, the magnetic field produced is n times as large as that produced by a single turn, because the current in each turn has the same direction and the fields add up.
• Right-Hand Thumb Rule (for Straight Wires): Often confused with the loop rule, here the thumb points along the current in a straight wire, and the curled fingers show the circular magnetic field lines around it. Hence, statement 4 (referring to a "Left-hand" version) is scientifically incorrect as the left hand is not used for this convention.
• Biot-Savart Law: This is the fundamental mathematical law used to derive the magnetic field for any current configuration, including circular loops and solenoids.

The correct answer is Beryllium.

Key Points

• Beryllium (Be) is a chemical element with the atomic number 4. Its electronic configuration is 1s² 2s². Because it has two electrons in its outermost shell (valence shell), it easily loses these two electrons to achieve a stable electronic configuration similar to Helium, resulting in a valency of +2. Hence, option 1 is correct.
• Lithium (Li) is the third element in the periodic table with an atomic number of 3. Its electronic configuration is 1s² 2s¹. It has only one electron in its valence shell, which it loses to form a Li⁺ ion, giving it a valency of +1. Hence, option 2 is incorrect.
• Hydrogen (H) has the atomic number 1 and an electronic configuration of 1s¹. It requires one electron to complete its duplet or can lose its single electron. Therefore, it typically exhibits a valency of 1 (+1 or -1). Hence, option 3 is incorrect.
• Helium (He) has an atomic number of 2 and a configuration of 1s². Since its K-shell is already completely filled (attaining a stable duplet), it is chemically inert and does not usually participate in chemical bonding. Consequently, its valency is 0. Hence, option 4 is incorrect.
• Elements in Group 2 of the modern periodic table, such as Magnesium (Mg) and Calcium (Ca), share the same +2 valency as Beryllium because they all possess two valence electrons.

Additional Information

• Valency Definition:
• Valency is defined as the combining capacity of an atom of an element. It is determined by the number of valence electrons present in the outermost shell of the atom.
• For metals in Groups 1, 2, and 13, the valency is equal to the number of valence electrons. For non-metals, it is often calculated as 8 minus the number of valence electrons.
• Beryllium Characteristics:
• Location: It is located in Group 2 and Period 2 of the Modern Periodic Table.
• Features: It is a steel-gray, strong, lightweight, and brittle alkaline earth metal. It has a relatively high melting point of 1287°C.
• Uses: Beryllium is used as an alloying agent in beryllium copper, in the construction of aerospace components, and as a neutron moderator in nuclear reactors.
• Periodic Table Trends:
• Moving down a group, the valency remains the same because the number of valence electrons is constant.
• Moving across a period from left to right, the valency first increases (from 1 to 4) and then decreases (down to 0 for noble gases).
• Noble Gases:
• Elements like Neon (Ne), Argon (Ar), and Krypton (Kr) belong to Group 18. They have a completely filled valence shell (octet), making them stable with a valency of zero.

Important Points

• Octet Rule: Atoms tend to gain, lose, or share electrons to achieve a stable configuration of eight electrons (or two for the first shell) in their valence shell, known as the Octet Rule.
• Electropositive Nature: Since Beryllium loses electrons to form cations (Be²⁺), it is considered an electropositive element.
• Atomic Radius: Beryllium has a smaller atomic radius compared to other elements in its group (like Magnesium), which results in a higher ionization energy within Group 2.

The correct answer is It will emerge without any deviation.

Key Points

• The optical centre is a point on the principal axis of a lens such that any ray of light passing through it does not undergo any deviation or refraction.
• For a thin lens, when a light ray passes through the optical centre, the incident ray and the emergent ray are collinear. Hence, option 1 is correct.
• If a light ray passes through the principal focus of a convex lens, it will emerge parallel to the principal axis on the other side. Hence, option 4 is incorrect.
• A light ray that is parallel to the principal axis will pass through the principal focus on the other side after refraction in a convex lens. Hence, option 2 is incorrect.
• The property of no deviation at the optical centre applies to both convex (converging) and concave (diverging) lenses.
• In ray diagrams, the optical centre is usually represented by the letter O or C, and it serves as the reference point for measuring all distances like object distance (u) and image distance (v).

Additional Information

• Convex Lens:
• A convex lens is thicker at the centre than at the edges. It is known as a converging lens because it converges parallel rays of light at a single point called the focus.
• The focal length of a convex lens is considered positive according to the Cartesian sign convention.
• It is used in magnifying glasses, cameras, and to correct Hypermetropia (long-sightedness).
• Concave Lens:
• A concave lens is thinner at the centre than at the edges. It is known as a diverging lens because it diverges parallel rays of light.
• The focal length of a concave lens is always negative.
• It is commonly used in flashlights, peepholes in doors, and to correct Myopia (short-sightedness).
• Refraction:
• Refraction is the bending of light as it passes from one transparent medium to another of different optical density.
• The refractive index of a material determines how much the light will bend. When light enters a denser medium, it bends towards the normal.

Important Points

• Lens Formula: The relationship between the object distance (u), image distance (v), and focal length (f) of a lens is given by 1/v - 1/u = 1/f.
• Power of a Lens: The power (P) of a lens is the reciprocal of its focal length (f) in meters. The SI unit of power is Dioptre (D).
• Lateral Displacement: While a ray through the optical centre of a thin lens has zero deviation, in a thick lens, there might be a very slight lateral displacement, although the emergent ray remains parallel to the incident ray.
• Sign Convention: According to the New Cartesian Sign Convention, the optical centre is the origin, and the direction of incident light is taken as positive.

The correct answer is Covalent bond.

Key Points

• A Covalent bond is a chemical bond that involves the sharing of electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs.
• Organic compounds are primarily defined by the presence of Carbon atoms. Carbon has four valence electrons, which allows it to form four covalent bonds with other atoms, a property known as tetravalency.
• The vast majority of organic molecules, including hydrocarbons, proteins, lipids, and carbohydrates, are held together by covalent bonds because carbon-carbon and carbon-hydrogen bonds are highly stable and strong. Hence, the covalent bond is the fundamental bond type in organic chemistry.
• Carbon also possesses the unique ability of catenation, which is the linkage of atoms of the same element into long chains or rings. This leads to the formation of millions of different organic compounds.
• Covalent bonds usually occur between non-metals. Since organic chemistry focuses on carbon, hydrogen, oxygen, and nitrogen (all non-metals), covalent bonding is the dominant force. Hence, statement 1 is correct.

Additional Information

• Ionic Bond:
• This bond is formed by the complete transfer of electrons from one atom to another, typically between a metal and a non-metal.
• It results in the formation of cations (positive) and anions (negative) that are held together by electrostatic forces.
• Examples include Sodium Chloride (NaCl) and Magnesium Oxide (MgO). These are inorganic in nature and typically form crystalline solids with high melting points.
• Metallic Bond:
• This bond occurs between atoms of metallic elements like Copper (Cu), Iron (Fe), or Gold (Au).
• It is characterized by a "sea of delocalized electrons" that move freely around a lattice of positive metal ions.
• This structure provides metals with their characteristic conductivity, malleability, and ductility.
• Dipolar Bond (Dipole-Dipole Interaction):
• This is a type of intermolecular force rather than a primary chemical bond within a molecule.
• It occurs between polar molecules that have permanent dipole moments, where the positive end of one molecule is attracted to the negative end of another.
• Hydrogen bonding is a specific, strong type of dipolar interaction often found in water (H2O) and DNA structures.
• Comparison of Bond Strength:
• Generally, Covalent and Ionic bonds are considered primary (strong) bonds, with bond energies typically ranging from 100 to 1000 kJ/mol.
• Dipolar interactions are much weaker secondary forces, usually around 5 to 25 kJ/mol, but they are crucial for determining the boiling points and solubility of organic compounds.

Important Points

• The number of known organic compounds exceeds 10 million, far surpassing the number of inorganic compounds, primarily due to the versatility of covalent bonding in carbon.
• In organic molecules, covalent bonds can be single, double, or triple bonds, depending on how many pairs of electrons are shared. For example, Ethane has a single bond, Ethene has a double bond, and Ethyne has a triple bond.
• While most organic compounds are covalent, some organic molecules can have ionic characters, such as Soap (Sodium Stearate), which contains a long covalent hydrocarbon chain and an ionic head.

The correct answer is Because the bullet exerts an equal and opposite force on the gun.

Key Points

• The phenomenon of a gun moving backward when a bullet is fired is known as recoil. This is a direct application of Newton’s Third Law of Motion, which states that for every action, there is an equal and opposite reaction.
• When the gunpowder in a cartridge explodes, it creates expanding gases that exert a huge force on the bullet, pushing it forward through the barrel. Hence, statement 2 is correct.
• Simultaneously, according to the law of action and reaction, the bullet exerts an exactly equal amount of force back onto the gun in the opposite direction.
• The forward motion of the bullet is the action, and the backward motion of the gun is the reaction.
• While the forces acting on the gun and the bullet are equal in magnitude, their resulting accelerations differ significantly because of the vast difference in their masses.

Additional Information

• Law of Conservation of Momentum:
• The recoil of a gun can also be explained using the Law of Conservation of Momentum. This law states that the total momentum of an isolated system remains constant if no external force acts on it.
• Before firing, the initial momentum of both the gun and the bullet is zero because they are at rest.
• After firing, the final momentum must also be zero. Therefore, the positive momentum of the bullet (moving forward) must be balanced by the negative momentum of the gun (moving backward).
• Mathematically: m × v + M × V = 0, where m and v are the mass and velocity of the bullet, and M and V are the mass and velocity of the gun.
• Recoil Velocity:
• The recoil velocity (V) of the gun is given by V = -(m × v) / M.
• Because the mass of the gun (M) is much larger than the mass of the bullet (m), the recoil velocity is much smaller than the bullet's velocity.
• This explains why the gun moves back with a jerk rather than flying back at the same speed as the bullet.
• Newton's Second Law in Context:
• According to Newton's Second Law (F = ma), for a constant force, acceleration is inversely proportional to mass.
• The bullet, having a very small mass, experiences a tremendous acceleration. The gun, having a large mass, experiences a small acceleration.

Important Points

• Analysis of Incorrect Options:
• Option 1: The bullet's acceleration is actually much greater than the gun’s, not less, due to its smaller mass. Hence, statement 1 is incorrect.
• Option 3: In reality, a gun's mass is always larger than the bullet's mass to minimize the recoil velocity and ensure the safety of the user. Hence, statement 3 is incorrect.
• Option 4: According to Newton’s Third Law, the force pushed by the gun on the bullet and the force pushed by the bullet on the gun are exactly equal in magnitude. Hence, statement 4 is incorrect.
• Practical Applications:
• Soldiers are trained to hold a gun tightly against their shoulder. This effectively increases the mass (M) by adding the shooter's body mass to the gun's mass, thereby reducing the recoil velocity and preventing injury.
• Muzzle brakes or recoil pads are often used in modern firearms to absorb or redirect some of the kinetic energy generated during the recoil.

The correct answer is Bee-keeping.

Key Points

• Bee-keeping, also known as Apiculture, is the practice of maintaining honey bee colonies in hives for the production of honey and beeswax.
• It is an important agricultural enterprise that requires low investment and provides high returns, making it a popular subsidiary occupation for farmers.
• Honey is a natural sweet substance produced by bees from the nectar of plants; it is widely used in Ayurveda and as a nutrient-rich food supplement.
• The process also yields beeswax, which is used in several industries, including pharmaceuticals, cosmetics, and the manufacture of polishes and candles.
• Commercial honey production often utilizes the Italian bee (Apis mellifera) because of its high honey collection capacity and its ability to stay in a given beehive for long periods.
• Hence, the correct answer is Bee-keeping.

Additional Information

• Culture Fishery:
• This refers to the rearing and breeding of fish in controlled water bodies such as ponds, tanks, and reservoirs.
• It includes Aquaculture (freshwater) and Mariculture (marine water) to increase the production of protein-rich aquatic food.
• Poultry:
• This branch of animal husbandry involves raising domesticated birds like chickens, ducks, and turkeys for their eggs and meat.
• Common breeds used in India include the indigenous Aseel and exotic Leghorn.
• Cattle Farming:
• This is the management of cows (Bos indicus) and buffaloes (Bos bubalis) for two main purposes: milk production and agricultural labor.
• Animals kept for milk are called milch animals, while those used for tilling and irrigation are called draught animals.
• Pasturage:
• The quality and taste of honey depend significantly on the pasturage, which refers to the flowers available to the bees for nectar and pollen collection.

Important Points

• Bee Species in India:
• Apis cerana indica: Commonly known as the Indian bee, used for local honey production.
• Apis dorsata: Known as the Rock bee, it is a large bee that produces significant amounts of honey but is difficult to domesticate.
• Apis florea: Known as the Little bee, it builds small hives in the open.
• Colony Hierarchy:
• A honey bee colony consists of three types of bees: the Queen bee (responsible for laying eggs), Drones (males for mating), and Worker bees (sterile females that perform all hive tasks).
• Apiary:
• An apiary is a location where beehives are kept. Placing apiaries near orchards or crop fields is beneficial as bees act as excellent pollinators, increasing crop yield.

The correct answer is Your weight acting downward on the chair.

Key Points

• This phenomenon is a direct application of Newton's Third Law of Motion, which states that for every action, there is always an equal and opposite reaction.
• When you sit on a chair, your body exerts a downward force on the chair. This force is primarily due to your weight, which is the gravitational pull of the Earth acting on your mass.
• The chair, in turn, exerts an upward force on you to support your weight. This upward force is known as the Normal Force.
• In the context of action-reaction pairs, the two forces must act on different objects. Here, the action is your body pushing down on the chair, and the reaction is the chair pushing up on your body. Hence, the reaction to the chair's upward force is your weight acting downward on it.
• Both forces are equal in magnitude but opposite in direction. If the chair could not exert an equal upward force, it would break or you would accelerate downwards. Hence, the statement regarding weight as the reaction force is correct.

Additional Information

• Newton's First Law (Inertia):
• It states that an object will remain at rest or move at a constant velocity unless acted upon by an external unbalanced force. When sitting still on a chair, the net force on you is zero because the forces are balanced.
• Newton's Second Law (F=ma):
• This law defines force as the product of mass and acceleration. Since you are not moving (acceleration is 0) while sitting, the sum of the upward normal force and downward gravitational force equals zero.
• Contact Forces:
• The interaction between your clothes and the chair involves friction, but this is a lateral (horizontal) force that prevents sliding and is not the reaction to the vertical upward force.
• The force of air on the chair (atmospheric pressure) exists but is negligible in this specific mechanical interaction between the person and the furniture.
• Gravity vs. Weight:
• The gravitational pull of the Earth is a non-contact force acting on your body. Its reaction force is actually the gravitational pull your body exerts back on the Earth, not the force between you and the chair.

Important Points

• Action-Reaction Pair Characteristics:
• They always occur in pairs and act simultaneously.
• They never act on the same object; one force acts on Object A (the chair) and the other acts on Object B (the person).
• The magnitude of the Normal Force is exactly equal to the Weight (mg) of the person only when the surface is horizontal and there are no other vertical forces acting.
• In physics, Weight is measured in Newtons (N), just like any other force, whereas Mass is measured in Kilograms (kg).

The correct answer is It is a universal constant, same everywhere.

Key Points

• The Universal Gravitational Constant, denoted by the symbol G, is a fundamental physical constant that appears in Sir Isaac Newton's Law of Universal Gravitation. It defines the strength of the gravitational attraction between two objects.
• Statement 1: It is equal to the acceleration due to gravity. This is not true because G is a constant of proportionality, whereas g (acceleration due to gravity) is the acceleration an object undergoes due to the gravitational pull of a massive body like Earth. Hence, statement 1 is incorrect.
• Statement 2: It is a universal constant, same everywhere. This is true because the value of G does not change regardless of the location in the universe, the time of measurement, or the nature of the bodies involved. Hence, statement 2 is correct.
• Statement 3: It varies from place to place on Earth. While the value of g (approximately 9.8 m/s²) varies based on latitude, altitude, and depth, the value of G remains absolutely constant at every point on Earth and in space. Hence, statement 3 is incorrect.
• Statement 4: It depends on the medium between two masses. Gravitational force is unique compared to electromagnetic forces because it is independent of the intervening medium. Whether two masses are in a vacuum, water, or solid rock, the constant G and the resulting force remain the same. Hence, statement 4 is incorrect.
• Hence, the correct answer is It is a universal constant, same everywhere.

Additional Information

• Newton's Law of Universal Gravitation:
• The law states that every point mass attracts every other point mass by a force acting along the line intersecting both points.
• The mathematical formula is F = G(m1m2 / r²), where F is the force, m1 and m2 are the masses, and r is the distance between their centers.
• Numerical Value and Units:
• The accepted value of G is approximately 6.674 x 10^-11 N m² kg&sup-2.
• The SI units for G are Newton meter squared per kilogram squared.
• In base SI units, it is expressed as m³ kg&sup-1 s&sup-2.
• Dimensions of G:
• The dimensional formula for the universal gravitational constant is [M^-1 L^3 T^-2].
• Historical Context:
• Although Newton formulated the law in 1687, the first accurate measurement of G was performed by Henry Cavendish in 1798 using a torsion balance experiment.
• This experiment was famously known as "weighing the Earth" because knowing G allowed for the calculation of the Earth's mass.

Important Points

• Difference between G and g:
• G is a scalar quantity, whereas g is a vector quantity (it has a direction toward the center of the mass).
• G is Universal, while g is local (it is different on the Moon, Mars, or at the Equator vs. the Poles).
• The value of g is derived using G through the formula: g = GM / R², where M is the mass of the planet and R is its radius.
• Weakest Force:
• Gravity is the weakest of the four fundamental forces of nature (the others being Electromagnetism, Strong Nuclear, and Weak Nuclear forces).
• Because G is so small (10^-11), gravitational attraction is only significant when at least one of the objects has a massive scale, such as a planet or star.

The correct answer is Both Statement A and Statement B are correct.

Key Points

• Yeast is a unicellular organism belonging to the Kingdom Fungi that reproduces asexually through a process known as budding.
• In this process, a small bulb-like projection or outgrowth appears on the parent cell body, which is scientifically termed a bud. Hence, statement A is correct.
• As the process progresses, the nucleus of the parent yeast cell divides, and one of the daughter nuclei migrates into the developing bud.
• Under favorable conditions and rapid growth, the bud may not immediately detach from the parent cell; instead, it starts producing its own bud, leading to the formation of a chain of buds. Hence, statement B is correct.
• Eventually, these buds detach from the parent cell and mature into new, independent yeast cells, allowing for rapid population growth in a short time.

Additional Information

• Characteristics of Yeast:
• Yeast, such as Saccharomyces cerevisiae, is widely used in the bakery and brewery industries due to its ability to perform fermentation.
• It converts sugar into alcohol and carbon dioxide, which is the fact behind bread rising and the production of ethanol.
• Types of Asexual Reproduction:
• Binary Fission: Observed in organisms like Amoeba, where a single cell divides into two equal daughter cells.
• Fragmentation: Seen in Spirogyra (an alga), where the body breaks into smaller pieces, and each piece grows into a new individual.
• Spore Formation: Common in fungi like Rhizopus (bread mould), where spores are released into the air and germinate under favorable conditions of moisture and temperature.
• Vegetative Propagation:
• This is a form of asexual reproduction in plants where new plants are produced from roots, stems, leaves, and buds (e.g., Potato tubers, Bryophyllum leaves).

Important Points

• The budding process in Yeast is distinct because the division of cytoplasm is unequal, resulting in a smaller bud and a larger parent cell, unlike binary fission where the division is roughly equal.
• The formation of a chain of buds is a characteristic feature of yeast when nutrients are abundant, specifically glucose.
• In 2024, yeast remains a primary model organism in biological research for studying cell cycle and genetics due to its simple unicellular structure and eukaryotic nature.

The correct answer is A reaction where an insoluble solid is formed.

Key Points

• A precipitation reaction is a chemical reaction that occurs in an aqueous solution when two ionic bonds combine, resulting in the formation of an insoluble salt.
• The insoluble solid that emerges from the liquid solution is called the precipitate. This occurs when the concentration of dissolved ions exceeds the solubility product of the resulting compound. Hence, statement 4 is correct.
• These reactions typically involve the exchange of cations and anions between two soluble salts in a liquid environment, often water.
• A common example is the reaction between Silver Nitrate (AgNO3) and Sodium Chloride (NaCl). When mixed, they produce a white precipitate of Silver Chloride (AgCl) and soluble Sodium Nitrate.
• The general chemical representation for a precipitation reaction is A(aq) + B(aq) → C(s) + D(aq), where (s) denotes the solid state of the precipitate formed.

Additional Information

• Color Change Reactions: A reaction involving a color change only (Option 1) is often a characteristic of Redox reactions or Acid-Base reactions using indicators. While a precipitate might have a color, color change alone does not define the formation of a solid. Hence, statement 1 is incorrect.
• Gas Evolution: A reaction in which a gas is formed (Option 2) is known as a gas evolution reaction. An example is the reaction of Carbonates with Acids to release Carbon Dioxide (CO2). This is a distinct category from precipitation. Hence, statement 2 is incorrect.
• Exothermic Reactions: A reaction where heat is released (Option 3) is classified as an exothermic reaction. While many chemical processes release energy, heat release is not the defining criterion for a precipitation reaction. Hence, statement 3 is incorrect.
• Double Displacement: Most precipitation reactions are double displacement reactions (also known as metathesis), where the reactants "swap" partners to create a new insoluble product.
• Solubility Rules: Chemists use solubility rules to predict whether a precipitate will form. For instance, salts containing Alkali Metals or Ammonium ions are generally soluble, whereas many Sulfates and Chlorides of heavy metals are insoluble.

Important Points

• Industrial Uses: Precipitation is widely used in wastewater treatment to remove heavy metals like Lead and Mercury by turning them into insoluble solids that can be filtered out.
• Analytical Chemistry: These reactions are essential in qualitative inorganic analysis to detect the presence of specific ions in a sample based on the color and solubility of the precipitate formed.
• Biological Significance: In the human body, the formation of kidney stones is essentially a precipitation reaction where Calcium Oxalate becomes insoluble in the urinary tract.
• Pigment Production: Many industrial pigments used in paints and dyes are manufactured through controlled precipitation reactions to ensure uniform particle size and color intensity.

The correct answer is Endoplasmic reticulum.

Key Points

• The Endoplasmic Reticulum (ER) is a complex network of tiny tubular structures that are scattered throughout the cytoplasm of eukaryotic cells.
• Electron microscopic studies reveal that the ER divides the intracellular space into two distinct compartments: the luminal (inside the ER) and the extra-luminal (cytoplasm) compartments.
• This organelle was first observed by Keith Porter, Albert Claude, and Ernest Fullam in the year 1945.
• The ER is often continuous with the outer membrane of the nucleus and extends throughout the cell, providing a large surface area for various chemical reactions.
• It plays a vital role as a cytoskeletal framework and acts as a channel for the transport of materials between different parts of the cytoplasm or between the nucleus and the cytoplasm.

Additional Information

• Golgi apparatus:
• First described by Camillo Golgi in 1898, it consists of flat, disc-shaped sacs called cisternae of 0.5μm to 1.0μm diameter.
• Its primary function is the packaging and processing of proteins and lipids, especially for secretion.
• It is the site for the formation of glycoproteins and glycolipids.
• Lysosome:
• These are membrane-bound vesicular structures formed by the process of packaging in the Golgi apparatus.
• They are rich in almost all types of hydrolytic enzymes (hydrolases – lipases, proteases, carbohydrases) which are active at an acidic pH.
• They are often called suicidal bags because they can digest the entire cell if it becomes damaged or dead.
• Mitochondria:
• Known as the powerhouse of the cell, these double-membrane-bound organelles are responsible for aerobic respiration.
• They produce cellular energy in the form of ATP (Adenosine Triphosphate).
• Mitochondria possess their own DNA and 70S ribosomes, making them semi-autonomous organelles.

Important Points

• Types of Endoplasmic Reticulum:
• Rough Endoplasmic Reticulum (RER): It bears ribosomes on its surface. It is actively involved in protein synthesis and secretion. It is extensive and continuous with the outer membrane of the nucleus.
• Smooth Endoplasmic Reticulum (SER): It lacks ribosomes. It is the major site for the synthesis of lipids and steroid hormones. In animal cells, lipid-like steroid hormones are synthesized in SER.
• Detoxification: In the liver cells of vertebrates, the SER plays a crucial role in detoxifying many poisons and drugs.
• Sarcoplasmic Reticulum: A specialized type of smooth ER found in muscle cells that regulates calcium ion concentration, which is essential for muscle contraction.

The correct answer is To make hard water soft.

Key Points

• Washing soda, chemically known as sodium carbonate, is primarily added to water during the laundry process to make hard water soft. Hard water contains high concentrations of calcium (Ca2+) and magnesium (Mg2+) ions, which interfere with the cleaning action of soap.
• When sodium carbonate is added to hard water, it reacts with the dissolved calcium and magnesium salts. This chemical reaction leads to the formation of insoluble carbonates (precipitates), effectively removing the mineral ions from the water.
• The removal of these minerals prevents the formation of scum, a sticky precipitate that occurs when soap reacts with hard water minerals instead of dissolving to create lather.
• By softening the water, washing soda allows the soap or detergent to work more efficiently, ensuring that the clothes are cleaned thoroughly and reducing the amount of detergent needed. Hence, the correct option is "To make hard water soft."

Additional Information

• Hard Water vs. Soft Water:
• Hard water is characterized by a high mineral content, specifically bicarbonates, chlorides, and sulfates of calcium and magnesium.
• Soft water has low concentrations of these ions, allowing for easy lathering with soap. Rainwater is a natural example of soft water.
• Types of Hardness:
• Temporary Hardness: This is caused by the presence of magnesium and calcium hydrogen carbonates. It can be removed simply by boiling or by using Clark’s method (adding lime).
• Permanent Hardness: This is caused by chlorides and sulfates of magnesium and calcium. It cannot be removed by boiling. Methods like the washing soda process, Calgon’s process, and ion-exchange are required to treat it.
• Solvay Process: This is the industrial method used for the production of sodium carbonate. The process involves sodium chloride, ammonia, and carbon dioxide as raw materials.
• Chemical Nature: Sodium carbonate is a basic salt. In its hydrated form, it is known as sodium carbonate decahydrate with the chemical formula Na2CO3.10H2O. When heated, it loses its water of crystallization to become anhydrous soda ash.

Important Points

• Uses of Washing Soda: Beyond laundry, it is used in the glass, soap, and paper industries. It is also used in the manufacture of sodium compounds like borax.
• Cleaning Agent: It serves as a domestic cleaning agent for floors and surfaces due to its ability to cut through grease and oil.
• pH Regulation: In some industrial applications, it is used as a pH regulator or buffer to maintain alkalinity in solutions.
• Difference from Baking Soda: While washing soda is sodium carbonate (Na2CO3), baking soda is sodium bicarbonate (NaHCO3). Washing soda is much more alkaline and is not used in food preparation.

The correct answer is smaller than the smallest resistance.

Key Points

• In a parallel combination of resistors, the reciprocal of the equivalent resistance (Rp) is equal to the sum of the reciprocals of the individual resistances. The formula is expressed as 1/Rp = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn.
• Because the total reciprocal resistance is the sum of several positive values, the resulting equivalent resistance will always be smaller than the smallest resistance present in the circuit. Hence, option 3 is correct.
• When resistors are connected in parallel, the potential difference (voltage) across each resistor remains the same and is equal to the voltage of the source.
• The total current (I) in the circuit is the sum of the currents flowing through each individual branch, represented as I = I1 + I2 + I3 + ... + In.
• Adding more resistors in parallel effectively increases the number of paths for the electric current to flow, which decreases the total resistance of the circuit.

Additional Information

• Series Combination:
• In a series circuit, resistors are connected end-to-end. The equivalent resistance (Rs) is the sum of all resistances (Rs = R1 + R2 + ...).
• In this configuration, the equivalent resistance is always greater than the largest resistance in the group. This corresponds to options 1 and 2, which are true for series but not parallel combinations.
• The current remains constant through all components in a series circuit, while the voltage is divided among them.
• Ohm's Law:
• Formulated by Georg Simon Ohm in 1827, it states that the current through a conductor is directly proportional to the voltage across it, provided physical conditions remain constant (V = IR).
• The unit of resistance is the Ohm, symbolized by the Greek letter Ω (Omega).
• Applications of Parallel Circuits:
• Household wiring is almost always done in parallel. This ensures that every appliance receives the standard 220V (in India) and continues to function even if one appliance fails or is turned off.
• It allows for independent switching of different electrical devices, which would not be possible in a series arrangement.
• Conductance:
• Conductance (G) is the reciprocal of resistance (G = 1/R). In a parallel circuit, the total conductance is simply the sum of individual conductances (Gp = G1 + G2 + ...).
• The SI unit of conductance is the Siemens (S), formerly known as the mho.

Important Points

• If 'n' identical resistors each of resistance 'R' are connected in parallel, the equivalent resistance becomes R/n.
• If those same 'n' identical resistors are connected in series, the equivalent resistance becomes nR.
• A short circuit occurs when a path of zero resistance is connected in parallel with a component, causing all current to bypass that component and flow through the path of least resistance.

The correct answer is Cells are tightly arranged..

Key Points

• Connective tissues are the most abundant and widely distributed tissues in the bodies of complex animals, named for their special function of linking and supporting other tissues and organs of the body.
• A primary characteristic of connective tissue is that the cells are loosely spaced and embedded in an intercellular matrix. Unlike epithelial tissue, which forms continuous sheets, the cells in connective tissue do not touch each other and are separated by a large amount of extracellular material. Hence, statement 4 is incorrect.
• The intercellular matrix is a non-living material that can vary significantly in its physical properties depending on the specific type of tissue. It may be jelly-like, fluid, dense, or rigid. Hence, statement 2 is correct.
• In all connective tissues except blood, the cells secrete fibers of structural proteins called collagen or elastin, which provide strength, elasticity, and flexibility to the tissue. These cells are essentially embedded within the matrix they help create. Hence, statement 1 is correct.
• Blood is a unique type of fluid connective tissue. It contains blood cells (RBCs, WBCs, and platelets) suspended in a fluid matrix called plasma. It serves as the main circulating fluid that helps in the transport of various substances. Hence, statement 3 is correct.
• Hence, the incorrect statement (and correct answer) is that cells are tightly arranged.

Additional Information

• Classification of Connective Tissue:
• Loose Connective Tissue: Cells and fibers are loosely arranged in a semi-fluid ground substance. Examples include Areolar tissue (found beneath the skin) and Adipose tissue (specialized for fat storage).
• Dense Connective Tissue: Fibers and fibroblasts are compactly packed. This includes Dense Regular (like tendons and ligaments) and Dense Irregular tissues.
• Specialized Connective Tissue: This category includes Cartilage, Bones, and Blood.
• Functions and Features:
• Bone: Has a hard and non-pliable ground substance rich in calcium salts and collagen fibers which give bone its strength. It provides the structural frame for the body.
• Cartilage: The intercellular material is solid and pliable and resists compression. It is found in the tip of the nose, outer ear joints, and between adjacent bones of the vertebral column.
• Adipose Tissue: Located mainly beneath the skin, the cells of this tissue are specialized to store fats. The excess of nutrients which are not utilized immediately are converted into fats and stored here.

Important Points

• Fibroblasts: These are the most common cells in connective tissue proper that produce and secrete the extracellular matrix and fibers like collagen.
• Tendons and Ligaments: Tendons are dense regular connective tissues that attach skeletal muscles to bones, while ligaments attach one bone to another.
• Cellularity vs. Matrix: One of the easiest ways to distinguish between tissue types is the ratio of cells to matrix. Epithelial tissues have high cellularity and minimal matrix, whereas Connective tissues have low cellularity and an extensive matrix.
• Vascularity: Most connective tissues are highly vascular (rich blood supply), with the notable exception of cartilage, which is avascular and heals slowly.

The correct answer is CaSO4 . 1/2H2O.

Key Points

• Plaster of Paris is a white powdery chemical compound officially known as Calcium Sulfate Hemihydrate.
• The chemical formula for Plaster of Paris is CaSO4 . 1/2H2O, which means that half a molecule of water is attached to one molecule of calcium sulfate.
• It is produced by heating Gypsum (CaSO4 . 2H2O) at a temperature of 373 K (or 100 degrees Celsius). During this process, gypsum loses three-fourths of its water of crystallization.
• The name "Plaster of Paris" originates from the fact that gypsum was found in abundance near Paris, particularly at Montmartre, and was first used there on a large scale for construction and plastering.
• A notable property of this substance is its ability to turn back into a hard solid mass of gypsum upon being mixed with water, a process that happens within a few minutes.

Additional Information

• Gypsum (CaSO4 . 2H2O):
• It is a soft sulfate mineral composed of calcium sulfate dihydrate.
• When gypsum is heated above 400 K, it loses all its water of crystallization to form Anhydrous Calcium Sulfate (CaSO4), which is also known as Dead Burnt Plaster because it does not set like Plaster of Paris when mixed with water.
• Common Applications:
• Medical Use: Doctors use Plaster of Paris as orthopedic casts to immobilize fractured bones, ensuring they remain in the correct position during the healing process.
• Art and Decor: It is widely used for making statues, toys, sculptures, and decorative items for interior design.
• Construction: Used for creating fireproof coatings and smooth finishes on ceilings and walls in buildings.
• Calcium Carbonate (CaCO3):
• This is a chemical compound found commonly in rocks such as limestone, chalk, and marble.
• It is a key ingredient in agricultural lime and is used as a calcium supplement or an antacid in medicine.
• Storage Conditions:
• Plaster of Paris must be stored in moisture-proof containers because even a small amount of atmospheric moisture can cause it to undergo a hydration reaction and turn into a hard mass of gypsum, making it useless for future applications.

The correct answer is Beyond the centre of curvature.

Key Points

• A concave mirror forms a real, inverted, and diminished image when the object is placed beyond the centre of curvature (C).
• In this case, the image is formed:
• Between the focus (F) and the centre of curvature (C).
• Real
• Inverted
• Diminished (smaller than the object)
• If the object is placed between the focus (F) and the pole (P), the image formed is virtual, erect, and magnified.
• If the object is placed at the focus (F), the reflected rays become parallel and the image is formed at infinity.
• Hence, the correct answer is Option 1 – Beyond the centre of curvature.

Additional Information

The correct answer is Suspension is a heterogeneous mixture.

Key Points

• A suspension is defined as a heterogeneous mixture where the solute particles do not dissolve in the solvent but instead remain suspended throughout the bulk of the medium. Hence, statement 1 is correct.
• The particles of a suspension are quite large, typically with a diameter greater than 1000 nm (or 10^-6 meters), which makes them easily visible to the naked eye. Hence, statement 3 is incorrect.
• Suspensions are inherently unstable. If a suspension is allowed to stand undisturbed, the heavy solute particles settle down at the bottom of the container due to the force of gravity. Hence, statement 2 is incorrect.
• Due to the significant size of the particles, they can be separated from the liquid or gas medium using simple physical methods like filtration. The particles are trapped by the filter paper while the liquid passes through. Hence, statement 4 is incorrect.
• When particles are suspended and have not yet settled, they are capable of scattering a beam of light passing through the mixture, making the path of light visible; this phenomenon is known as the Tyndall effect.

Additional Information

• Types of Mixtures based on Particle Size:
• True Solution: This is a homogeneous mixture where the particle size is less than 1 nm. These particles do not scatter light and cannot be seen even with a microscope. Examples include salt in water and sugar in water.
• Colloidal Solution: These appear homogeneous but are actually heterogeneous. The particle size ranges between 1 nm and 1000 nm. They show the Tyndall effect and do not settle over time. Examples include milk, fog, and blood.
• Suspension: As discussed, these are heterogeneous with particles larger than 1000 nm. Common examples include muddy water, chalk powder in water, and dust particles in the air.
• Filtration Process:
• In a suspension, the solid phase is dispersed in the liquid phase. Because the pores of standard filter paper are smaller than the suspended particles, filtration is an effective separation technique.
• Once the particles settle down, the suspension effectively ceases to exist as the mixture is no longer uniform, and it loses its ability to scatter light.

Important Points

• Stability: The stability of a mixture is a key identifying feature. Solutions and colloids are considered stable because their particles remain dispersed indefinitely. Suspensions are unstable because the components eventually separate into distinct layers.
• Optical Properties: While suspensions show the Tyndall effect when the particles are floating, once the particles precipitate or settle at the bottom, the remaining liquid above typically stops scattering light.
• Naked Eye Visibility: The threshold for naked eye visibility is generally around 0.1 millimeters. Since suspension particles are often larger than this or clump together, they are easily distinguishable without magnification, unlike the components of homogeneous mixtures.

The correct answer is The body must move at a constant speed..

Key Points

• In physics, work is said to be done when a force acts upon an object to cause a displacement. The mathematical definition is W = Fs cos(θ), where F is the force, s is the displacement, and θ is the angle between the force and the direction of motion.
• A force must be applied on the body: This is a fundamental requirement. If the net force acting on the body is zero, no work can be performed according to the formula. Hence, statement 4 is a necessary condition.
• The body must get displaced: Even if a massive force is applied, work is zero if there is no displacement (s = 0). For example, pushing against a stationary brick wall results in no work done on the wall. Hence, statement 2 is a necessary condition.
• The force must have a component along the direction of displacement: If the force is applied perpendicularly to the direction of motion (90 degrees), the value of cos(90°) becomes zero, resulting in no work. A component of the force must act in the same or opposite direction of the displacement. Hence, statement 1 is a necessary condition.
• The body must move at a constant speed: This is not necessary for work to be done. In fact, work is often done to accelerate an object, thereby changing its speed. Work can be done whether the speed is increasing, decreasing, or constant (provided other forces are balanced). Hence, statement 3 is NOT a necessary condition and is the correct answer.

Additional Information

• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy. If work is done on an object, its velocity (and thus its speed) typically changes, unless multiple forces cancel each other out.
• Nature of Work:
• Positive Work: Occurs when the force and displacement are in the same direction (angle between 0° and 90°).
• Negative Work: Occurs when the force acts in the opposite direction to the displacement (angle between 90° and 180°), such as the force of friction acting on a moving car.
• Zero Work: Occurs when the force is perpendicular to the displacement or when either force or displacement is zero.
• SI Unit: The standard unit of work is the Joule (J). One Joule is defined as the work done by a force of one Newton causing a displacement of one meter.
• Scalar Quantity: Work is a scalar quantity, meaning it has magnitude but no specific direction, even though it is calculated using two vector quantities (force and displacement).

Important Points

• Circular Motion: In uniform circular motion, the centripetal force acts toward the center while the displacement is tangential. Since they are perpendicular, the work done by the centripetal force is always zero.
• Gravity and Work: When a person walks horizontally carrying a load on their head, the work done by gravity on the load is zero because the gravitational force (downward) is perpendicular to the horizontal displacement.
• Power: It is important to distinguish work from power. Power is the rate at which work is done, measured in Watts (W), representing Joules per second.

The correct answer is Serving food in banana leaves or reusable plates.

Key Points

• A 'Zero Waste' initiative is a philosophy and a design principle for the 21st Century that includes the "no-waste" approach, aiming to guide people in changing their lifestyles and practices to emulate sustainable natural cycles.
• Serving food in banana leaves or reusable plates is the most effective practice because it eliminates the generation of trash. Banana leaves are completely biodegradable and compostable, while reusable plates (made of stainless steel, ceramic, or glass) can be washed and used thousands of times, preventing material from entering the waste stream. Hence, this statement is correct.
• Using single-use plastic cutlery involves items like spoons and forks that are discarded after one use. These are typically made from polystyrene or polypropylene, which take hundreds of years to decompose in landfills and often end up polluting oceans. Hence, this statement is incorrect.
• Disposing food waste in regular trash bins prevents organic matter from being composted. When organic waste is sent to landfills, it undergoes anaerobic decomposition and produces methane, a potent greenhouse gas that contributes to climate change. Hence, this statement is incorrect.
• Wrapping snacks in plastic foil or aluminum foil creates immediate waste once the snack is consumed. Plastic wraps are difficult to recycle due to their thinness and chemical composition, frequently leading to microplastic contamination in the environment. Hence, this statement is incorrect.

Additional Information

• The 5 R's of Waste Management: To achieve zero waste, individuals and institutions follow the hierarchy of Refuse (saying no to single-use items), Reduce (consuming less), Reuse (using items repeatedly), Recycle (processing materials to make new products), and Rot (composting organic matter).
• Single-Use Plastic Ban in India: The Government of India implemented a nationwide ban on identified Single-Use Plastic (SUP) items with effect from July 1, 2022. This includes items with low utility and high littering potential, such as plastic sticks for balloons, plastic flags, and thermocol for decoration.
• Solid Waste Management Rules, 2016: These rules in India emphasize source segregation of waste into three streams: Biodegradable (wet), Non-biodegradable (dry), and Domestic Hazardous waste. Schools and bulk generators are mandated to process biodegradable waste through composting or biomethanation within their premises.
• Circular Economy: A Zero Waste approach is a key component of a Circular Economy, which moves away from the traditional linear economy (take-make-dispose) model. It focuses on keeping resources in use for as long as possible to extract the maximum value.

Important Points

• Environmental Impact of Plastics: Plastics are petroleum-based synthetic polymers. Microplastics (particles less than 5mm) have been found in drinking water, food chains, and even human blood, posing significant health risks.
• Banana Leaves Benefits: Beyond being eco-friendly, banana leaves contain polyphenols (natural antioxidants) like EGCG, which can be absorbed by the food served on them. They are widely used in South India, Philippines, and Thailand as a traditional, hygienic, and cost-effective serving medium.
• Waste Audits: For a school to successfully implement a 'Zero Waste' day, it often conducts a Waste Audit, which is a formal process used to quantify the amount and types of waste being generated, helping to identify areas for waste reduction.

The correct answer is Fragmentation.

Key Points

• Fragmentation is a type of asexual reproduction in which a multicellular organism breaks into two or more fragments upon maturation.
• In this process, each cut or broken body part develops into a fully functional, independent new organism.
• This method is typically seen in organisms with relatively simple body designs, such as Spirogyra (a filamentous green algae) and Planaria (flatworms). Hence, statement 4 is correct.
• In Spirogyra, when the filament reaches a certain length, it simply breaks into smaller pieces, and each piece continues to grow through cell division to form a new individual.
• While similar to regeneration, fragmentation is considered a deliberate reproductive strategy in these organisms, allowing for rapid population growth in favorable environments.

Additional Information

• Spore Formation:
• This is a method of reproduction used by many Fungi (like Rhizopus or bread mold), Ferns, and Mosses.
• Organisms produce microscopic, unicellular reproductive bodies called spores covered by a hard protective coat to withstand unfavorable conditions. Hence, statement 1 is incorrect.
• Budding:
• In this process, a small projection or bud grows out from the body of the parent organism due to repeated mitotic cell divisions.
• The bud matures and eventually detaches to live as an independent organism. Common examples include Yeast and Hydra. Hence, statement 2 is incorrect.
• Binary Fission:
• This is a form of asexual reproduction where a single-celled organism divides into two equal halves.
• It is commonly seen in Prokaryotes (Bacteria) and Protists like Amoeba, Paramecium, and Leishmania. Hence, statement 3 is incorrect.
• Asexual Reproduction:
• It involves only a single parent and does not require the fusion of gametes (sperm and egg).
• The resulting offspring are genetically identical to the parent and are often referred to as clones.
• Other types include Vegetative Propagation, which is used by plants like Potato, Ginger, and Bryophyllum to grow new plants from vegetative parts like roots, stems, or leaves.

Important Points

• Difference between Fragmentation and Regeneration: While Fragmentation is a reproductive method for simple organisms, Regeneration is the ability of an organism to regrow lost body parts (like a Lizard's tail or Starfish arms).
• In highly complex multicellular organisms, fragmentation is not possible because cells are organized into tissues, which are organized into organs, and finally organ systems, requiring specialized cells for reproduction.
• The process of fragmentation is often triggered by physical disturbances or natural maturation cycles in aquatic environments.

The correct answer is 0.004 s.

Key Points

• Frequency (f) is defined as the number of complete vibrations or cycles that occur in one second. It is measured in Hertz (Hz), where 1 Hz equals one vibration per second.
• The Time Period (T) is the time required for the wave to complete one full vibration or cycle.
• There is an inverse relationship between frequency and time period, which is expressed by the mathematical formula: T = 1 / f.
• In this specific problem, the frequency is given as 250 Hz. By applying the formula, the calculation becomes T = 1 / 250.
• Dividing 1 by 250 gives a numerical value of 0.004. Therefore, the time taken for one complete vibration is 0.004 s. Hence, the statement regarding the time period being 0.004 s is correct.

Additional Information

• Sound Waves:
• Sound is a mechanical wave that requires a medium (solid, liquid, or gas) to propagate. It travels as a longitudinal wave, where particles of the medium vibrate parallel to the direction of wave travel.
• The speed of sound depends on the nature of the medium and the temperature. For example, sound travels at approximately 343 meters per second in air at 20 degrees Celsius, but much faster in water (about 1,480 meters per second) and solids like steel (about 5,960 meters per second).
• Audible Range:
• The human ear can typically hear frequencies ranging from 20 Hz to 20,000 Hz (20 kHz).
• Frequencies below 20 Hz are known as infrasonic sound, while frequencies above 20,000 Hz are called ultrasonic sound.
• Characteristics of Sound:
• Pitch: This is determined by the frequency of the sound wave. A higher frequency results in a higher pitch.
• Loudness: This depends on the amplitude of the vibration. It is measured in decibels (dB).
• Quality (Timbre): This allows the ear to distinguish between sounds of the same pitch and loudness produced by different sources.

Important Points

• Mathematical Relations: The velocity (v) of a sound wave is the product of its frequency (f) and its wavelength (lambda). This is written as v = f × lambda.
• Hertz: The unit Hertz is named after the German physicist Heinrich Rudolf Hertz, who was the first to provide conclusive proof of the existence of electromagnetic waves in the year 1887.
• Time Period Conversion: In physics problems, if the time period is given in milliseconds (ms), it must be converted to seconds (s) by dividing by 1,000 to ensure consistency with SI units. For instance, 4 ms is equal to 0.004 s.
• Oscillation: A complete back-and-forth motion is called an oscillation. The time period is effectively the duration of one oscillation.

Shortcut Trick

Age of new person = New Average + (Previous members × Increase in average)

Age of Teacher = 22 + (38 × (22 − 21))

Age of Teacher = 22 + (38 × 1)

Age of Teacher = 22 + 38 = 60 years

∴ The correct answer is 60.

Alternate Method

Given:

Initial students = 38, Initial Average = 21 years

New Average (including teacher) = 22 years

Formula Used:

Average = Total Sum of values ÷ Number of values

Total Sum = Average × Number of values

⇒ Total age of 38 students = 38 × 21 = 798 years

⇒ Total number of people including teacher = 38 + 1 = 39

⇒ Total age of 39 people = 39 × 22 = 858 years

⇒ Age of the teacher = (Total age of 39 people) − (Total age of 38 students)

⇒ Age of the teacher = 858 − 798 = 60 years

∴ The correct answer is 60.

Additional Information

Weighted Average

When two groups are combined, the combined average is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Net Deviation

The sum of deviations of all observations from the average is always zero.

Average Inclusion Rule

If the new value is greater than the original average, the overall average increases; if smaller, it decreases.

Shortcut Trick

In 'Buy X Get Y Free' offers, the discount is calculated on the total number of items.

Total Items = Paid Items + Free Items = 9 + 1 = 10

Discount % = (Free Items ÷ Total Items) × 100

Discount % = (1 ÷ 10) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given: Buy = 9 sweaters, Free = 1 sweater.

Formula: Discount % = [Free Items ÷ (Buy Items + Free Items)] × 100

Let the Marked Price (MP) of each sweater be ₹1.

⇒ Total MP for 10 sweaters = 10 × 1 = ₹10

⇒ Customer pays for only 9 sweaters.

⇒ Selling Price (SP) for 10 sweaters = 9 × 1 = ₹9

⇒ Discount Amount = MP − SP = 10 − 9 = ₹1

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (1 ÷ 10) × 100 = 10%

∴ The correct answer is 10%.

Additional Information

Successive Discounts

When two discounts d₁ and d₂ are applied, effective discount = d₁ + d₂ − (d₁ × d₂) ÷ 100.

Markup Percentage

Markup % is the percentage by which MP is higher than CP: [(MP − CP) ÷ CP] × 100.

Profit and Discount Relationship

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

Shortcut Trick

Difference in Annual Rates = 7.5% − 3.5% = 4%

Difference in interest for 2 years = 4% × 2 = 8%

Required difference = 8% of ₹2,20,000

Difference = 0.08 × 2,20,000 = ₹17,600

∴ The correct answer is ₹17,600.

Alternate Method

Given:

Principal (P) = ₹2,20,000

Rate of Bank A (R₁) = 3.5% | Rate of Bank B (R₂) = 7.5%

Time (T) = 2 years

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

⇒ SI from Bank A = (2,20,000 × 3.5 × 2) ÷ 100

⇒ SI_A = 2,200 × 7 = ₹15,400

⇒ SI from Bank B = (2,20,000 × 7.5 × 2) ÷ 100

⇒ SI_B = 2,200 × 15 = ₹33,000

⇒ Difference = SI_B − SI_A = 33,000 − 15,400

⇒ Difference = ₹17,600

∴ The correct answer is ₹17,600.

Additional Information

Simple Interest (SI) Difference Formula

When Principal (P) and Time (T) are constant, the difference in interest is: ΔSI = (P × ΔR × T) ÷ 100, where ΔR is the difference in interest rates.

SI vs CI for First Year

For the first year, if the interest is compounded annually, the Simple Interest and Compound Interest on the same principal and rate are always equal.

SI and CI Difference for 2 Years

The difference between Compound Interest and Simple Interest for 2 years is given by: CI − SI = P(R ÷ 100)².

Shortcut Trick

Income = 100%, Savings = 27.5%, Expenditure = 72.5%.

ΔSavings = ΔIncome − ΔExpenditure

ΔIncome = 34% of 100 = 34 units

ΔExpenditure = 20% of 72.5 = 14.5 units

ΔSavings = 34 − 14.5 = 19.5 units

Actual Increase = 19.5% of 49,000 = 19.5 × 490 = ₹9,555.

∴ The correct answer is increase by ₹9,555.

Alternate Method

Given: Income = ₹49,000 | Savings % = 27.5% | Income Increase = 34% | Expenditure Increase = 20%

Formula: Income = Expenditure + Savings

⇒ Initial Savings = 27.5% of 49,000

⇒ Initial Savings = (27.5 ÷ 100) × 49,000 = 13,475

⇒ Initial Expenditure = 49,000 − 13,475 = 35,525

⇒ Increase in Income = 34% of 49,000 = 0.34 × 49,000 = 16,660

⇒ Increase in Expenditure = 20% of 35,525 = 0.20 × 35,525 = 7,105

⇒ Increase in Savings = Increase in Income − Increase in Expenditure

⇒ Increase in Savings = 16,660 − 7,105 = 9,555

∴ The correct answer is increase by ₹9,555.

Additional Information

Income-Expenditure Identity

Income is always the sum of Expenditure and Savings (I = E + S). Any change in Income must be balanced by the sum of changes in Expenditure and Savings.

Percentage Change in Components

If Income increases by x% and Expenditure increases by y%, the percentage change in Savings is given by (I×x − E×y)/S.

Effective Savings Ratio

When the income is very large, using ratios of savings to expenditure (e.g., 27.5 : 72.5 → 11 : 29) simplifies calculations significantly.

Shortcut Trick

Convert percentages and fractions into a single multiplier.

20% × 90% × 5/9 ⇒ (1/5) × (9/10) × (5/9)

Multiplying the terms: (1 × 9 × 5) / (5 × 10 × 9) = 1/10

If 1/10 of the number is 345, then the total number is 345 × 10.

∴ The correct answer is 3450.

Alternate Method

Given:

Result = 345

Percentages = 20% and 90%

Fraction = 5/9

Formula Used:

Value = Number × (P₁/100) × (P₂/100) × Fraction

Let the number be x.

⇒ x × (5/9) × (90/100) × (20/100) = 345

⇒ x × (5/9) × (9/10) × (1/5) = 345

⇒ x × (5 × 9 × 1) / (9 × 10 × 5) = 345

⇒ x × (1/10) = 345

⇒ x = 345 × 10

⇒ x = 3450

∴ The correct answer is 3450.

Additional Information

Percentage to Fraction Conversion

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

Successive Percentage Application

When multiple percentages are applied to a single base, the net multiplier is the product of individual fraction equivalents: Net % = (P₁ × P₂ × ... × P_n).

Reverse Percentage Calculation

If the final value is given and the original is required, multiply the final value by the inverse of each multiplier: Original = Final × (1/M₁) × (1/M₂).

Shortcut Trick

Check for Pythagorean triplet: 36² + 77² = 1296 + 5929 = 7225 = 85².

Since it is a right-angled triangle, the side perpendicular to the base is the altitude.

For base = 36 cm, the corresponding altitude = 77 cm.

∴ The correct answer is 77 cm.

Alternate Method

Given sides of triangle: a = 77 cm, b = 36 cm, c = 85 cm.

First, verify if the triangle is right-angled using Pythagoras Theorem:

⇒ 36² + 77² = 1296 + 5929 = 7225

⇒ 85² = 7225

⇒ Since 36² + 77² = 85², the triangle is right-angled with hypotenuse 85 cm.

⇒ Area of triangle = (1/2) × base × height

⇒ Area = (1/2) × 36 × 77

⇒ For any base, Area = (1/2) × base × corresponding altitude

⇒ (1/2) × 36 × altitude = (1/2) × 36 × 77

⇒ altitude = 77 cm.

∴ The correct answer is 77 cm.

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, and c, such that a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (36, 77, 85).

Heron's Formula

For a triangle with sides a, b, and c, Area = √[s(s−a)(s−b)(s−c)], where semi-perimeter s = (a + b + c) / 2.

Altitude of a Triangle

The perpendicular segment from a vertex to the opposite side. In a right-angled triangle, the legs are altitudes to each other.

Shortcut Trick

Total deduction = 10 + 45 + 39 = ₹94

Adjusted total amount = 567 − 94 = ₹473

Total ratio units = 3 + 6 + 2 = 11

11 units = ₹473 ⇒ 1 unit = ₹43

Nisha’s actual share = (6 × 43) + 45 = 258 + 45 = ₹303

∴ The correct answer is ₹303.

Alternate Method

Given: Total sum = ₹567, Deductions from Arjun, Nisha, and Gaurav are ₹10, ₹45, and ₹39 respectively. Ratio after deduction = 3 : 6 : 2.

⇒ Let the reduced shares be 3x, 6x, and 2x.

⇒ Total reduced sum = 3x + 6x + 2x = 11x

⇒ 11x = 567 − (10 + 45 + 39)

⇒ 11x = 473

⇒ x = 473 ÷ 11 = 43

⇒ Nisha’s reduced share = 6x = 6 × 43 = ₹258

⇒ Nisha’s actual share = 258 + 45 = ₹303

∴ The correct answer is ₹303.

Additional Information

Ratio Distribution Rule

If an amount A is divided among P and Q in the ratio m:n, then share of P = [m ÷ (m+n)] × A.

Effective Sum adjustment

When values are subtracted or added to individual shares, the total sum must be adjusted by the net change before applying the ratio units.

Duplicate and Sub-duplicate Ratio

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

Shortcut Trick

Direct formula for distance moved away from tower: Distance (D) = h × (cot θ₂ − cot θ₁)

Given: Height (h) = 10 m, θ₁ (initial) = 60°, θ₂ (final) = 30°

D = 10 × (cot 30° − cot 60°)

D = 10 × (√3 − 1/√3)

D = (10√3 − 10/√3) m

∴ The correct answer is (10√3 − 10/√3) m.

Alternate Method

Given: Height of tower (AB) = 10 m. Initial angle of depression = 60°. Final angle of depression = 30°.

Formula: tan θ = Perpendicular / Base

⇒ In ΔABC, tan 60° = AB / BC

⇒ √3 = 10 / BC

⇒ BC = 10 / √3 m

⇒ In ΔABD, tan 30° = AB / BD

⇒ 1 / √3 = 10 / BD

⇒ BD = 10√3 m

⇒ Distance moved (CD) = BD − BC

⇒ CD = 10√3 − 10/√3 m

∴ The correct answer is (10√3 − 10/√3) m.

Additional Information

Angle of Depression

It is the angle formed between the horizontal line from the observer and the line of sight to an object below. By alternate interior angles, it equals the angle of elevation from the object.

Trigonometric Identities

Common values for tangent: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3. Note that cot θ = 1 / tan θ.

Distance Calculation for Moving Objects

When an object moves from θ₁ to θ₂, the distance is D = h |cot θ₁ − cot θ₂|, where h is the constant vertical height.

Shortcut Trick

Difference in profit percentages = 35% − 30% = 5%

This 5% difference in profit is due to the ₹48 increase in selling price.

5% of Cost Price = ₹48

100% of Cost Price = (48 ÷ 5) × 100 = 9.6 × 100 = ₹960

∴ The correct answer is ₹960.

Alternate Method

Given:

Initial profit = 30%

New profit = 35%

Increase in Selling Price (SP) = ₹48

Formula Used:

Selling Price (SP) = CP × (100 + Profit%) ÷ 100

Calculations:

⇒ Let the Cost Price (CP) of the article be x.

⇒ Initial Selling Price (SP₁) = 1.30x

⇒ New Selling Price (SP₂) = 1.35x

⇒ According to the question, SP₂ − SP₁ = 48

⇒ 1.35x − 1.30x = 48

⇒ 0.05x = 48

⇒ x = 48 ÷ 0.05

⇒ x = 4800 ÷ 5

⇒ x = 960

∴ The correct answer is ₹960.

Additional Information

Profit and Loss Formulas

Profit = SP − CP and Loss = CP − SP. Profit% or Loss% is always calculated on the Cost Price (CP) unless stated otherwise.

Marked Price and Discount

Marked Price (MP) is the price listed on the product. Discount = MP − SP. Discount% = (Discount ÷ MP) × 100.

Successive Percentage Change

If there are two successive profit percentages a% and b%, the effective profit percentage is given by [a + b + (a × b)/100]%.

Shortcut Trick

If x² + 1/x² = k, then (x − 1/x) = √(k − 2).

Here, k = 123.

∴ (x − 1/x) = √(123 − 2) = √121 = 11.

∴ The correct answer is 11.

Alternate Method

Given: x² + 1/x² = 123

Formula used: (a − b)² = a² + b² − 2ab

⇒ Let a = x and b = 1/x

⇒ (x − 1/x)² = x² + (1/x)² − 2 × x × (1/x)

⇒ (x − 1/x)² = x² + 1/x² − 2

⇒ Substitute the given value of x² + 1/x²:

⇒ (x − 1/x)² = 123 − 2

⇒ (x − 1/x)² = 121

⇒ x − 1/x = ±√121

⇒ x − 1/x = 11 (taking the positive value from options)

∴ The correct answer is 11.

Additional Information

Standard Algebraic Identity

If x² + 1/x² = k, then (x + 1/x) = √(k + 2).

Cubic Relation

If (x − 1/x) = a, then the value of x³ − 1/x³ = a³ + 3a.

Fourth Power Relation

If x + 1/x = a, then x⁴ + 1/x⁴ = (a² − 2)² − 2.

Shortcut Trick

Speed ratio (Usual : New) = 1 : 2/3 = 3 : 2.

Since distance is constant, Time ratio = 2 : 3.

Difference in time units = 3 − 2 = 1 unit.

Given, 1 unit = 18 minutes.

Usual time = 2 units = 2 × 18 = 36 minutes.

∴ The correct answer is 36 minutes.

Alternate Method

Given: New Speed = 2/3 × Usual Speed | Late time = 18 minutes

Formula: Distance = Speed × Time

⇒ Let usual speed = S and usual time = T.

⇒ Distance (D) = S × T

⇒ New speed = (2/3)S and New time = (T + 18)

⇒ D = (2/3)S × (T + 18)

⇒ S × T = (2/3)S × (T + 18)

⇒ T = (2/3)(T + 18)

⇒ 3T = 2T + 36

⇒ T = 36 minutes

∴ The correct answer is 36 minutes.

Additional Information

Inverse Proportionality of Speed and Time

When the distance is constant, speed (S) is inversely proportional to time (T). If speed becomes x/y of original, time becomes y/x of original.

Average Speed Formula

If an equal distance is covered at two different speeds x and y, the average speed for the whole journey is (2 × x × y) ÷ (x + y).

Time Delay Concept

If a person is late by t₁ and early by t₂ in two different scenarios, the total time difference used for calculations is (t₁ + t₂).

Shortcut Trick

Simplify the comparison: 2/40 reduces to 1/20.

Comparing 1/20 to 1/20 is a 1 : 1 ratio.

Any value compared to itself is 100%.

∴ The correct answer is 100%.

Alternate Method

Given values: Fraction A = 1/20 and Fraction B = 2/40

Formula: Percentage = (Fraction A ÷ Fraction B) × 100

⇒ First Fraction = 1/20

⇒ Second Fraction = 2/40 = 1/20 (after dividing numerator and denominator by 2)

⇒ Required % = [(1/20) ÷ (1/20)] × 100

⇒ Required % = 1 × 100

⇒ Required % = 100%

∴ The correct answer is 100%.

Additional Information

Percentage of a Value

To find what percentage x is of y, use the expression: (x ÷ y) × 100.

Equivalent Fractions

Fractions that look different but have the same value are equivalent; e.g., 1/2 = 2/4 = 50/100.

Decimal Conversion

1/20 can be written as 0.05. Therefore, 0.05 is (0.05 ÷ 0.05) × 100 = 100% of itself.

Shortcut Trick

Formula: HCF of fractions = HCF(Numerators) ÷ LCM(Denominators)

Numerators are 1, 3, 27, 81 → HCF is 1

Denominators are 2, 4, 16, 64 → LCM is 64 (since 2, 4, 16 are factors of 64)

HCF = 1/64

∴ The correct answer is 1/64.

Alternate Method

Given: Fractions are 1/2, 3/4, 27/16, and 81/64.

Formula Used: HCF of (a/b, c/d, e/f) = HCF(a, c, e) / LCM(b, d, f)

⇒ Step 1: Find HCF of numerators {1, 3, 27, 81}

⇒ Factors of 1 = 1

⇒ Factors of 3 = 1, 3

⇒ Factors of 27 = 1, 3, 9, 27

⇒ Factors of 81 = 1, 3, 9, 27, 81

⇒ Common factor is 1 ∴ HCF = 1

⇒ Step 2: Find LCM of denominators {2, 4, 16, 64}

⇒ 2 = 2¹ | 4 = 2² | 16 = 2⁴ | 64 = 2⁶

⇒ LCM = 2⁶ = 64

⇒ Required HCF = 1/64

∴ The correct answer is 1/64.

Additional Information

LCM of Fractions

The LCM of fractions is calculated as: LCM of Fractions = LCM(Numerators) ÷ HCF(Denominators).

Simplification Rule

Always simplify fractions to their simplest form (coprime numerators and denominators) before calculating HCF or LCM.

HCF × LCM Property

For any two numbers A and B, the product of the numbers is equal to the product of their HCF and LCM: A × B = HCF(A,B) × LCM(A,B).

HCF of Co-prime Numbers

If a set of numbers includes the number 1 or any two numbers that are co-prime, the HCF of the entire set is always 1.

Shortcut Trick

Let the original fraction be x/y.

According to the conditions: 9(x − 9) = 2y and 3x = y + 12.

From the second eq, y = 3x − 12. Substitute this in the first eq.

9x − 81 = 2(3x − 12) ⇒ 9x − 81 = 6x − 24 ⇒ 3x = 57 ⇒ x = 19.

Then, y = 3(19) − 12 = 45. Sum = 19 + 45 = 64.

∴ The correct answer is 64.

Alternate Method

Given: Let the fraction be x/y.

Condition 1: (x − 9)/y = 2/9

Condition 2: x/(y + 12) = 1/3

From Condition 1:

⇒ 9(x − 9) = 2y

⇒ 9x − 81 = 2y → (Eq. 1)

From Condition 2:

⇒ 3x = y + 12

⇒ y = 3x − 12 → (Eq. 2)

Substitute y from (Eq. 2) into (Eq. 1):

⇒ 9x − 81 = 2(3x − 12)

⇒ 9x − 81 = 6x − 24

⇒ 3x = 57

⇒ x = 19

Substitute x = 19 in (Eq. 2):

⇒ y = 3(19) − 12 = 45

Sum of numerator and denominator:

⇒ x + y = 19 + 45 = 64

∴ The correct answer is 64.

Additional Information

Proper and Improper Fractions

A proper fraction has a numerator less than the denominator (e.g., 2/9), while an improper fraction has a numerator greater than or equal to the denominator.

Equivalent Fractions

Fractions that represent the same value even though they look different, obtained by multiplying or dividing both N and D by the same non-zero number.

Simplification of Fractions

Reducing a fraction to its lowest terms by dividing both the numerator and denominator by their Highest Common Factor (HCF).

Shortcut Trick

Present Ratio (Mother : Daughter) = 3 : 1 (Difference = 2 units)

Ratio after 12 years = 2 : 1 (Difference = 1 unit)

Equate differences: Multiply future ratio by 2 → 4 : 2

Ratio change: 3 to 4 (Mother) or 1 to 2 (Daughter) = 1 unit

1 unit = 12 years

Mother's present age = 3 units = 3 × 12 = 36 years

∴ The correct answer is 36 years.

Alternate Method

Given:

Mother's age = 3 × Daughter's age

After 12 years, Mother's age = 2 × Daughter's age

Formula Used:

Linear equations in one variable: Age after n years = Present Age + n

Let the present age of the daughter be x years.

⇒ Mother's present age = 3x

⇒ After 12 years, Mother's age = 3x + 12

⇒ After 12 years, Daughter's age = x + 12

⇒ 3x + 12 = 2 × (x + 12)

⇒ 3x + 12 = 2x + 24

⇒ 3x − 2x = 24 − 12

⇒ x = 12

⇒ Mother's present age = 3 × 12 = 36 years.

∴ The correct answer is 36 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains constant regardless of the number of years that pass.

Ratio Balancing Method

When ages at two different times are given as ratios, scaling the ratios to have the same part-difference allows for solving without algebra.

Average Age Property

If the current average age of a group is A, then after 'n' years, the new average age will be A + n.

Shortcut Trick

Ratio of sides = 2 : 3 ⇒ Area parts = 2 × 3 = 6

6 units² = 2646 ⇒ 1 unit² = 441

1 unit = √441 = 21 cm

Perimeter units = 2 × (2 + 3) = 10 units

Perimeter = 10 × 21 = 210

∴ The correct answer is 210.

Alternate Method

Given:

Ratio of sides = 2 : 3

Area of rectangle = 2646 cm²

Formula Used:

Area = Length × Breadth

Perimeter = 2 × (Length + Breadth)

Let the sides of the rectangle be 2x and 3x.

⇒ Area = 2x × 3x

⇒ 2646 = 6x²

⇒ x² = 2646 ÷ 6

⇒ x² = 441

⇒ x = √441 = 21 cm

⇒ Length = 3 × 21 = 63 cm

⇒ Breadth = 2 × 21 = 42 cm

⇒ Perimeter = 2 × (63 + 42)

⇒ Perimeter = 2 × 105 = 210 cm

∴ The correct answer is 210.

Additional Information

Rectangle Dimensions

The area of a rectangle is the product of its adjacent sides (Length × Breadth), while the perimeter is the sum of all its boundary sides.

Diagonal Property

The diagonal of a rectangle forms a right-angled triangle with the sides; hence, Diagonal = √(Length² + Breadth²).

Ratio and Area

If the sides are in the ratio a : b, the area is proportional to a × b × x², where x is the common multiplier.

Shortcut Trick

Assumed Average = 90

Deviation for 17 students = 17 × (70 − 90) = −340

Net change in average for 78 students = −340 ÷ 78 ≈ −4.36

Correct average = 90 − 4.36 = 85.64...

∴ The correct answer is 85.6.

Alternate Method

Given:

Total number of students = 78

Number of students in first group (n₁) = 17, Average score (A₁) = 70

Number of students in second group (n₂) = 78 − 17 = 61, Average score (A₂) = 90

Formula Used:

Weighted Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)

⇒ Total score of first group = 17 × 70 = 1190

⇒ Total score of second group = 61 × 90 = 5490

⇒ Combined total score = 1190 + 5490 = 6680

⇒ Class average = 6680 ÷ 78

⇒ Average = 85.6410...

Rounding to one decimal place, we get 85.6.

∴ The correct answer is 85.6.

Additional Information

Weighted Average Formula

It is used to find the average of groups with different sizes: Avg = (n₁w₁ + n₂w₂) ÷ (n₁ + n₂).

Assumed Mean Method

Calculation is simplified by taking a temporary average and adjusting by (Sum of deviations ÷ Total count).

Rounding Rule

To round to one decimal place, look at the second decimal digit; if it is 5 or more, round up the first decimal digit.

Shortcut Trick

√6 × √15 = √90 = 3√10

Expression = (3√10 − 5√10) + 3√10

⇒ −2√10 + 3√10 = √10

∴ The correct answer is √10.

Alternate Method

Given: (√6 × √15 − 5√10) + 3√10

Formula Used: √a × √b = √ab

⇒ (√(6 × 15) − 5√10) + 3√10

⇒ (√90 − 5√10) + 3√10

⇒ (√(9 × 10) − 5√10) + 3√10

⇒ (3√10 − 5√10) + 3√10

⇒ −2√10 + 3√10

⇒ (3 − 2)√10

⇒ √10

∴ The correct answer is √10.

Additional Information

Product Rule for Surds

For any positive real numbers a and b, the square root of their product is √a × √b = √ab.

Simplifying Radicals

A radical is in simplest form when the radicand has no perfect square factors other than 1, such as √90 = √(9 × 10) = 3√10.

Addition of Like Radicals

Only like radicals (radicals with the same index and same radicand) can be added or subtracted using the rule a√x ± b√x = (a ± b)√x.

Shortcut Trick

Total wages for 5 carpenters for 2 days = ₹1,220

Calculate total carpenter-days = 5 carpenters × 2 days = 10 man-days

1-day wage for 1 carpenter = Total wages ÷ Total man-days

Wage = 1220 ÷ 10 = 122

∴ The correct answer is ₹122.

Alternate Method

Given:

Number of carpenters (M₁) = 5

Number of days (D₁) = 2

Total Wages (W₁) = ₹1,220

Formula Used:

Wages ∝ (Number of workers × Time)

W₁ / (M₁ × D₁) = W₂ / (M₂ × D₂)

⇒ Total work units = 5 × 2 = 10 units

⇒ 10 units = ₹1,220

⇒ 1 unit = 1220 ÷ 10

⇒ 1 unit = ₹122

∴ The correct answer is ₹122.

Additional Information

Chain Rule for Wages

When multiple variables (men, days, hours) are involved, the formula is (M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂.

Work & Wages Concept

Wages are distributed in the ratio of the work done by each individual. For equal efficiency, it is proportional to the time spent.

Unitary Method

Always find the value for one unit (one person for one day) by dividing the total amount by the product of the number of people and days.

Shortcut Trick

Use the direct ratio formula between Marked Price (MP) and Cost Price (CP):

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

MP/CP = (100 + 82)/(100 − 48) = 182/52 = 3.5/1

Mark-up % (X) = (Ratio − 1) × 100

X = (3.5 − 1) × 100 = 250%

∴ The correct answer is 250.

Alternate Method

Given:

Discount % = 48%

Profit % = 82%

Formula Used:

SP = CP × (100 + P%)/100

MP = SP × 100 / (100 − D%)

Mark-up % (X) = [(MP − CP) / CP] × 100

Let Cost Price (CP) = 100 units

⇒ Selling Price (SP) = CP + 82% of CP

⇒ SP = 100 + 82 = 182 units

⇒ SP = MP × (100 − Discount%)/100

⇒ 182 = MP × (100 − 48)/100

⇒ 182 = MP × 52/100

⇒ MP = (182 × 100) / 52

⇒ MP = 350 units

⇒ Mark-up = MP − CP = 350 − 100 = 250 units

⇒ X% = (Mark-up / CP) × 100

⇒ X% = (250 / 100) × 100 = 250%

∴ The correct answer is 250.

Additional Information

Relation between CP and MP

The most efficient way to link mark-up, discount, and profit is: MP/CP = (100 + P%)/(100 − D%).

Successive Percentage Method

Profit % can be calculated using the net effect of mark-up (m) and discount (d): Net % = m − d − (md/100).

Mark-up Calculation

Mark-up value is always calculated on the Cost Price (CP), whereas Discount is always calculated on the Marked Price (MP).

Shortcut Trick

Ratio of Volumes of cylinders (V₁ : V₂) = (r₁ : r₂)² × (h₁ : h₂)

Given: r₁ : r₂ = 3 : 4 and h₁ : h₂ = 5 : 2

V₁ : V₂ = (3)² × 5 : (4)² × 2

V₁ : V₂ = (9 × 5) : (16 × 2) = 45 : 32

∴ The correct answer is 45 : 32.

Alternate Method

Given:

Ratio of radii (r₁ : r₂) = 3 : 4

Ratio of heights (h₁ : h₂) = 5 : 2

Formula used: Volume of a Cylinder (V) = πr²h

⇒ Let the radii be 3x and 4x, and heights be 5y and 2y.

⇒ Volume of first cylinder (V₁) = π × (3x)² × 5y

⇒ V₁ = π × 9x² × 5y = 45πx²y

⇒ Volume of second cylinder (V₂) = π × (4x)² × 2y

⇒ V₂ = π × 16x² × 2y = 32πx²y

⇒ Ratio V₁ : V₂ = 45πx²y : 32πx²y

⇒ Ratio = 45 : 32

∴ The correct answer is 45 : 32.

Additional Information

Volume of a Cylinder

The amount of space occupied by a cylinder is given by V = πr²h, where r is the radius and h is the height.

Curved Surface Area (CSA)

The area of the curved surface only is CSA = 2πrh.

Total Surface Area (TSA)

The sum of the area of the curved surface and the two circular bases is TSA = 2πr(h + r).

Ratio Property for Cylinders

If radii ratio is a:b and heights ratio is c:d, then Volume ratio = (a² × c) : (b² × d) and CSA ratio = (a × c) : (b × d).

Shortcut Trick

LCM of 3, 6, and 4 = 12 units (Total Work)

Efficiencies: A = 12/3 = 4, B = 12/6 = 2, C = 12/4 = 3

Combined Efficiency (A + B + C) = 4 + 2 + 3 = 9 units/day

Thrice the work = 12 × 3 = 36 units

Time taken = 36 ÷ 9 = 4 days

∴ The correct answer is 4.

Alternate Method

Given: Time taken by A = 3 days, B = 6 days, C = 4 days.

Formula: Efficiency = Total Work ÷ Time

⇒ Total efficiency of A + B + C = 4 + 2 + 3 = 9 units/day

⇒ Standard Total Work = 12 units

⇒ New Work to be completed = 3 × 12 = 36 units

⇒ Time required = Total Work ÷ Combined Efficiency

⇒ Time required = 36 ÷ 9 = 4 days

∴ The correct answer is 4.

Additional Information

Work and Efficiency

Efficiency is inversely proportional to time when work is constant. Efficiency = Work ÷ Time.

Combined Work Formula

If three people do work in x, y, and z days, the time taken together is (x × y × z) ÷ (xy + yz + zx).

Pipes and Cisterns

The concept of time and work is identical to pipes filling a tank, where efficiency represents the flow rate.

Shortcut Trick

Identify the common term r in both ratios.

Ratio 1: p : q : r = 8 : 5 : 12

Ratio 2: r : s = 36 : 18

Equate r by multiplying Ratio 1 by 3 (since 12 × 3 = 36).

Combined Ratio = (8 × 3) : (5 × 3) : (12 × 3) : 18 = 24 : 15 : 36 : 18.

∴ The correct answer is 24:15:36:18.

Alternate Method

Given:

p : q : r = 8 : 5 : 12

r : s = 36 : 18

To join the ratios, the value of the common variable (r) must be the same in both sets.

⇒ In first ratio, r = 12

⇒ In second ratio, r = 36

⇒ L.C.M. of 12 and 36 = 36

⇒ Multiply the first ratio by (36 ÷ 12) = 3

⇒ p : q : r = (8 × 3) : (5 × 3) : (12 × 3) = 24 : 15 : 36

⇒ Now, r : s = 36 : 18

⇒ Combining both: p : q : r : s = 24 : 15 : 36 : 18

∴ The correct answer is 24:15:36:18.

Additional Information

Compound Ratio

The compound ratio of two ratios a : b and c : d is calculated as (a × c) : (b × d).

Inverse Ratio

The inverse ratio of a : b is 1/a : 1/b, which simplifies to b : a.

Proportion Rule

If four quantities a, b, c, d are in proportion (a : b = c : d), then the product of extremes equals the product of means: a × d = b × c.

Shortcut Trick

Relative speed (opposite direction) = 61 + 47 = 108 km/hr

Speed in m/s = 108 × 5/18 = 30 m/s

Total distance = 295 + 215 = 510 m

Time = 510 / 30 = 17 seconds

∴ The correct answer is 17 seconds.

Alternate Method

Given: Length of first train L₁ = 295 m, Length of second train L₂ = 215 m

Speed of first train S₁ = 61 km/hr, Speed of second train S₂ = 47 km/hr

Formula: Time taken to cross = (Total Length) ÷ (Relative Speed)

⇒ Total distance to be covered = 295 + 215 = 510 m ⇒ Relative speed in opposite direction = S₁ + S₂ ⇒ Relative speed = 61 + 47 = 108 km/hr ⇒ Converting speed to m/s = 108 × (5/18) = 6 × 5 = 30 m/s ⇒ Time = Distance ÷ Relative Speed ⇒ Time = 510 ÷ 30 = 17 seconds

∴ The correct answer is 17 seconds.

Additional Information

Relative Speed in Opposite Direction

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: S_rel = S₁ + S₂.

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the difference between their speeds: S_rel = |S₁ − S₂|.

Unit Conversion

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

Train passing stationary objects

When a train passes a pole or a person, the distance covered is its own length. When it passes a platform, the distance is (Train length + Platform length).

Shortcut Trick

Work done by (A + B) in x hours = Work done by B in (x + 5) hours.

⇒ (Efficiency_A + Efficiency_B) × x = Efficiency_B × (x + 5)

⇒ 15x + 17x = 17x + 85 (Using Efficiency A=15, B=17 from inverse of time)

⇒ 15x = 85 ⇒ x = 85/15 = 17/3 = 5 2/3 hours.

∴ The correct answer is 5 2/3.

Alternate Method

Given:

Time taken by Pipe A = 17 hours

Time taken by Pipe B = 15 hours

Formula Used:

Total Work = LCM of individual times; Efficiency = Total Work ÷ Time

Combined efficiency of A + B = 15 + 17 = 32 units/hr

⇒ Quantity emptied by (A + B) in x hours = 32 × x = 32x units

⇒ Quantity emptied by B alone in (x + 5) hours = 17 × (x + 5) units

⇒ Since the quantity is the same: 32x = 17(x + 5)

⇒ 32x = 17x + 85

⇒ 32x − 17x = 85

⇒ 15x = 85

⇒ x = 85 ÷ 15 = 17 ÷ 3

⇒ x = 5 2/3 hours

∴ The correct answer is 5 2/3.

Additional Information

Pipes and Cisterns

Outlet pipes are considered to do negative work. If an outlet pipe empties a tank in T hours, its efficiency is −1/T per hour.

Efficiency and Time Relationship

The efficiency of a pipe is inversely proportional to the time it takes to fill or empty the tank (Efficiency ∝ 1/Time).

Combined Work Equation

When multiple pipes work together, the total efficiency is the algebraic sum of their individual efficiencies (Net E = E₁ + E₂ + ...).

BODMAS Table: 

Given equation: 81 × 9 − 11 ÷ 2 + 6 = ?

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Now, replacing the symbols:

⇒ 81 ÷ 9 + 11 × 2 − 6 = ?

⇒ 81 ÷ 9 + 11 × 2 − 6 = ?

⇒ 9 + 11 × 2 − 6 = ?

⇒ 9 + 22 − 6 = ?

⇒ 9 + 22 − 6 = ?

⇒ 31 − 6 = ?

⇒ 25.

Hence, the correct answer is "Option 2".

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(1) ILE

I + 3 = L

L - 7 = E

(2) LOH

L + 3 = O

O - 7 = H

(3) GJC

G + 3 = J

J - 7 = C

(4) ORL

O + 3 = R

R - 6 = L

Hence, the correct answer is “Option 4”.

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Pattern followed here is,

Hence, the correct answer is “Option 1”.

Given:

Amit is 14th from the top.

Amit is 55th from the bottom.

Total = Rank from Top + Rank from Bottom - 1

Number of students = 14 + 55 - 1

= 69 - 1

= 68 students.

Hence, the correct answer is "Option 4".

The logic followed here is:

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

Now,

For, (24,480,10)

⇒ (24 × 2) × 10 = 480

For, (12, 552, 23)

⇒ (12 × 2) × 23 = 552

Similarly,

Option (1): (4, 80, 10)

⇒ (4 × 2) ×  10 = 80

Hence, the closest matching answer is "Option 1".

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(1) AE-HD

(2) JN-QL

(3) DH-KF

(4) GK-NI

Hence, the correct answer is “Option 1”.

Least possible Venn diagram,

Conclusions:

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(I) Some plastics are glasses →Does not follow (There is no direct relation given between plastics and glasses. so, it is possible not definite).

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(II) All metals are glasses → Does not follow (All glasses are metals, and  vice versa is possible not definite).

Neither conclusion (I) nor (II) follows.

Hence, the correct answer is “Option 2”.

Seven boxes B, C, D, I, J, K and L are kept one over the other but not necessarily in the same order.

1. Only two boxes are kept between J and B.
2. Only K is kept above I.
3. Only two boxes are kept between K and J.
4. C is kept at some place below L but at some place above D. 

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One box kept between C and L.

Hence, the correct answer is “Option 4”.

Given:

‘bat fan world’ is coded as ‘vd ru ua’

And

 ‘world olive union’ is coded as ‘ru bi rl’. 

‘world’ coded in that language as "ru"

Hence, the correct answer is “Option 2”.

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Given equation: D + F - G × H ÷ J → D is daughter of F, F is brother of G, G is wife of H and H is father of J.

D is J's Mother's Brother's Daughter.

Hence, the correct answer is “Option 4”.

<br> <br>

HM 16 is related to JQ 19,

And 

CH 22 is related to EL 25.

Similarly,

 MR 27 related to,

Hence, the correct answer is “Option 3”.

<br> <br>

RL 12 is related to VQ 16

And 

NH 15 is related to RM 19

Similarly,

 IC 24 related to,

Hence, the correct answer is “Option 4”.