RRB Group D Official Paper (Held On: 26 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is Kinetic Energy.

Key Points

• Kinetic Energy is defined as the energy possessed by an object due to its motion. Any object that has mass and is moving possesses this form of energy.
• For a body of mass ‘m’ moving with a uniform velocity ‘v’, the Kinetic Energy (K.E.) is mathematically expressed by the formula: K.E. = 1/2 mv².
• Since the question specifies the body has a mass and is in motion with a velocity, it directly satisfies the conditions for possessing Kinetic Energy. Hence, the correct answer is Kinetic Energy.
• Kinetic Energy is a scalar quantity, which means it has magnitude but no specific direction. The SI unit of energy, including kinetic energy, is the Joule (J), which is equivalent to 1 kg·m²/s².
• The amount of Kinetic Energy depends on two factors: the mass of the object and the square of its velocity. If the velocity is zero, the Kinetic Energy will also be zero.

Additional Information

• Potential Energy:
• This is the energy held by an object because of its position relative to others, stresses within itself, or its electric charge.
• The most common form is Gravitational Potential Energy, calculated as P.E. = mgh, where g is the acceleration due to gravity and h is the height. Hence, statement 3 is incorrect in the context of a body moving with uniform velocity on a level plane.
• Heat Energy:
• Also known as Thermal Energy, it is the result of the microscopic motion of atoms and molecules within a substance.
• It is transferred between systems due to a temperature difference. Hence, statement 1 is incorrect.
• Sound Energy:
• This form of energy is associated with the vibration of matter. It travels through a medium as mechanical waves.
• It is typically produced when a force causes an object or substance to vibrate. Hence, statement 2 is incorrect.
• Mechanical Energy:
• The total mechanical energy of an object is the sum of its Kinetic Energy and Potential Energy.
• In the absence of friction or other non-conservative forces, the total mechanical energy of a system remains constant.

Important Points

• Work-Energy Theorem: This principle states that the net work done by all forces acting on a body is equal to the change in its Kinetic Energy. This provides a direct link between force, displacement, and motion.
• Velocity Dependency: Because the formula involves v², doubling the velocity of an object will increase its Kinetic Energy by four times (2² = 4). Conversely, doubling the mass only doubles the kinetic energy.
• Types of Kinetic Energy: Beyond simple translation (moving from point A to B), kinetic energy can be rotational (spinning) or vibrational.
• Law of Conservation of Energy: Energy can neither be created nor destroyed, but only transformed from one form to another. For example, a falling object converts its Potential Energy into Kinetic Energy as its velocity increases.

The correct answer is Green/white seeds.

Key Points

• Gregor Mendel, known as the father of genetics, conducted hybridization experiments on garden peas (Pisum sativum) for seven years between 1856 and 1863.
• He selected seven pairs of contrasting visible characters to study the laws of inheritance. One of these traits was stem height, where he compared Tall and Short (Dwarf) plants. Hence, statement 1 is correct.
• Mendel studied the shape of the seeds, using Round (dominant) and Wrinkled (recessive) as the contrasting pair. Hence, statement 2 is correct.
• For flower color, he observed Violet and White flowers, where violet was the dominant trait and white was the recessive trait. Hence, statement 3 is correct.
• Regarding seed color, Mendel used Yellow (dominant) and Green (recessive) seeds. He did not use "white" as a contrasting character for seed color, as white was specifically a trait for flowers. Hence, statement 4 is incorrect.
• Other traits included pod shape (Inflated/Constricted), pod color (Green/Yellow), and flower position (Axial/Terminal).

Additional Information

• Selection of Pea Plant: Mendel chose the garden pea (Pisum sativum) because it was easy to grow, had a short life cycle, and possessed clearly distinguishable contrasting characters. The plants also allowed for both self-pollination and cross-pollination.
• Mendel's Laws: Based on his observations, he proposed the Law of Dominance, the Law of Segregation, and the Law of Independent Assortment.
• Dominant and Recessive: In each of the seven traits, one form always masked the other in the first generation (F1). The trait that appeared was called Dominant, while the hidden trait was Recessive.
• Statistical Analysis: Mendel was one of the first biologists to use mathematics and statistical analysis to explain biological phenomena, which was a revolutionary approach in the 19^th century.
• Publication: His work was published in 1866 under the title "Experiments on Plant Hybridization," but its significance was only recognized in 1900 after his death.

Important Points

• Seed Color vs Pod Color: It is important to note the distinction between seed color and pod color in Mendel's study. For Seed Color, Yellow is dominant and Green is recessive. However, for Pod Color, Green is dominant and Yellow is recessive.
• Number of Traits: Mendel strictly worked with 14 true-breeding pea plant varieties, which corresponded to 7 pairs of contrasting traits.
• Genotype and Phenotype: Mendel's experiments laid the foundation for understanding Genotype (genetic makeup) and Phenotype (physical appearance), terms that were coined later by other scientists.
• Recognition: His findings were rediscovered in 1900 by three scientists: Hugo de Vries, Carl Correns, and Erich von Tschermak, working independently.

The correct answer is Leghorn breed.

Key Points

• Leghorn is a world-famous breed of chicken that originated in Tuscany, in central Italy. It is one of the most popular breeds used in commercial poultry farming worldwide.
• It is specifically utilized for egg production and is classified as a Layer breed. These birds are known for their high fecundity, with a single hen capable of laying an average of 280 to 300 white eggs per year.
• In poultry management, Leghorns are preferred because they have an excellent feed-to-egg conversion ratio, meaning they require less feed to produce a large number of eggs compared to other heavier breeds.
• The breed was first exported to North America in 1828 and subsequently to other parts of the world. They are characterized by their white plumage, yellow legs, and white earlobes.
• Other options like Sardines, Prawns, and Catla are all aquatic organisms and are associated with fisheries, pisciculture, or aquaculture, rather than poultry farming.

Additional Information

• Poultry Farming:
• It is the practice of raising domesticated birds such as chickens, turkeys, ducks, and geese for the purpose of farming meat or eggs for food.
• Breeds are generally categorized into Broilers (raised for meat) and Layers (raised for eggs). Common examples include Rhode Island Red and Plymouth Rock.
• Sardines:
• These are small, epipelagic fish that belong to the family Clupeidae. They are rich in omega-3 fatty acids and are found in marine environments.
• They are a major component of the global fishing industry and are used for human consumption, fish oil, and animal feed.
• Prawns:
• Prawns are crustaceans belonging to the suborder Dendrobranchiata. Their cultivation is known as shrimp farming or aquaculture.
• They can be found in both freshwater and saltwater habitats and are a significant export commodity for many countries, including India.
• Catla:
• Scientifically named Labeo catla, it is an economically important South Asian freshwater fish in the carp family.
• It is a surface feeder and is one of the three Indian Major Carps (along with Rohu and Mrigal) used extensively in pisciculture.

Important Points

• Indigenous Breeds: In India, Aseel is a well-known indigenous breed used for meat and traditional cockfighting due to its high stamina and pugnacity.
• Dual-Purpose Breeds: Some breeds like the Rhode Island Red are used for both meat and egg production, making them versatile for small-scale poultry farmers.
• Disease Management: Poultry farming requires strict biosecurity measures to prevent outbreaks of diseases like Avian Influenza (Bird Flu) and Ranikhet disease.

The correct answer is nanometres.

Key Points

• Atomic radius is defined as the total distance from an atom's nucleus to the outermost orbital of electrons.
• Because atoms are incredibly small entities, their dimensions cannot be measured using standard units like centimetres, decimetres, or millimetres, which are meant for macroscopic objects.
• The nanometre (nm) is a unit of length in the metric system, equal to one-billionth of a metre (10⁻⁹ m). It is the standard unit used to express dimensions on an atomic scale.
• Typical atomic radii range from approximately 0.03 nm to 0.3 nm. For example, the hydrogen atom has a radius of about 0.053 nm.
• The choice of nanometres allows scientists to work with manageable numerical values when describing the size of atoms and molecules.

Additional Information

• Periodic Trends in Atomic Radius:
• Across a Period: The atomic radius generally decreases from left to right. This happens because the effective nuclear charge increases, pulling the electron cloud closer to the nucleus.
• Down a Group: The atomic radius increases as you move down a column. This is due to the addition of new principal energy levels or electron shells, which increases the distance between the nucleus and the valence electrons.
• Types of Atomic Radii:
• Covalent Radius: Half the distance between the nuclei of two identical atoms joined by a single covalent bond.
• Metallic Radius: Half the distance between the nuclei of two adjacent metal atoms in a metallic lattice.
• Van der Waals Radius: Half the distance between the nuclei of two non-bonded, isolated atoms in their closest possible approach.
• Alternative Units:
• Another very common unit used for atomic dimensions is the Angstrom (Å). One Angstrom is equal to 10⁻¹⁰ metres or 0.1 nanometres.
• The picometre (pm) is also frequently used in modern chemistry, where 1 pm = 10⁻¹² metres. 1 nanometre is equal to 1000 picometres.

Important Points

• Scale Comparisons:
• Centimetre: 10⁻² metres. Used for everyday objects like stationery.
• Millimetre: 10⁻³ metres. Used for small mechanical components.
• Nanometre: 10⁻⁹ metres. Essential for Nanotechnology and Quantum Physics.
• Measurement Techniques: Scientists determine the atomic radius using techniques such as X-ray crystallography or electron diffraction, which provide data on the distance between atomic nuclei in various substances.

The correct answer is Only A and C.

Key Points

• When a chemical compound is formed from a metal and a non-metal, the symbol or name of the metal is written first in the chemical formula. This follows the IUPAC (International Union of Pure and Applied Chemistry) nomenclature rules where the electropositive element (metal) precedes the electronegative element (non-metal). Hence, statement A is correct.
• Conversely, writing the non-metal name or symbol first is incorrect according to standard chemical conventions. In a formula like NaCl, the Sodium (metal) is placed before Chlorine (non-metal). Writing it as ClNa would be systematically wrong. Hence, statement B is incorrect.
• For any ionic compound to be stable, the total valency or electrical charges on the ions must be perfectly balanced. The sum of the positive charges from cations and negative charges from anions must equal zero to maintain electrical neutrality. Hence, statement C is correct.
• Hence, the correct answer is Only A and C.

Additional Information

• Rules for Writing Chemical Formulae:
• The valencies or charges on the ion must be balanced. In compounds formed with polyatomic ions, the ion is enclosed in a bracket before writing the number to indicate the ratio, if the number is greater than one.
• When a compound consists of a metal and a non-metal, the name or symbol of the metal is written first. For example: Calcium Oxide (CaO), Sodium Chloride (NaCl), Iron Sulphide (FeS), and Copper Oxide (CuO).
• In these examples, Oxygen, Chlorine, and Sulphur are non-metals and are written on the right, whereas Calcium, Sodium, Iron, and Copper are metals and are written on the left.
• Valency:
• Valency is defined as the combining capacity of an atom or an element. It determines how the atoms of one element will combine with the atoms of another element to form a chemical compound.
• For example, the valency of Magnesium is 2 and Chlorine is 1. To balance the charges in Magnesium Chloride, two chlorine atoms are needed for every one magnesium atom, resulting in the formula MgCl2.

Important Points

• Criss-cross Method:
• A common method to write the formula of a binary compound is the criss-cross method. In this, the valency of each atom is written below its symbol, and then the valencies are crossed over to the other atom as subscripts.
• Example: For Aluminium Oxide, the symbol for Aluminium is Al (valency +3) and Oxygen is O (valency -2). Crossing them gives Al2O3.
• Naming Non-metals:
• In binary compounds (compounds containing two different elements), the non-metal is usually given the suffix "-ide". Examples include Oxide, Chloride, Nitride, and Bromide.

The correct answer is More power.

Key Points

• The Power of a lens is defined as the ability of the lens to converge or diverge light rays falling on it.
• It is mathematically defined as the reciprocal of the focal length (f) of the lens. The formula is given by P = 1/f, where f is measured in meters.
• Because of this inverse relationship, as the focal length of a lens decreases, its power increases. Hence, a lens with a shorter focal length has more power.
• A lens with a shorter focal length has more curvature, which allows it to bend light rays more sharply, leading to a higher degree of convergence or divergence.
• Conversely, a lens with a longer focal length is flatter and bends light rays less, meaning it has less power. Hence, statement 4 is incorrect while statement 3 is correct.

Additional Information

• SI Unit of Power: The standard unit for measuring the power of an optical lens is the Dioptre, represented by the symbol D. 1 Dioptre is equal to 1 m⁻¹.
• Convex Lens (Converging): These lenses have a positive focal length according to the sign convention. Consequently, the power of a convex lens is always positive. They are used to treat Hypermetropia (farsightedness).
• Concave Lens (Diverging): These lenses have a negative focal length. Therefore, the power of a concave lens is always negative. These are typically used to correct Myopia (nearsightedness).
• Nature of Light Bending: Power is a measure of the "strength" of a lens. A lens that can focus light at a very short distance from its optical center is considered more powerful than one that focuses light far away.

Important Points

• Lens Combination: When multiple thin lenses are placed in contact with each other, the net power (P) of the system is the algebraic sum of the powers of the individual lenses (P = P1 + P2 + P3...).
• Refractive Index: The power of a lens also depends on the refractive index of the material it is made of and the medium in which it is placed. If a lens is placed in a medium with a high refractive index, its focal length changes, thereby altering its power.
• Eye Health: Optometrists prescribe corrective glasses by mentioning the power in Dioptres. For example, a prescription of +2.0 D indicates a convex lens, whereas -2.0 D indicates a concave lens.
• Magnification: While power relates to the bending of light, magnification refers to the ratio of the height of the image to the height of the object. High-power lenses often provide higher magnification in simple microscopes.

The correct answer is moisture and oxygen.

Key Points

• Rusting is a chemical process that leads to the corrosion of iron. It occurs when iron surfaces react with oxygen and moisture (water or water vapor) present in the atmosphere.
• For rusting to take place, both oxygen and moisture are essential. If iron is kept in completely dry air or in distilled water without air, it will not rust. Hence, statement 4 is correct.
• While dirt may collect on the surface and hold moisture, it is not a primary chemical reactant in the formation of rust. Hence, statement 1 is incorrect.
• High temperature can increase the speed of the chemical reaction (rusting), but it is not the fundamental cause of the process itself. Hence, statement 2 is incorrect.
• Pollution, specifically the presence of gases like sulfur dioxide or nitrogen oxides, can accelerate the rate of corrosion, but they are not the basic requirements like air and water. Hence, statement 3 is incorrect.
• The chemical product of this reaction is Hydrated Iron (III) Oxide, which appears as a reddish-brown, flaky substance on the surface of the metal.

Additional Information

• The Chemistry of Rusting:
• Rusting is an oxidation reaction where iron (Fe) loses electrons to oxygen.
• The presence of electrolytes (like salt in seawater) can significantly speed up the rusting process because they increase the conductivity of water.
• Prevention Methods:
• Galvanization: This involves coating iron or steel with a thin layer of Zinc to prevent oxygen and water from reaching the iron.
• Painting and Oiling: These create a protective barrier on the metal surface to exclude moisture and air.
• Alloying: Mixing iron with other metals like Chromium and Nickel produces Stainless Steel, which is highly resistant to rusting.
• Metal Properties:
• Iron is a transition metal with the chemical symbol Fe and an atomic number of 26.
• Unlike iron, some metals like Aluminum form a protective oxide layer that prevents further corrosion, whereas rust on iron is porous and allows the reaction to continue deeper into the metal.

Important Points

• Environmental Factors:
• Humidity: Higher levels of humidity in coastal areas lead to faster rusting compared to desert regions.
• Acid Rain: The presence of acids in rain (due to pollution) lowers the pH level, which drastically increases the corrosion rate of iron structures and monuments.
• Economic Impact:
• Rusting causes massive economic losses globally every year by damaging bridges, buildings, ships, and industrial machinery, necessitating regular maintenance and replacement.

The correct answer is Xanthium.

Key Points

• Seed dispersal is the process by which seeds are carried away from the parent plant to new locations to reduce competition for sunlight, water, and minerals.
• Certain plants have evolved specific structures like hooks, spines, or barbs to facilitate dispersal by animals. This specific mechanism is known as epizoochory.
• Xanthium (commonly known as Cocklebur) produces seeds that are encased in a spiny bur. These spines end in hooks that easily attach to the fur of animals or the clothing of humans passing by.
• Once attached, these seeds can be transported over long distances before they eventually fall off or are removed, allowing the plant to colonize new habitats. Hence, the correct answer is Xanthium.
• Other plants mentioned, like Madar and Drumstick, do not use hooks; Madar uses hairy seeds for wind dispersal, and Drumstick uses winged seeds.

Additional Information

• Dispersal by Wind (Anemochory):
• Plants that disperse seeds through wind often have lightweight seeds or special structures to catch the breeze.
• Drumstick seeds have wings that help them glide through the air.
• Madar (Calotropis) and Sunflower (Auchenes) often possess hairy outgrowths (pappus) that act like small parachutes to stay buoyant in the wind.
• Maple is another common example of a plant with winged seeds.
• Dispersal by Water (Hydrochory):
• Seeds or fruits dispersed by water usually develop floating abilities.
• The Coconut has a fibrous outer coat that allows it to float on seawater and travel between islands or coastal regions.
• Lotus fruits have a spongy part that helps them stay afloat in freshwater bodies.
• Dispersal by Animals (Zoochory):
• This can happen through ingestion (animals eating fleshy fruits and excreting seeds) or attachment.
• Urena and Xanthium are primary examples of seeds with hooks for attachment.
• Birds often disperse seeds of fleshy fruits like guava or berries after consuming them.
• Dispersal by Explosion (Autochory):
• Some plants have fruits that burst open with a sudden jerk, scattering seeds far from the mother plant.
• Examples include Balsam, Castor, and Pea pods.

Important Points

• The Xanthium plant belongs to the family Asteraceae. It is often considered a weed in many parts of the world, including India and North America.
• The hooks on Xanthium were the original inspiration for the invention of Velcro by George de Mestral in the 1940s, after he observed how the burs stuck to his dog's fur.
• Effective seed dispersal prevents overcrowding and allows plants to inhabit different geographical locations and climatic zones.
• The number of seeds produced and the efficiency of the dispersal agent (wind, water, or animals) directly impact the survival rate of the plant species.

The correct answer is circular motion.

Key Points

• Circular motion is defined as the movement of an object along the circumference of a circle or rotation along a circular path. When a stone tied to a string is whirled, it travels in a circle centered at the point where the string is held. Hence, statement 1 is correct.
• In this scenario, the string provides a tension force that acts toward the center of the circle. This force is known as the centripetal force, which continuously changes the direction of the stone's motion without necessarily changing its speed.
• Even if the stone is whirled at a constant speed, its velocity is constantly changing because velocity is a vector quantity that depends on both magnitude and direction. Since the direction changes at every point along the circle, the motion is considered accelerated.
• The path of the stone is not a straight line; therefore, it does not exhibit uniform straight-line motion or linear motion. Hence, statements 2 and 3 are incorrect.
• While the motion can be non-uniform if the speed changes, the most fundamental description of a stone whirled in a circle is circular motion. Hence, statement 4 is less specific than statement 1.

Additional Information

• Centripetal Force: This is the external force required to maintain circular motion. For a stone on a string, the tension (T) equals the mass (m) times the square of the velocity (v) divided by the radius (r) of the path (T = mv²/r).
• Uniform Circular Motion (UCM): If the speed of the stone remains constant as it moves in a circle, it is called Uniform Circular Motion. In UCM, the acceleration is always directed toward the center (centripetal acceleration).
• Tangential Velocity: At any instant, the direction of the stone's motion is along the tangent to the circular path at that point. If the string suddenly breaks, the stone will fly off in a straight line along that tangent due to inertia.
• Angular Velocity: This refers to the rate of change of the angle subtended by the object at the center of the circle, usually measured in radians per second (rad/s).

Important Points

• Real-world Examples: Other common examples of circular motion include the Moon revolving around the Earth, electrons orbiting a nucleus, and a car taking a turn on a curved road.
• Work Done: In Uniform Circular Motion, the work done by the centripetal force is always zero because the force is perpendicular to the displacement (tangent) at every point.
• Inertia of Direction: The tendency of an object to resist changes in its direction is what necessitates the continuous application of force (tension) to keep the stone in its circular path.

The correct answer is direction and magnitude.

Key Points

• A vector quantity is defined as a physical quantity that is completely specified by both a magnitude (its size or numerical value) and a direction. Since the magnetic field exerts a force that depends on the direction of motion of a charge, it is classified as a vector quantity.
• The magnetic field, represented by the symbol B, is a region around a magnetic material or a moving electric charge within which the force of magnetism acts.
• The direction of the magnetic field at any point is the direction in which the north pole of a compass needle points when placed at that location.
• In physics, quantities like force, velocity, acceleration, and displacement are all vectors because they cannot be fully described without mentioning which way they are acting.
• The SI unit of magnetic field strength (magnetic flux density) is the Tesla (T), named after Nikola Tesla. Another common unit used in the CGS system is the Gauss (G), where 1 Tesla equals 10,000 Gauss.

Additional Information

• Scalar Quantities:
• Unlike vectors, scalar quantities possess only magnitude and no direction. Examples include mass, density, time, temperature, and electric potential. Hence, options 1, 2, and 3 are incorrect as they involve scalar properties.
• Magnetic Field Lines:
• These are imaginary lines used to represent a magnetic field. They emerge from the North Pole and enter the South Pole outside a magnet, forming continuous closed loops.
• The density of these lines indicates the magnitude of the field; where lines are closer, the field is stronger.
• Sources of Magnetic Fields:
• Magnetic fields are produced by moving electric charges (currents) and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property known as spin.
• In 1820, Hans Christian Ørsted discovered that an electric current creates a magnetic field, a discovery that unified electricity and magnetism into electromagnetism.

Important Points

• Lorentz Force: The vector nature of the magnetic field is evident in the Lorentz Force equation, where the force F on a charge q moving with velocity v is given by F = q(v x B). The cross product indicates that the force is perpendicular to both the velocity and the field.
• Right-Hand Thumb Rule: This rule is used to find the direction of the magnetic field around a straight current-carrying wire. If the thumb points in the direction of the current, the curled fingers show the direction of the magnetic field lines.
• Earth's Magnetic Field: The Earth acts like a giant bar magnet. Its magnetic field protects the planet from solar wind and is essential for navigation using a magnetic compass.
• Magnetic Flux: It is the total magnetic field passing through a given area and is a scalar quantity, measured in Webers (Wb).

The correct answer is two; once.

Key Points

• Fish possess a heart that consists of only two chambers: one atrium and one ventricle. Hence, the first part of the statement is correct.
• In the circulatory system of a fish, the heart pumps deoxygenated blood toward the gills for oxygenation.
• Once the blood is oxygenated at the gills, it does not return to the heart immediately; instead, it flows directly to the rest of the body tissues.
• Because the blood passes through the heart only once during a complete circuit of the body, this type of system is known as single circulation. Hence, the second part of the statement is correct.
• This single circulatory system is efficient enough to meet the metabolic demands of ectothermic (cold-blooded) aquatic organisms like fish.

Additional Information

• Three-Chambered Hearts:
• Amphibians and most reptiles (such as lizards and snakes) have a three-chambered heart consisting of two atria and one ventricle.
• In these animals, blood follows a double circulation pattern, but it is incomplete because oxygenated and deoxygenated blood mix to some extent in the single ventricle.
• Four-Chambered Hearts:
• Birds and mammals (including humans) possess a four-chambered heart with two atria and two ventricles.
• This structure ensures a complete separation of oxygenated and deoxygenated blood, allowing for highly efficient oxygen delivery required for endothermic (warm-blooded) lifestyles.
• Double Circulation:
• In double circulation, blood passes through the heart twice during one complete cycle: once through the pulmonary circuit (to the lungs) and once through the systemic circuit (to the body).

Important Points

• Venous Heart: The heart of a fish is often referred to as a venous heart because it only ever contains deoxygenated blood.
• Accessory Structures: While the main heart has two chambers, fish also have associated structures like the sinus venosus (which receives blood) and the conus arteriosus or bulbus arteriosus (which helps regulate blood flow out of the ventricle).
• Metabolic Rate: The single circulation system results in lower blood pressure in the systemic organs because the pressure drops significantly as blood passes through the narrow capillaries of the gills.
• Evolutionary Context: The transition from single circulation in fish to double circulation in land vertebrates is a key evolutionary milestone that supported higher metabolic rates and thermoregulation.

The correct answer is A book sliding on a rough surface and stopping.

Key Points

• Inertia is the inherent property of an object by virtue of which it resists any change in its state of rest or uniform motion in a straight line. It was first conceptualized by Galileo Galilei and later formalized by Isaac Newton.
• Dust falling off a carpet when beaten: When a carpet is beaten with a stick, the carpet fibers move forward suddenly. However, the dust particles tend to remain in their state of rest due to inertia of rest. As a result, the dust separates from the carpet and falls down. Hence, statement 1 is an example of inertia.
• A book sliding on a rough surface and stopping: According to Newton's First Law, an object in motion will stay in motion unless acted upon by an external unbalanced force. In this case, the force of friction provided by the rough surface acts as that external force which opposes the motion and brings the book to a halt. The stopping is a result of an external force overcoming inertia, not an example of the property of inertia itself. Hence, statement 2 is NOT an example of inertia.
• A ball rolling on a frictionless surface forever: This is a classic theoretical example of inertia of motion. In the absence of any external force like friction or air resistance, the ball's inertia ensures that it maintains its state of uniform motion indefinitely. Hence, statement 3 is an example of inertia.
• Passenger falling forward when a bus stops suddenly: When a moving bus stops, the lower part of the passenger's body, which is in contact with the bus, comes to rest immediately. However, the upper part of the body continues to move forward due to inertia of motion, causing the passenger to lurch forward. Hence, statement 4 is an example of inertia.

Additional Information

• Newton's First Law of Motion: Often referred to as the Law of Inertia, it states that every body continues in its state of rest or of uniform motion in a straight line unless it is compelled by some external force to change that state.
• Mass and Inertia: The mass of an object is a quantitative measure of its inertia. An object with a larger mass has more inertia and requires a greater force to change its state of motion compared to an object with lesser mass. For example, it is harder to stop a moving truck than a bicycle moving at the same speed.
• Types of Inertia:
• Inertia of Rest: The inability of a body to change by itself its state of rest. Example: When a branch of a tree is shaken, the fruits fall off because they stay at rest while the branch moves.
• Inertia of Motion: The inability of a body to change by itself its state of uniform motion. Example: An athlete runs some distance before taking a long jump to gain momentum and utilize inertia.
• Inertia of Direction: The inability of a body to change by itself its direction of motion. Example: When a car rounds a sharp curve, the passengers are thrown toward the outside of the curve.
• Force of Friction: This is a contact force that always opposes the relative motion between two surfaces. On a rough surface, the magnitude of friction is high, which leads to the rapid deceleration of moving objects, effectively counteracting their inertia.

Important Points

• In everyday life, we rarely see objects moving "forever" (like in option 3) because invisible forces such as gravity, friction, and air resistance are almost always present on Earth.
• The concept of inertia was crucial in transitioning from Aristotelian physics (which believed a force was needed to keep something moving) to Classical Mechanics (which recognizes that a force is only needed to change motion).
• In space (a vacuum), the effects of inertia are much more visible because the opposing force of friction is absent, allowing satellites and celestial bodies to maintain their paths for vast periods.

The correct answer is The shape and size of cells are related to the specific function they perform..

Key Points

• The shape and size of cells are not uniform; they exhibit significant diversity depending on the biological role and physiological requirements of the organism. Cells are structured to maximize their efficiency in performing specific functions. Hence, statement 4 is correct.
• While plants and animals share some basic cellular components, their shape and size are not identical. Plant cells typically have a fixed, rectangular shape due to the cell wall, whereas animal cells are often irregular or round and lack a cell wall. Hence, statement 1 is incorrect.
• The morphology of a cell is determined by genetic factors and functional needs rather than being solely dependent on the availability of water. Although turgor pressure (water content) affects cell volume, it is not the only deciding factor for shape and size. Hence, statement 2 is incorrect.
• The shape and size of cells do not remain constant; they vary across different tissues and species. For instance, a nerve cell is drastically different from a skin cell in the same organism, and unicellular organisms like Amoeba can even change their shape constantly. Hence, statement 3 is incorrect.
• Specific examples of functional adaptation include human nerve cells (neurons), which are long and branched to transmit electrical impulses over long distances, and Red Blood Cells (RBCs), which are biconcave to increase surface area for oxygen transport.

Additional Information

• Cell Size Variability:
• The smallest cell known is Mycoplasma (a type of bacteria), measuring approximately 0.1 to 0.5 micrometers.
• The largest isolated single cell is the egg of an ostrich, which can measure approximately 170 mm x 130 mm.
• In the human body, the smallest cell is generally considered the sperm cell, while the largest cell is the ovum (egg cell). The longest cell is the nerve cell.
• Cell Shape Types:
• Cells can be spherical, spindle-shaped, elongated, cuboidal, or columnar.
• White Blood Cells (WBCs) in humans are amoeboid, meaning they can change shape to squeeze through capillary walls and engulf pathogens.
• Guard cells in plant stomata are kidney-shaped or bean-shaped to regulate the opening and closing of pores for gas exchange.

Important Points

• Discovery of the Cell: The cell was first discovered and named by Robert Hooke in 1665 using a primitive microscope to observe cork slices.
• Cell Theory: Proposed by Matthias Schleiden and Theodor Schwann in 1838-39, it states that all living things are composed of cells. Rudolf Virchow later added in 1855 that all cells arise from pre-existing cells (Omnis cellula-e cellula).
• Functional Correlation: The surface area to volume ratio is a critical factor limiting cell size. A smaller cell has a larger surface area relative to its volume, allowing for faster diffusion of nutrients and waste.

The correct answer is Hybridisation.

Key Points

• Hybridisation is a traditional method of crop variety improvement that involves the crossing of two or more plants with different genotypes to produce a new variety.
• The goal of this process is to combine desirable phenotypic traits—such as high yield, disease resistance, and nutritional quality—found in different parent plants into a single hybrid progeny.
• This method can be categorized based on the relationship between parents: Intervarietal (between two different varieties), Interspecific (between two different species of the same genus), and Intergeneric (between two different genera).
• The mechanical process of hybridisation involves several critical steps: selection of parents, emasculation (the removal of stamens to prevent self-pollination), bagging to prevent contamination, and artificial cross-pollination.
• By creating new genetic combinations, hybridisation has been a cornerstone of the Green Revolution, leading to the development of high-yielding varieties (HYV) of crops like wheat and rice. Hence, the correct answer is Hybridisation.

Additional Information

• Tissue Culture:
• This is an in vitro technique where cells, tissues, or organs are grown in a synthetic nutrient medium under aseptic conditions.
• It allows for the rapid cloning of plants, known as micropropagation, and is used to produce disease-free plants from meristems.
• Genetic Engineering:
• This modern biotechnology involves the direct manipulation of an organism's DNA to introduce a specific foreign gene (transgene).
• It results in Genetically Modified Organisms (GMOs), such as Bt Cotton, which possesses insecticidal properties derived from the bacterium Bacillus thuringiensis.
• Mutation Breeding:
• This method utilizes physical mutagens (like gamma rays or X-rays) or chemical mutagens to induce sudden, heritable changes in the plant's genetic sequence.
• It is used to create genetic variation when the desired trait is not available in the existing germplasm of the crop.

Important Points

• Selection: Before crossing, breeders perform mass selection or pure-line selection to ensure the parental plants are of the highest quality and possess the specific traits required for the improvement program.
• Heterosis: Also known as hybrid vigor, this is the phenomenon where the hybrid offspring exhibits superior biological functions (like faster growth or higher yield) compared to either of its parents.
• Objectives of Crop Improvement: Scientists focus on biotic resistance (against pests and pathogens), abiotic resistance (against drought, salinity, and heat), and shorter maturity duration to allow farmers to grow multiple crops in a year.

p-end=""57"" data-start=""0"">The correct answer is 10 Ω.

 

Key Points

• In an electrical circuit, when resistors are connected series, they are arranged in a single line or a daisy-chain fashion, meaning there is only one path for the current to flow through each component.
• The Equivalent Resistance (R_eq) or total resistance of a series combination is calculated by finding the arithmetic sum of the individual resistances.
• The mathematical formula used for this calculation is R_total = R₁ + R₂ + R₃ + ... + R_n.
• Applying the formula to the given problem where the resistors are 2 Ω, 3 Ω, and 5 Ω:
• R_total = 2 Ω + 3 Ω + 5 Ω.
• R_total = 10 Ω.
• This indicates that the combined effect of these three resistors in series is equivalent to a single resistor of 10 Ω.

Additional Information

• Current in Series Circuits:
• The Electric Current (I) remains constant throughout a series circuit. Since there is only one path, the same number of electrons must pass through every point in the circuit per unit time.
• Voltage Distribution:
• According to Ohm’s Law (V = IR), the Potential Difference (Voltage) across each resistor is proportional to its resistance. The sum of the individual voltages equals the total source voltage: V_total = V₁ + V₂ + V₃.
• Parallel Resistors:
• In a Parallel Connection, resistors are connected across the same two nodes. The Total Resistance (R_p) is found using the reciprocal formula: 1/R_p = 1/R₁ + 1/R₂ + 1/R₃. For the given values, the parallel resistance would be approximately 0.967 Ω, which is much lower than the series total.
• Applications and Examples:
• Series circuits are often found in Voltage Dividers and simple Christmas tree lights (older versions), where if one bulb burns out, the entire string goes dark because the circuit is broken.

Important Points

• Resistance Characteristics:
• In any Series Combination, the total resistance is always greater than the largest individual resistance in the circuit. Here, 10 Ω is greater than the largest resistor of 5 Ω.
• The SI Unit of resistance is the Ohm (Ω), named after the physicist Georg Simon Ohm who published his work on the relationship between voltage, current, and resistance in 1827.
• Impact of Adding Resistors:
• Adding more resistors in series increases the Total Impedance of the circuit, which subsequently decreases the total current flowing from the power source, assuming the voltage remains constant.

The correct answer is Moves from the principal focus (F) towards the pole (P).

Key Points

• A convex mirror is a spherical mirror whose reflecting surface is curved outwards. It is also known as a diverging mirror because it diverges light rays after reflection.
• When an object is placed at infinity, the rays of light are parallel to the principal axis. These rays appear to diverge from the principal focus (F) after reflection. Therefore, the image is formed at focus (F) behind the mirror and is virtual, erect, and highly diminished.
• As the object is moved from infinity towards the pole (P) of the mirror, the virtual image shifts its position from the principal focus (F) toward the pole (P).
• During this transition, the size of the image gradually increases, although it always remains smaller than the object (diminished). Hence, the statement that the image moves from the principal focus (F) towards the pole (P) is correct.
• The image formed by a convex mirror is always situated between the pole (P) and the focus (F), regardless of the object's position (unless the object is at infinity).

Additional Information

• Image Characteristics: For all positions of an object in front of a convex mirror, the image formed is always virtual, erect, and diminished. This makes these mirrors extremely useful for specific applications where a wide field of view is required.
• Mirror Formula: The relationship between object distance (u), image distance (v), and focal length (f) is given by 1/v + 1/u = 1/f. In a convex mirror, the focal length (f) and image distance (v) are always considered positive according to the Cartesian sign convention, while the object distance (u) is negative.
• Magnification: The linear magnification (m) is defined as the ratio of the height of the image to the height of the object (m = h'/h = -v/u). For a convex mirror, the value of m is always positive and less than 1, indicating a virtual, erect, and diminished image.
• Applications:
• Rear-view mirrors: They are used in vehicles because they provide an erect image and have a wider field of view as they are curved outwards.
• Security Mirrors: Used in shops and ATM centers to monitor a large area.
• Road Safety: Installed at sharp turns or blind corners to allow drivers to see oncoming traffic from the other side.

Important Points

• Unlike a concave mirror, which can form both real and virtual images depending on the object's position, a convex mirror can never form a real image of a real object.
• The focal length of a spherical mirror is exactly half of its radius of curvature (R), expressed as f = R/2.
• As the object distance u decreases (moves closer to the mirror), the image distance v also decreases (moves closer to the mirror surface), but v can never exceed the value of f when the object is real.

The correct answer is The cube will float partially submerged, displacing water equal to its weight.

Key Points

• Archimedes' Principle states that any object, totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
• A solid cube of wood has a density less than that of water (which is 1000 kg/m³ or 1 g/cm³). Because its density is lower, the upward buoyant force will balance its weight before it is fully submerged.
• Statement 1: The cube will completely sink to the bottom. This occurs only if the density of the object is greater than the density of the fluid. Since wood is less dense than water, this is impossible. Hence, statement 1 is incorrect.
• Statement 2: The cube will float with its entire volume above the water surface. For a buoyant force to exist, some volume of fluid must be displaced. An object cannot float "on top" of the water without any part of it being inside the water. Hence, statement 2 is incorrect.
• Statement 3: The cube will float partially submerged, displacing water equal to its weight. According to the Law of Flotation, a floating body displaces a weight of liquid exactly equal to its own weight. Because the wood is less dense than water, the volume of water required to match the wood's weight is less than the total volume of the cube. Hence, statement 3 is correct.
• Statement 4: The cube will remain suspended in the middle of water. This phenomenon, known as neutral buoyancy, occurs only when the density of the object is exactly equal to the density of the fluid. Hence, statement 4 is incorrect.

Additional Information

• Density and Floating:
• If the density of the object is less than the fluid, it floats.
• If the density of the object is more than the fluid, it sinks.
• If the densities are equal, the object floats fully submerged at any depth (suspended).
• Relative Density (Specific Gravity):
• This is the ratio of the density of a substance to the density of water at 4°C.
• Wood typically has a relative density ranging from 0.3 to 0.9. If a piece of wood has a relative density of 0.7, then 70% of its volume will be submerged under water while 30% remains above the surface.
• Displacement:
• The term displacement refers to the actual volume or weight of the fluid that is pushed aside by the object. For ships made of steel (which is denser than water), the shape is designed to displace a weight of water equal to the ship's weight before the water level reaches the top of the hull.

Important Points

• Buoyant Force (Upthrust): It is the upward force exerted by a fluid that opposes the weight of an immersed object. It is calculated as Fb = V × ρ × g, where V is the submerged volume, ρ is fluid density, and g is acceleration due to gravity.
• Center of Buoyancy: For a floating body to be in equilibrium, the center of gravity and the center of buoyancy must lie on the same vertical line.
• Apparent Weight: When an object is in a fluid, its apparent weight is equal to its true weight minus the buoyant force. For a floating object, the apparent weight is zero because the buoyant force perfectly balances the weight.
• Iceberg Example: Since the density of ice is approximately 917 kg/m³ and seawater is about 1025 kg/m³, about 90% of an iceberg stays submerged, illustrating the high displacement needed to balance its mass.

The correct answer is they are very unreactive.

Key Points

• The reactivity series (or activity series) is a list of metals arranged in descending order of their chemical reactivity. Metals at the bottom are the least likely to lose electrons and form compounds.
• Metals like Gold (Au), Silver (Ag), and Platinum (Pt) are located at the very bottom of this series. Their chemical stability means they do not easily react with atmospheric gases or moisture.
• Statement 1: Metals low in the activity series do not react vigorously with acids; in fact, Gold and Platinum are so stable they only dissolve in a specific mixture called Aqua Regia. Hence, statement 1 is incorrect.
• Statement 2: Highly reactive metals (like Sodium or Potassium) react violently with oxygen and water, whereas metals like Gold remain untarnished and pure over thousands of years. Hence, statement 2 is incorrect.
• Statement 3: These metals are very unreactive (chemically inert), which allows them to exist in their elementary form or native state within the earth's crust rather than as oxides or sulfides. Hence, statement 3 is correct.
• Statement 4: The melting point of a metal is a physical property that does not dictate its chemical occurrence in nature. For instance, Gold has a high melting point of 1,064°C. Hence, statement 4 is incorrect.

Additional Information

• Classification of Metals by Reactivity:
• High Reactivity: Metals like Potassium (K), Sodium (Na), and Calcium (Ca) are so reactive they are never found free in nature. They are extracted using electrolysis of their molten chlorides or oxides.
• Medium Reactivity: Metals like Zinc (Zn), Iron (Fe), and Lead (Pb) are found in the earth's crust mainly as oxides, sulfides, or carbonates. They are typically extracted via reduction with Carbon.
• Low Reactivity: Metals like Gold and Platinum are found in the native state. Copper (Cu) and Silver (Ag) are also found in the free state but can occasionally occur as sulfide or oxide ores.
• Noble Metals: These are metals that are resistant to corrosion and oxidation in moist air. Gold is the most prominent example, often used in jewelry and electronics because it does not corrode or react with most chemicals.
• Extraction: Because gold and silver are found in their metallic form, the extraction process often involves physical separation methods like hydraulic washing or chemical leaching (cyanidation) to separate the pure metal from the surrounding gangue (earthy impurities).

Important Points

• Standard Electrode Potential: In scientific terms, metals low in the activity series have positive reduction potentials, meaning they are difficult to oxidize (lose electrons).
• Occurrence: Gold is typically found in alluvial deposits (placer gold) or in quartz veins. It has been mined for over 5,000 years due to its ease of identification in its pure, yellow metallic form.
• Silver: Although found free, silver is slightly more reactive than gold and may develop a black coating of Silver Sulfide (Ag2S) when exposed to sulfur compounds in the air (tarnishing).

The correct answer is Silver sulphide.

Key Points

• The dark or black layer that develops on Silver articles when exposed to air for a long time is Silver sulphide.
• This chemical process is specifically known as Tarnishing. It occurs because Silver reacts with Hydrogen Sulphide (H2S) gas present in the Atmosphere.
• The chemical formula for the black coating formed is Ag2S. Unlike the corrosion of iron, Silver sulphide is relatively stable and forms a thin film that protects the underlying metal from further deep Corrosion, although it ruins the luster.
• The reaction can be summarized as: Silver (Ag) reacting with Sulphur compounds and Oxygen to produce Silver sulphide and water/hydrogen.
• This layer is highly insoluble in most solvents and requires specific chemical treatments, such as Electrochemical reduction or Acidic cleaning, to be removed and restore the metal's shine.

Additional Information

• Iron oxide:
• Commonly known as Rust, it forms when Iron reacts with Oxygen and Moisture (water) in the air.
• Its chemical formula is typically represented as Fe2O3.xH2O (Hydrated Iron III Oxide).
• Unlike silver tarnishing, Rust is flaky and porous, which allows the Corrosion process to continue until the metal is completely eaten away.
• Copper oxide:
• When Copper is exposed to the atmosphere, it initially reacts with oxygen to form Copper (I) oxide or Copper (II) oxide, which are reddish or black.
• Over a longer period, Copper reacts with Carbon dioxide and moisture to form a green coating called Patina or basic Copper carbonate.
• This Green layer is famously seen on the Statue of Liberty.
• Zinc oxide:
• Zinc reacts with oxygen to form Zinc oxide (ZnO), which is usually a white or greyish-white powder.
• This layer is very effective at preventing further Oxidation, which is why Zinc is widely used in the process of Galvanization to protect Steel and Iron.

Important Points

• Factors affecting Corrosion: The rate of formation of Silver sulphide increases significantly in areas with high Industrial pollution, as the concentration of Sulphur dioxide and Hydrogen sulphide is higher.
• Prevention: Silver articles are often kept in Acid-free tissue paper or stored with Anti-tarnish strips that absorb Sulphur gases from the surrounding air to prevent the formation of the dark layer.
• Cleaning: A common household method to remove Silver sulphide is placing the silver item in a bowl lined with Aluminum foil containing hot water and Baking soda; the sulphur atoms transfer from the Silver to the Aluminum, restoring the Silver.
• Noble Metals: While Silver tarnishes, other noble metals like Gold and Platinum do not react easily with Atmospheric sulphur or oxygen, which is why they do not develop such layers under normal conditions.

The correct answer is The Moon has no atmosphere.

Key Points

• Sound waves are mechanical waves that require a material medium such as solid, liquid, or gas to propagate from one point to another.
• On the Moon, there is no atmosphere, which means the space is essentially a vacuum. In a vacuum, there are no air molecules or particles to carry vibrations.
• Sound is produced by vibrating objects, and these vibrations must be transmitted through a medium by the collision of particles. Since there are no particles on the Moon to collide and pass on the energy, sound cannot travel. Hence, the reason for not being able to hear sound on the Moon is that it has no atmosphere.
• Even if a massive explosion or a loud noise occurs on the lunar surface, an observer would not hear anything because there is no gaseous medium to facilitate the movement of longitudinal waves.
• The absence of sound is a characteristic of the lunar environment that necessitates specialized equipment for communication between astronauts.

Additional Information

• Nature of Sound Waves:
• Sound waves are longitudinal waves where the particles of the medium move parallel to the direction of the wave's propagation.
• These waves consist of compressions (high-pressure regions) and rarefactions (low-pressure regions).
• Communication in Space:
• Since sound cannot travel in a vacuum, astronauts use radio waves for communication.
• Radio waves are electromagnetic waves, which do not require a material medium and can travel through the vacuum of space at the speed of light.
• Speed of Sound:
• The speed of sound varies depending on the medium. It is approximately 343 meters per second in dry air at 20°C.
• Sound travels faster in liquids (about 1,480 m/s in water) and fastest in solids (about 5,000 m/s in steel) due to the closer proximity of atoms.
• Lunar Conditions:
• While the Moon has craters and experiences extreme temperature variations (ranging from -173°C at night to 127°C during the day), these factors do not directly inhibit sound; only the lack of air does.
• The Moon's gravity is about 1/6^th of Earth's gravity, which is too weak to retain a substantial atmosphere.

Important Points

• Vacuum Properties: A vacuum is a space devoid of matter. The outer space between celestial bodies is a near-perfect vacuum, which prevents the transmission of acoustic energy.
• Audible Range: For humans, the audible frequency range is typically between 20 Hz and 20,000 Hz. On the Moon, even if a source produces these frequencies, they cannot reach the ear.
• Scientific Exploration: During the Apollo missions (such as Apollo 11 in 1969), it was practically demonstrated that voice communication was only possible through electronic headsets converting sound into radio signals.

The correct answer is Christmas pudding.

Key Points

• J.J. Thomson proposed his model of an atom in 1904, shortly after the discovery of the electron.
• Thomson’s model described the atom as a sphere of positive charge with negatively charged electrons embedded in it.
• To help visualize this, he used the analogy of a Christmas pudding, where the positive charge is like the pudding (the dough) and the electrons are like the raisins or dry fruits scattered throughout it.
• Another common comparison used by Thomson was a watermelon; the positive charge represents the red edible part, while the electrons are represented by the seeds embedded within the fruit.
• The most significant feature of this model was the proposal that the positive and negative charges are equal in magnitude, making the atom as a whole electrically neutral.

Additional Information

• Discovery of the Electron: J.J. Thomson discovered the electron in 1897 through his experiments with cathode ray tubes. This discovery proved that atoms were not indivisible, contradicting Dalton’s atomic theory.
• Rutherford’s Model: In 1911, Ernest Rutherford proposed the Planetary Model or Solar System analogy. He suggested that the atom has a dense, positive nucleus at its center, with electrons revolving around it like planets around the Sun.
• Bohr’s Model: Niels Bohr refined the atomic structure in 1913 by introducing the concept of quantized energy levels, where electrons move in fixed orbits without radiating energy.
• The Nobel Prize: Sir Joseph John Thomson was awarded the Nobel Prize in Physics in 1906 for his theoretical and experimental investigations on the conduction of electricity by gases.

Important Points

• Shortcomings: Although Thomson’s model was a breakthrough, it could not explain the results of the alpha-particle scattering experiment conducted by Ernest Rutherford in 1911.
• The gold foil experiment showed that most of the alpha particles passed straight through the atom, implying that atoms are mostly empty space, which directly contradicted the "solid pudding" idea of Thomson.
• The Christmas pudding model is also widely known as the Plum Pudding Model in scientific literature.
• This model was the first to recognize that subatomic particles exist within the atom, paving the way for modern atomic physics.

The correct answer is Binary fission.

Key Points

• Amoeba is a unicellular eukaryotic organism that primarily reproduces through an asexual method known as Binary fission.
• During Binary fission, the process begins with the division of the nucleus, which is technically termed karyokinesis. The genetic material is replicated and separated into two equal halves.
• Following nuclear division, the cytoplasm starts to constrict and eventually divides, a process known as cytokinesis.
• This results in the formation of two identical daughter cells from a single parent cell, ensuring the continuity of the species without the need for gametes. Hence, the correct answer is Binary fission.
• This method typically occurs under favorable environmental conditions where food and temperature are optimal for growth and survival.

Additional Information

• Fragmentation:
• This is a form of asexual reproduction where the body of a multicellular organism breaks into two or more fragments upon maturation.
• Each fragment subsequently grows into a complete, new individual. A classic example of this is Spirogyra (a green algae) and Planaria. Hence, statement 1 is incorrect.
• Budding:
• In this process, a small outgrowth or bud forms on the parent body due to repeated cell division at one specific site.
• The bud matures while staying attached to the parent and eventually detaches to become an independent organism. Common examples include Yeast (unicellular) and Hydra (multicellular). Hence, statement 2 is incorrect.
• Spore Formation:
• This method involves the production of specialized reproductive structures called spores, usually encased in a protective thick wall to survive unfavorable conditions.
• When these spores land on a suitable moist surface, they germinate and grow into new individuals. This is frequently seen in fungi like Rhizopus (bread mold) and Mucor, as well as in ferns and mosses. Hence, statement 4 is incorrect.
• Multiple Fission:
• While Amoeba usually undergoes binary fission, it can perform multiple fission during extreme or unfavorable conditions.
• In this state, the Amoeba withdraws its pseudopodia and secretes a hard three-layered cyst. Inside the cyst, the nucleus divides many times to produce many daughter nuclei, eventually releasing multiple pseudopodiospores when conditions improve.

Important Points

• Asexual Reproduction is characterized by the involvement of only one parent and the absence of gametic fusion, resulting in offspring that are genetically identical (clones) to the parent.
• Binary fission in Amoeba can take place in any plane because the organism does not have a fixed body shape, unlike Leishmania, which undergoes binary fission in a definite orientation (longitudinal) relative to its flagellum.
• The simplest organisms, such as bacteria and protozoa, utilize fission as their primary mode of multiplication, which allows for rapid population growth in a short time period.

The correct answer is To create an effective barrier regulating permeability of substances.

Key Points

• Epithelial tissues are the covering or protective tissues in the animal body. They cover most organs and cavities within the body.
• The cells of epithelial tissue are tightly packed and form a continuous sheet. This structural arrangement ensures that there is almost no intercellular space between the cells.
• The primary reason for this tight packing is to act as a selective barrier. Since it forms the outermost layer of the body and lines internal organs, any substance entering or leaving the body must cross at least one layer of epithelium.
• The permeability of these epithelial cells is vital for regulating the exchange of materials between the body and the external environment, and between different internal compartments. Hence, the statement in option 1 is correct.
• All epithelium is usually separated from the underlying tissue by an extracellular fibrous basement membrane, which provides structural support and further regulates filtration.

Additional Information

• Types of Epithelial Tissue:
• Simple Squamous Epithelium: Extremely thin and flat cells forming a delicate lining. Found in oesophagus and the lining of the mouth.
• Stratified Squamous Epithelium: Skin epithelial cells arranged in many layers to prevent wear and tear.
• Columnar Epithelium: Pillar-like cells found where absorption and secretion occur, such as the inner lining of the intestine.
• Cuboidal Epithelium: Cube-shaped cells that provide mechanical support. Found in kidney tubules and salivary gland ducts.
• Ciliated Epithelium: Columnar tissues with hair-like projections (cilia) that move mucus forward to clear it, found in the respiratory tract.
• Functions: Apart from acting as a barrier, they are involved in protection, secretion, absorption, and excretion.
• Avascularity: Epithelial tissues are avascular, meaning they lack their own blood supply. They rely on the diffusion of nutrients from the underlying connective tissue.

Important Points

• Rapid Blood Flow: This is a function associated with Connective Tissue (specifically Blood and Vascular system), not epithelium. Epithelial cells are too tightly packed to allow blood to flow through them directly. Hence, option 2 is incorrect.
• Movement of Muscles: This is the specialized function of Muscular Tissue, which contains contractile proteins like actin and myosin that contract and relax to cause movement. Hence, option 3 is incorrect.
• Transmission of Electrical Signals: This is the primary function of Nervous Tissue. Neurons are designed to receive and conduct impulses (electrical signals) from one part of the body to another. Hence, option 4 is incorrect.
• Cell Junctions: The "little space" between epithelial cells is maintained by specialized structures called tight junctions, adherens junctions, and desmosomes, which seal the cells together to prevent leakage.

The correct answer is They revolve around the nucleus in fixed orbits..

Key Points

• Electrons are subatomic particles that revolve around the central nucleus of an atom in specific circular paths known as orbits or shells. These orbits represent discrete energy levels. Hence, statement 1 is correct.
• The nucleus at the center of the atom contains protons and neutrons, which are collectively called nucleons. Electrons are located in the extranuclear region outside the nucleus. Hence, statement 2 is incorrect.
• The mass of an electron is approximately 9.11 x 10^-31 kg, which is about 1/1836 times the mass of a proton or a hydrogen atom. Because it is so small, its mass is usually considered negligible. Hence, statement 3 is incorrect.
• Electrons carry a negative electric charge of magnitude 1.602 x 10^-19 Coulombs. In contrast, protons are positively charged. Hence, statement 4 is incorrect.
• Hence, the correct answer is They revolve around the nucleus in fixed orbits.

Additional Information

• Discovery of the Electron:
• The electron was discovered by Sir J.J. Thomson in 1897 during his experiments with cathode ray tubes.
• He observed that the rays were deflected by electric and magnetic fields, suggesting they consisted of negatively charged particles.
• Bohr's Model of an Atom:
• Proposed by Niels Bohr in 1913, this model refined the idea of electron orbits.
• He stated that electrons move in fixed circular orbits without radiating energy. These orbits are designated by letters K, L, M, N or principal quantum numbers n = 1, 2, 3, 4.
• Charge and Mass:
• The charge-to-mass ratio (e/m) was first calculated by J.J. Thomson.
• The precise negative charge of the electron was later determined by Robert Millikan through the Oil Drop Experiment in 1909.
• Valence Electrons:
• Electrons in the outermost shell are called valence electrons. They are responsible for the chemical properties and valency of an element.
• The maximum number of electrons that can be accommodated in a shell is given by the formula 2n².

Important Points

• Atomic Neutrality: An atom is electrically neutral because the number of electrons (negative charge) is equal to the number of protons (positive charge) in the nucleus.
• Subatomic Mass: The atomic mass of an element is almost entirely due to the nucleus (protons + neutrons), as the electron mass is so tiny (0.0005485 amu).
• Quantum Nature: In modern Quantum Mechanics (developed by Schrödinger in 1926), electrons are described as existing in orbitals, which are regions of high probability rather than strict circular tracks.

The correct answer is They do not form ions in solution.

Key Points

• Electrical conductivity in a liquid medium depends entirely on the presence of mobile charged particles, specifically ions (cations and anions) or free electrons.
• Glucose (C₆H₁₂O₆) and Alcohol (such as Ethanol, C₂H₅OH) are categorized as covalent compounds. In these molecules, atoms are held together by the sharing of electrons rather than the transfer of electrons.
• When Glucose or Alcohol is dissolved in water, the molecules do not dissociate or break apart into ions; they remain as neutral, intact molecules throughout the solution.
• Because there is a total absence of free ions to carry the electric current between the positive and negative electrodes, these solutions act as insulators. Hence, statement 3 is correct.
• The property of evaporating quickly (volatility) is related to intermolecular forces and vapor pressure, not to the ability to carry a charge in an aqueous state. Hence, statement 1 is incorrect.
• Substances that are acidic in nature (like Hydrochloric acid) conduct electricity because they release Hydrogen ions (H+) in water. Since Glucose and Alcohol do not release such ions, they are not acidic and do not conduct. Hence, statement 2 is incorrect.
• These substances are organic molecules composed of Carbon, Hydrogen, and Oxygen, which are non-metals. They do not possess the metallic lattice or free electrons found in metals. Hence, statement 4 is incorrect.

Additional Information

• Electrolytes vs. Non-electrolytes:
• Electrolytes are substances that produce ions when dissolved in a solvent (usually water) or when melted. Common examples include Sodium Chloride (NaCl), Sulfuric Acid (H₂SO₄), and Sodium Hydroxide (NaOH).
• Non-electrolytes are substances that do not provide ions in a solution. Examples include Glucose, Urea, and Sucrose. These solutions do not conduct electricity.
• The Role of Ions:
• In an electrolytic cell, the anions (negative ions) move toward the anode (+), and cations (positive ions) move toward the cathode (-). This movement of charges completes the electrical circuit.
• Without the formation of these ions, the flow of electric current through the liquid is physically impossible.
• Chemical Bonding:
• Most organic compounds involve covalent bonding. Unless these compounds have polar groups that can ionize (like the carboxyl group in Acetic Acid), they generally remain non-conductive.
• Glucose and Alcohol contain hydroxyl (-OH) groups, but unlike bases, these -OH groups are covalently bonded to carbon and do not dissociate as hydroxide ions (OH-) in water.
• Distilled Water:
• Distilled water itself is a very poor conductor of electricity because it is a covalent compound with negligible ionization. It only becomes conductive when salts, acids, or bases are added to provide the necessary ions.

Shortcut Trick

When the distance is constant, the Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂).

Average Speed = (2 × 9 × 45) ÷ (9 + 45)

Average Speed = 810 ÷ 54 = 15 km/hr.

∴ The correct answer is 15 km/hr.

Alternate Method

Given:

Speed to school (S₁) = 9 km/hr

Speed from school (S₂) = 45 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Let the distance between home and school be 45 km (LCM of 9 and 45).

⇒ Time taken going (T₁) = 45 ÷ 9 = 5 hours

⇒ Time taken returning (T₂) = 45 ÷ 45 = 1 hour

⇒ Total Distance = 45 + 45 = 90 km

⇒ Total Time = 5 + 1 = 6 hours

⇒ Average Speed = 90 ÷ 6 = 15 km/hr

∴ The correct answer is 15 km/hr.

Additional Information

Average Speed (General Case)

It is the ratio of Total Distance traveled to the Total Time taken. Formula: V_avg = (d₁ + d₂ + ...) ÷ (t₁ + t₂ + ...)

Average Speed (Equal Time Intervals)

If a body travels at speed v₁ and v₂ for equal time intervals, the average speed is the Arithmetic Mean: (v₁ + v₂) ÷ 2.

Relative Speed

For two objects moving in the same direction, relative speed is |S₁ − S₂|. For opposite directions, it is (S₁ + S₂).

Shortcut Trick

Identify the minimum difference between any two numbers: 32 − 28 = 4 and 28 − 24 = 4.

The HCF must be a factor of this difference (4).

Check if 4 divides 32, 28, and 24 completely → Yes.

∴ The correct answer is 4.

Alternate Method

Given: Numbers are 32, 28, and 24.

Formula Used: Prime Factorization Method for finding Highest Common Factor (HCF).

⇒ Prime factors of 32 = 2 × 2 × 2 × 2 × 2 = 2⁵

⇒ Prime factors of 28 = 2 × 2 × 7 = 2² × 7

⇒ Prime factors of 24 = 2 × 2 × 2 × 3 = 2³ × 3

⇒ The HCF is the product of the lowest power of common prime factors.

⇒ Common prime factor is 2, and its lowest power is 2².

⇒ HCF = 2 × 2 = 4

∴ The correct answer is 4.

Additional Information

HCF Properties

The HCF of two or more numbers is the greatest number that divides each of them exactly. It is also known as GCD (Greatest Common Divisor).

Relationship between HCF and LCM

For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b.

Division Method

To find HCF of large numbers, divide the larger by the smaller until the remainder is zero; the last divisor is the HCF.

Shortcut Trick

To find the age of the teacher when the average age increases, use the formula:

Teacher age = Old Average + (New Total Strength × Increase in Average)

Teacher age = 27 + (40 × 1) = 27 + 40 = 67

∴ The correct answer is 67 years.

Alternate Method

Given:

Number of students = 39

Initial average age = 27 years

Total members including teacher = 39 + 1 = 40

New average age = 28 years

Formula Used:

Total Sum = Average × Number of observations

Teacher age = (New Total Sum) − (Initial Total Sum)

Calculations:

⇒ Total age of 39 students = 39 × 27

⇒ Total age of 39 students = 1053 years

⇒ Total age including teacher = 40 × 28

⇒ Total age including teacher = 1120 years

⇒ Teacher age = 1120 − 1053

⇒ Teacher age = 67 years

∴ The correct answer is 67 years.

Additional Information

Net Deviation Method

The sum of deviations of all observations from their average is always zero. If the average increases by 1 for 40 people, the extra value brought by the new member must be 40 × 1.

Weighted Average Formula

Average_combined = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n is the number of items and A is the average.

Replacement Rule

Age of new person = Age of replaced person + (Total count × Change in Average).

Shortcut Trick

कुल वर्तमान आयु = 42 वर्ष | 5 वर्ष बाद कुल आयु = 42 + (2 × 5) = 52 वर्ष

5 वर्ष बाद आयु का अनुपात (माता : पुत्री) = 3 : 1 | कुल अनुपात = 4

4 यूनिट = 52 → 1 यूनिट (पुत्री की 5 वर्ष बाद आयु) = 13 वर्ष

पुत्री की वर्तमान आयु = 13 − 5 = 8 वर्ष | माता की 5 वर्ष बाद आयु = 3 × 13 = 39 वर्ष

माता की 4 वर्ष बाद आयु = 39 − 1 = 38 वर्ष | अभीष्ट अनुपात = 8 : 38 = 4 : 19

∴ The correct answer is 4 : 19.

Alternate Method

दिया गया है: माता और पुत्री की वर्तमान आयु का योग = 42 वर्ष

सूत्र: अनुपात = मान 1 : मान 2

माना पुत्री की वर्तमान आयु = x वर्ष, तो माता की वर्तमान आयु = (42 − x) वर्ष

5 वर्ष बाद की स्थिति:

⇒ (42 − x + 5) = 3 × (x + 5)

⇒ 47 − x = 3x + 15

⇒ 4x = 32 ⇒ x = 8 वर्ष (पुत्री की वर्तमान आयु)

⇒ माता की वर्तमान आयु = 42 − 8 = 34 वर्ष

⇒ माता की 4 वर्ष बाद आयु = 34 + 4 = 38 वर्ष

⇒ अभीष्ट अनुपात = पुत्री (वर्तमान) : माता (4 वर्ष बाद) = 8 : 38

⇒ सरल करने पर = 4 : 19

∴ The correct answer is 4 : 19.

Additional Information

आयु योग नियम (Sum of Ages)

यदि n व्यक्तियों की वर्तमान आयु का योग S है, तो t वर्ष बाद उनकी आयु का योग S + (n × t) होगा।

आयु अंतर (Age Difference)

दो व्यक्तियों के बीच आयु का अंतर हमेशा स्थिर रहता है, चाहे वह वर्तमान हो, भविष्य हो या भूतकाल।

अनुपात परिवर्तन (Ratio Concept)

यदि A:B का वर्तमान अनुपात x:y है, तो k वर्ष बाद अनुपात (x+k):(y+k) होगा (जहाँ x और y वास्तविक मान हैं)।

Shortcut Trick

For any regular polygon, the sum of an interior angle and its corresponding exterior angle is always 180°.

Exterior angle = 180° − 156° = 24°

Number of sides (n) = 360° ÷ Exterior angle

n = 360 ÷ 24 = 15

∴ The correct answer is 15 sides.

Alternate Method

Given:

Interior angle of a regular polygon = 156°

Formula Used:

Interior angle of a regular polygon = [(n − 2) × 180°] ÷ n

Where n = number of sides of the polygon.

Calculations:

⇒ 156 = [(n − 2) × 180] ÷ n

⇒ 156n = 180n − 360

⇒ 180n − 156n = 360

⇒ 24n = 360

Shortcut Trick

Using the ratio method for successive changes:

Initial : Final ratio = (100 : 50) × (100 : 70) × (100 : 50) × (100 : 165)

⇒ Ratio = (2 : 1) × (10 : 7) × (2 : 1) × (20 : 33) = 800 : 231

⇒ Net decrease = (800 − 231) ÷ 800 × 100 = 569 ÷ 8 = 71.125%

∴ The correct answer is Decreases by 71.13%.

Alternate Method

Given: Successive changes are −50%, −30%, −50%, and +65%.

Formula Used: Final Price = Initial Price × (1 + r₁/100) × (1 + r₂/100) ...

Let the original price of fuel be 100.

⇒ Price after 1^st month = 100 × (1 − 0.50) = 50

⇒ Price after 2^nd month = 50 × (1 − 0.30) = 35

⇒ Price after 3^rd month = 35 × (1 − 0.50) = 17.5

⇒ Price after 4^th month = 17.5 × (1 + 0.65) = 17.5 × 1.65 = 28.875

⇒ Net percentage change = (Initial − Final) / Initial × 100

⇒ Percentage change = (100 − 28.875) ÷ 100 × 100 = 71.125%

∴ The correct answer is Decreases by 71.13%.

Additional Information

Successive Percentage Change

For two changes a% and b%, effective change = a + b + (ab/100). This can be extended to multiple changes.

Ratio Method for Percentage

Express each percentage change as a fraction (e.g., 50% decrease = 1/2). Final Value = Initial Value × (Product of multipliers).

Base Value Selection

Taking 100 as the initial base simplifies percentage calculations as the absolute change directly equals the percentage change.

Shortcut Trick

The unit digit of the edge length 45 is 5.

The Volume of a cube is (Edge)³, so its unit digit will be 5 × 5 × 5 = 125, which ends in 5.

Among the given options, only 91125 ends with the digit 5.

∴ The correct answer is 91125 m³.

Alternate Method

Given:

Edge length of the cube (a) = 45 m

Formula Used:

Volume of a cube = a³ (where a is the edge of the cube)

Calculations:

⇒ Volume = 45³

⇒ Volume = 45 × 45 × 45

⇒ Volume = 2025 × 45

⇒ Volume = 91125 m³

∴ The correct answer is 91125 m³.

Additional Information

Total Surface Area of a Cube

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

Diagonal of a Cube

The longest distance between two opposite vertices is calculated as a√3.

Lateral Surface Area

The area of the four side faces (excluding top and bottom) is 4a².

Shortcut Trick

Let the number be N.

Given: 20% × 10% × (3/4) × N = 882

(1/5) × (1/10) × (3/4) × N = 882

3N / 200 = 882

N = (882 × 200) / 3 = 294 × 200 = 58800

∴ The correct answer is 58800.

Alternate Method

Given: 20% of 10% of 3/4 of a number = 882

⇒ Let the number be x.

⇒ (20/100) × (10/100) × (3/4) × x = 882

⇒ (1/5) × (1/10) × (3/4) × x = 882

⇒ (3/200) × x = 882

⇒ x = (882 × 200) ÷ 3

⇒ x = 294 × 200

⇒ x = 58800

∴ The correct answer is 58800.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 20% = 1/5 and 10% = 1/10.

Successive Percentage

When multiple percentages are applied to a number, they are multiplied together as factors: x × a% × b%.

Base Change Rule

If a number is increased or decreased, the new base for the next percentage calculation is the resulting value from the previous step.

Shortcut Trick

Total amount to be divided after deductions = 2037 − (23 + 42 + 52) = 2037 − 117 = ₹1920.

Ratio of remaining shares = 14 : 19 : 15. Sum of ratios = 14 + 19 + 15 = 48 units.

48 units = ₹1920 ⇒ 1 unit = ₹40.

Tina remaining share = 19 units × 40 = ₹760.

Tina actual share = ₹760 + 42 = ₹802.

∴ The correct answer is ₹802.

Alternate Method

Given: Total amount = ₹2,037. Deductions = ₹23, ₹42, ₹52. Ratio = 14:19:15.

Formula: Actual Share = (Ratio Share × Remaining Total / Sum of Ratios) + Deduction

⇒ Total Deduction = 23 + 42 + 52 = ₹117

⇒ Remaining Amount = 2037 − 117 = ₹1920

⇒ Let shares be 14x, 19x, and 15x.

⇒ 14x + 19x + 15x = 1920

⇒ 48x = 1920 ⇒ x = 40

⇒ Tina remaining share = 19 × 40 = ₹760

⇒ Tina actual share = 760 + 42 = ₹802

∴ The correct answer is ₹802.

Additional Information

Ratio and Proportion

Ratio is a comparison of two quantities of the same kind. If a sum S is divided in ratio a:b:c, the shares are [a/(a+b+c)]×S, [b/(a+b+c)]×S, and [c/(a+b+c)]×S respectively.

Inverse Variation

When two quantities are such that an increase in one leads to a decrease in the other, they are in inverse proportion (x × y = k).

Shortcut Trick

(1/3)^-2 = 9 and (1/2)^-3 = 8

{9 − 8} ÷ (1/4)^-2 = 1 ÷ 16 = 1/16

(-3/2)⁴ × (5/3)⁴ = [(−5/2)]⁴ = 625/16

Total = 1/16 + 625/16 = 626/16

∴ Answer = 313/8

Alternate Method

Given:

{ (1/3)^-2 − (1/2)^-3 } ÷ (1/4)^-2 + { (−3/2)⁴ × (5/3)⁴ }

Formula Used:

a⁻ⁿ = 1/aⁿ, (a/b)ⁿ × (c/d)ⁿ = [(ac)/(bd)]ⁿ

Calculation:

{ (1/3)-2 − (1/2)-3 } ÷ (1/4)-2 + { (−3/2)4 × (5/3)4 }

⇒ (3² − 2³) ÷ 4² + [ (−3/2 × 5/3)⁴ ]

⇒ (9 − 8) ÷ 16 + (−5/2)⁴

⇒ 1 ÷ 16 + 625/16

⇒ 626/16

⇒ 313/8

∴ The correct answer is 313/8.

Additional Information

Negative Powers

a⁻ⁿ = 1/aⁿ

Power Multiplication

(a/b)ⁿ × (c/d)ⁿ = [(ac)/(bd)]ⁿ

Even Power

Negative base becomes positive when power is even

Shortcut Trick

Loss of 5% means SP₁ is 95% of CP.

Gain of 5% means SP₂ must be 105% of CP.

Using the unitary method: 95% ⇒ ₹1805.

1% ⇒ 1805 ÷ 95 = ₹19.

105% ⇒ 19 × 105 = ₹1995.

∴ The correct answer is ₹ 1995.

Alternate Method

Given: Selling Price (SP₁) = ₹1805, Loss% = 5%, Required Gain% = 5%.

Formulas: CP = [SP × 100] ÷ [100 − Loss%], SP = [CP × (100 + Gain%)] ÷ 100.

⇒ First, find the Cost Price (CP): CP = [1805 × 100] ÷ (100 − 5) = [1805 × 100] ÷ 95.

⇒ CP = 19 × 100 = ₹1900.

⇒ Now, find the New Selling Price (SP₂) for 5% gain: SP₂ = [1900 × (100 + 5)] ÷ 100.

⇒ SP₂ = [1900 × 105] ÷ 100 = 19 × 105.

⇒ SP₂ = ₹1995.

∴ The correct answer is ₹ 1995.

Additional Information

Relation between SP and CP

When Profit occurs, SP = CP + Profit. When Loss occurs, SP = CP − Loss. CP is always considered 100% in percentage calculations.

Direct Ratio Method

If CP is constant, SP₁ ÷ (100 − L%) = SP₂ ÷ (100 + G%). This helps in solving questions involving change in profit/loss percentages quickly.

Shortcut Trick

The relationship between Cost Price (CP) and Marked Price (MP) is given by the ratio:

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

Given: Discount = 65% and Profit = 82%.

CP : MP = (100 − 65) : (100 + 82) = 35 : 182.

Dividing both sides by 7: 35 ÷ 7 : 182 ÷ 7 = 5 : 26.

∴ The correct answer is 5 : 26.

Alternate Method

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

⇒ Let CP be 100.

⇒ SP = 100 + 82% of 100 = 182.

⇒ SP is also equal to MP × (100 − 65) ÷ 100 = MP × 35 ÷ 100.

⇒ 182 = MP × 35 ÷ 100 ⇒ MP = (182 × 100) ÷ 35 = 520.

⇒ Ratio CP : MP = 100 : 520 = 10 : 52 = 5 : 26.

∴ The correct answer is 5 : 26.

Additional Information

Marked Price & Discount

The price at which an article is listed is the Marked Price (MP). Discount is always calculated on the MP.

Profit on Cost Price

Profit or loss is always calculated on the Cost Price (CP) unless otherwise stated.

Shortcut Trick

Ratio of Time = 1 : 1.5 = 2 : 3

Using weighted average: Average Speed = (S₁×T₁ + S₂×T₂) ÷ (T₁ + T₂)

52 = (70×2 + p×3) ÷ 5

260 = 140 + 3p ⇒ 3p = 120 ⇒ p = 40

∴ The correct answer is 40 km/hr.

Alternate Method

Given: Speed 1 = 70 km/hr, Time 1 = 1 hr, Speed 2 = p km/hr, Time 2 = 1.5 hr, Avg Speed = 52 km/hr.

Formula Used: Total Distance = Average Speed × Total Time

⇒ Total Distance = (70 × 1) + (p × 1.5) = 70 + 1.5p

⇒ Total Time = 1 + 1.5 = 2.5 hours

⇒ Average Speed = Total Distance ÷ Total Time

⇒ 52 = (70 + 1.5p) ÷ 2.5

⇒ 52 × 2.5 = 70 + 1.5p

⇒ 130 = 70 + 1.5p

⇒ 1.5p = 60

⇒ p = 60 ÷ 1.5 = 40

∴ The correct answer is 40 km/hr.

Additional Information

Average Speed Formula

Average Speed = Total Distance ÷ Total Time. It is not the simple arithmetic mean of speeds unless times are equal.

Harmonic Mean Case

If a car covers two equal distances at speeds x and y, then the average speed = 2xy ÷ (x + y).

Weighted Average Property

The average speed always lies between the minimum and maximum individual speeds of the journey segments.

Shortcut Trick

Total Work = LCM(12, 15, 20) = 60 units.

Efficiencies: (A+B) = 5, (B+C) = 4, (C+A) = 3.

2(A+B+C) = 5+4+3 = 12 → (A+B+C) = 6.

Efficiency of B = (A+B+C) − (C+A) = 6 − 3 = 3.

Efficiency of C = (A+B+C) − (A+B) = 6 − 5 = 1.

Ratio of Days = Inverse Ratio of Efficiency = 1 : 3.

∴ The correct answer is 1:3.

Alternate Method

Given: (A+B) = 12 days, (B+C) = 15 days, (C+A) = 20 days.

Formula Used: Efficiency = Total Work ÷ Time; Time ∝ 1 ÷ Efficiency.

⇒ 2(A + B + C) = 5 + 4 + 3 = 12

⇒ Efficiency of (A + B + C) = 6

⇒ Efficiency of B = (A + B + C) − (A + C) = 6 − 3 = 3 units/day

⇒ Efficiency of C = (A + B + C) − (A + B) = 6 − 5 = 1 unit/day

⇒ Days taken by B = 60 ÷ 3 = 20 days

⇒ Days taken by C = 60 ÷ 1 = 60 days

⇒ Ratio of days (B : C) = 20 : 60 = 1 : 3

∴ The correct answer is 1:3.

Additional Information

Inverse Relation Rule

For a constant amount of work, the time taken is inversely proportional to the efficiency of the worker (Time₁/Time₂ = Efficiency₂/Efficiency₁).

Combined Work Formula

If individuals work in pairs, the total efficiency is half the sum of paired efficiencies: (A+B+C) = [(A+B) + (B+C) + (C+A)] ÷ 2.

LCM Method

Taking the Least Common Multiple of time periods as the total work simplifies calculation by avoiding fractions.

Shortcut Trick

Selling Price = Marked Price × (100 − Discount%) ÷ 100

SP = 2,400 × (100 − 16) ÷ 100

SP = 2,400 × (84 ÷ 100) = 24 × 84

SP = 2,016

∴ The correct answer is ₹2,016.

Alternate Method

Given: Marked Price (MP) = ₹2,400, Discount Percentage = 16%.

Formulas: Discount = MP × (Discount% ÷ 100) | Selling Price (SP) = MP − Discount

⇒ Discount Amount = 2,400 × (16 ÷ 100)

⇒ Discount Amount = 24 × 16 = ₹384

⇒ Selling Price = 2,400 − 384

⇒ Selling Price = ₹2,016

∴ The correct answer is ₹2,016.

Additional Information

Marked Price & Discount

The Marked Price is the price printed on the label. Discount is always calculated on the Marked Price unless otherwise specified.

Profit/Loss vs. Discount

While Discount is the reduction from the Marked Price, Profit or Loss is determined by the difference between the Selling Price and the Cost Price.

Shortcut Trick

For divisibility by 72, the number must be divisible by 8 and 9.

Check 8: Last three digits 47x must be divisible by 8 ⇒ x = 2.

Check 9: Sum of digits (8+3+y+5+4+7+2) = 29 + y must be divisible by 9 ⇒ y = 7.

Required value: x² + y² = 2² + 7² = 4 + 49 = 53.

∴ The correct answer is 53.

Alternate Method

Given: Number 83y547x is divisible by 72.

Formula Used: Divisibility by 72 requires divisibility by 8 (co-prime factor 1) and 9 (co-prime factor 2).

Step 1: Divisibility by 8

⇒ The last three digits 47x must be divisible by 8.

⇒ 472 ÷ 8 = 59 (remainder 0).

⇒ x = 2.

Step 2: Divisibility by 9

⇒ The sum of digits 8 + 3 + y + 5 + 4 + 7 + 2 must be divisible by 9.

⇒ 29 + y must be divisible by 9.

⇒ The nearest multiple of 9 after 29 is 36.

⇒ 29 + y = 36 ⇒ y = 7.

Step 3: Final Calculation

⇒ Sum of squares = x² + y²

⇒ 2² + 7² = 4 + 49 = 53.

∴ The correct answer is 53.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if the last three digits are divisible by 8 or are zeros.

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its digits is divisible by 9.

Composite Divisibility

To check divisibility by a composite number like 72, break it into co-prime factors (8 and 9) and check for each.

Shortcut Trick

Net deviation of first 7 numbers from Average 92: (−62) + (−60) + (−46) + (−11) + (+18) + (−19) + (−72) = −252.

To maintain the overall average, the remaining 6 numbers must compensate for this loss.

Average compensation required = 252 ÷ 6 = +42.

Average of remaining numbers = 92 + 42 = 134.

∴ The correct answer is 134.

Alternate Method

Given: Total numbers = 13 | Average = 92 | First 7 numbers = 30, 32, 46, 81, 110, 73, 20.

Formula Used: Sum = Average × Number of terms | Average = Sum ÷ Number of terms.

⇒ Total sum of 13 numbers = 13 × 92 = 1196

⇒ Sum of first 7 numbers = 30 + 32 + 46 + 81 + 110 + 73 + 20 = 392

⇒ Sum of remaining 6 numbers = 1196 − 392 = 804

⇒ Average of remaining 6 numbers = 804 ÷ 6 = 134

∴ The correct answer is 134.

Additional Information

Deviation Method for Averages

The sum of deviations of all observations from their mean is always zero. This is used to find missing values or averages of subgroups quickly.

Weighted Average Formula

If two groups have sizes n₁, n₂ and averages A₁, A₂, then Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Effect of Change in Values

If every number in a set is increased or decreased by a constant k, the average of the set also increases or decreases by k.

Shortcut Trick

Required percentage = (Value 1 ÷ Value 2) × 100

Value 1 = 1/30, Value 2 = 2/70 = 1/35

Percentage = (1/30 ÷ 1/35) × 100 = (35/30) × 100 = (7/6) × 100

7/6 × 100 = 1.1666... × 100 = 116.67%

∴ The correct answer is 116.67%.

Alternate Method

Given: First fraction = 1/30, Second fraction = 2/70

Formula Used: Percentage = (Part/Whole) × 100

Calculations:

⇒ Let 1/30 be x% of 2/70.

⇒ 1/30 = (x/100) × (2/70)

⇒ 1/30 = (x/100) × (1/35)

⇒ x = (35/30) × 100

⇒ x = (7/6) × 100

⇒ x = 700/6 = 116.666...

⇒ Rounding off to two decimal places, x = 116.67

∴ The correct answer is 116.67%.

Additional Information

Fraction to Percentage Conversion

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

Reciprocal Fraction Values

1/6 is approximately 16.67%. Therefore, 7/6 can be calculated as 100% + 16.67% = 116.67%.

Rounding Off Rules

If the third decimal digit is 5 or greater, increase the second decimal digit by 1. Here, 116.666... becomes 116.67.

Shortcut Trick

Total days from 6 February 2023 to 20 April 2023 = 22 (Feb) + 31 (Mar) + 20 (Apr) = 73 days.

Time in years = 73 ÷ 365 = 1/5 year.

Interest for 1 year = 7% of ₹2,000 = ₹140.

Simple Interest for 1/5 year = 140 ÷ 5 = ₹28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate (R) = 7% per annum

Time Calculation:

February (non-leap year) = 28 − 6 = 22 days

March = 31 days

April = 20 days

⇒ Total Days = 22 + 31 + 20 = 73 days

⇒ Time (T) = 73/365 year = 1/5 year

Formula Used:

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

Calculations:

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (2000 × 7) ÷ (100 × 5)

⇒ SI = 14000 ÷ 500

⇒ SI = 140 ÷ 5

⇒ SI = ₹28

∴ The correct answer is ₹28.

Additional Information

Simple Interest (SI) Formula

Interest remains constant every year and is calculated only on the principal amount: SI = (P × R × T) ÷ 100.

Day Calculation Rule

While calculating the number of days, usually the day of deposit is excluded and the day of withdrawal is included.

Standard Conversions

In interest problems, 73 days is exactly 1/5 of a year (365 ÷ 73 = 5), and 146 days is 2/5 of a year.

Leap Year Check

If the year is divisible by 4, it is a leap year (29 days in Feb), otherwise, it has 28 days.

Shortcut Trick

Combined Efficiency = (Total Work ÷ A) + (Total Work ÷ B) + (Total Work ÷ C)

Using LCM (2, 24, 8) = 24 as Total Work.

Combined Efficiency = 12 + 1 + 3 = 16 units/day.

Time for twice the work = (2 × 24) ÷ 16 = 48 ÷ 16 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given:

Time taken by A = 2 days

Time taken by B = 24 days

Time taken by C = 8 days

Calculations:

⇒ Total Work (LCM of 2, 24, 8) = 24 units

⇒ Combined efficiency of A + B + C = 12 + 1 + 3 = 16 units/day

⇒ Total work to be done = 2 × 24 = 48 units

⇒ Time taken = 48 ÷ 16 = 3 days

∴ The correct answer is 3 days.

Additional Information

Total Work Formula

Total Work = Efficiency × Time. If work is doubled, the time taken at constant efficiency also doubles.

Combined Efficiency

When individuals work together, their individual efficiencies are added to find the net rate of work.

Work and Wages

Wages are generally distributed in the ratio of efficiency or the ratio of work done by each individual.

Shortcut Trick

Identify the symmetrical coefficients in the equations: 2R + 3B = 78 and 3R + 2B = 72.

Simply add both equations to find the combined sum: (2 + 3)R + (3 + 2)B = 78 + 72.

5 × (Red + Blue) = 150.

Divide the total sum by 5: 150 ÷ 5 = 30.

∴ The correct answer is ₹30.

Alternate Method

Let the cost of one red pen be ₹x and the cost of one blue pen be ₹y.

Given:

⇒ 2x + 3y = 78 --- (Equation 1)

⇒ 3x + 2y = 72 --- (Equation 2)

Calculations:

Adding Equation 1 and Equation 2:

⇒ (2x + 3y) + (3x + 2y) = 78 + 72

⇒ 5x + 5y = 150

Taking 5 as a common factor:

⇒ 5(x + y) = 150

Dividing both sides by 5:

⇒ x + y = 150 ÷ 5

⇒ x + y = 30

∴ The correct answer is ₹30.

Additional Information

Simultaneous Linear Equations

When equations are in the form ax + by = m and bx + ay = n, the quickest way to find (x + y) is to add them, and (x − y) is to subtract them.

Elimination Method

To find individual costs, multiply Equation 1 by 3 and Equation 2 by 2 to eliminate x, or use substitution once (x + y) is known.

Unitary Relationship

If nx + ny = K, then the cost of a single set (x + y) is always K ÷ n.

Shortcut Trick

Fourth proportion (x) = (3 × 16) ÷ 24 = 2.

Third proportion formula: b²/a = 20²/A.

Equating both: 2 = 400 ÷ A ⇒ A = 200.

∴ The correct answer is 200.

Alternate Method

Given: Numbers for fourth proportion = 24, 3, 16; Numbers for third proportion = A, 20.

Formula: Fourth proportion (d) of a, b, c is d = (b × c)/a. Third proportion (c) of a, b is c = b²/a.

⇒ Let the fourth proportion be x.

⇒ x = (3 × 16) ÷ 24

⇒ x = 48 ÷ 24 = 2

⇒ According to the question, Third proportion of A and 20 is also 2.

⇒ 2 = 20² ÷ A

⇒ 2 = 400 ÷ A

⇒ A = 400 ÷ 2 = 200

∴ The correct answer is 200.

Additional Information

Mean Proportion

The mean proportion between two numbers a and b is given by √(a × b).

Continued Proportion

If a, b, c are in continued proportion, then a/b = b/c, which implies b² = ac.

Ratio & Proportion Basics

Product of extremes = Product of means. For a : b :: c : d, then a × d = b × c.

Shortcut Trick

Mass = Density × Volume = ρ × π(R² − r²)h

Mass = 5 × (22/7) × (7² − 5²) × 42

Mass = 5 × 22 × (49 − 25) × 6 = 110 × 24 × 6 = 110 × 144

Mass = 15840 g = 15.84 kg

∴ The correct answer is 15.84 kg.

Alternate Method

Given:

Outer radius (R) = 7 cm

Inner radius (r) = 5 cm

Length (h) = 42 cm

Density (ρ) = 5 g/cm³

Formula Used:

Volume of hollow cylinder (V) = π(R² − r²)h

Mass = Volume × Density

Calculations:

⇒ Volume of metal (V) = (22/7) × (7² − 5²) × 42

⇒ V = (22/7) × (49 − 25) × 42

⇒ V = 22 × 24 × (42 ÷ 7)

⇒ V = 22 × 24 × 6 = 3168 cm³

⇒ Mass = 3168 × 5 = 15840 g

⇒ Mass in kg = 15840 ÷ 1000 = 15.84 kg

∴ The correct answer is 15.84 kg.

Additional Information

Total Surface Area of Hollow Cylinder

The total surface area includes inner and outer curved areas plus the areas of the two rings: 2πh(R + r) + 2π(R² − r²).

Curved Surface Area of Hollow Cylinder

The total curved surface area is the sum of the external and internal curved areas: 2πRh + 2πrh = 2πh(R + r).

Density-Mass Relation

Density is the mass per unit volume. For uniform materials, Mass = Volume × Density. Units must be consistent (e.g., g/cm³ with cm³ results in grams).

Shortcut Trick

Use the standard Pythagorean triplet (3, 4, 5).

Since sin θ = 3/5 = Perpendicular/Hypotenuse, the Base must be 4.

∴ cos θ = Base/Hypotenuse = 4/5.

∴ The correct answer is 4/5.

Alternate Method

Given: sin θ = 3/5

Formula Used: sin² θ + cos² θ = 1

⇒ cos² θ = 1 − sin² θ

⇒ cos² θ = 1 − (3/5)²

⇒ cos² θ = 1 − 9/25

⇒ cos² θ = (25 − 9)/25 = 16/25

⇒ cos θ = √(16/25) = 4/5 (as θ is in the 1^st quadrant, cos θ is positive)

∴ The correct answer is 4/5.

Additional Information

Trigonometric Identities

Fundamental identities include sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Quadrant Signs (ASTC Rule)

In the 1^st quadrant, All ratios are positive. In the 2^nd, only Sine/Cosec. In the 3^rd, Tan/Cot. In the 4^th, Cos/Sec.

Shortcut Trick

Identify terms: Let a = 2x and b = 3y. Given a + b = 8 and a² + b² = 32.

Find the value of ab: ab = [(a + b)² − (a² + b²)] ÷ 2 = [64 − 32] ÷ 2 = 16.

Apply sum of cubes formula: a³ + b³ = (a + b)(a² + b² − ab).

Substitute the values: 8x³ + 27y³ = 8 × (32 − 16) = 8 × 16 = 128.

∴ The correct answer is 128.

Alternate Method

Given:

2x + 3y = 8

4x² + 9y² = 32

Formula Used:

(a + b)² = a² + b² + 2ab

a³ + b³ = (a + b)³ − 3ab(a + b)

Calculations:

⇒ Let a = 2x and b = 3y

⇒ (2x + 3y)² = (2x)² + (3y)² + 2(2x)(3y)

⇒ (8)² = 32 + 2(6xy)

⇒ 64 = 32 + 12xy

⇒ 12xy = 64 − 32 = 32

⇒ 6xy = 16

⇒ Now, 8x³ + 27y³ = (2x + 3y)³ − 3(2x)(3y)(2x + 3y)

⇒ 8x³ + 27y³ = (8)³ − 3(6xy)(8)

⇒ 8x³ + 27y³ = 512 − 3(16)(8)

⇒ 8x³ + 27y³ = 512 − 384

⇒ 8x³ + 27y³ = 128

∴ The correct answer is 128.

Additional Information

Sum of Cubes Identity

The expression a³ + b³ can be factored as (a + b)(a² − ab + b²) or written as (a + b)³ − 3ab(a + b).

Difference of Cubes Identity

The expression a³ − b³ is calculated using (a − b)(a² + ab + b²) or (a − b)³ + 3ab(a − b).

Square of a Binomial

The identity (a + b)² = a² + 2ab + b² is essential for finding the product term ab when the sum and sum of squares are known.

Logic followed here is: (Positional value of the first letter of the 1^st cluster) + 3 = (Positional value of the first letter of the 2^nd cluster).

1) US - ZS →

2. AY - DY →

 

2. QO - TO →

3. SQ - VQ 

 

Therefore, the letter-cluster pair 'US - ZS' does not belong to the group.

Hence, the correct answer is "Option 1".

Given:

Total people in the row = 54

Mohit sits 22^nd from the right end.

Charu sits 15^th from the left end.

Step 1:

Find Mohit’s position from the left end.

Formula:

Position from left = Total people − Position from right + 1

= 54 − 22 + 1

= 33

So, Mohit is 33^rd from the left end.

Step 2:

Charu is 15^th from the left end.

People between Mohit and Charu:

= 33 − 15 − 1

= 17

Therefore, 17 people sit between Mohit and Charu.

Hence, the correct answer is "Option 4".

Logic followed here is: The resultant is determined by identifying the highest and lowest numbers in the given set and then summing their respective third digits.

For

The highest number 473 →

Third digit of the highest number = 3

The lowest number 125 →

Third digit of the lowest number = 5

The pattern observed is: (Third digit of the highest number) + (Third digit of the lowest number).

Similarly, let's find the resultant:

Sum of third digits = 3 + 5 = 8

Therefore, the resultant is 8.

Hence, the correct answer is "Option 4".

Logic followed here is: The given sequence is a combination of two alternating series.

Therefore, the number that should come in place of the question mark is 30.

Hence, the correct answer is "Option 1".

Given:

‘some good questions’ is coded as ‘jm ik cd’

‘many good problems’ is coded as ‘cd rc gh’

Common word analysis:

Therefore, “good” is coded as “cd”.

Hence, the correct answer is "Option 4".

Logic followed here is:

The next term in the series is determined by adding 2 to the positional values of the first and second letters, and subtracting 1 from the positional value of the third letter of the previous term in the English alphabetical order.

For

Next term = JWI

Therefore, the term in place of the question mark (?) is JWI.

Hence, the correct answer is "Option 1".

Given:

Number = 3215674

Rule:

Add 1 to each odd digit.

Subtract 1 from each even digit.

Step 1: Form the new number

 

New number formed = 4126583

Step 2: Find highest and lowest digits

Highest digit = 8

Lowest digit = 1

Difference = 8 − 1 = 7

Therefore, the required difference is 7.

Hence, the correct answer is "Option 3".

Statements:

All bananas are grapes.

No banana is a cucumber.

Venn Diagram Logic:

Conclusions:

(I) Some cucumbers are grapes: Does not follows (There is no definite relation given between cucumbers and grapes.)

(II) All grapes are cucumbers: Does not follows (Only bananas are grapes is given. It does not mean all grapes are cucumbers.)

Therefore, neither conclusion (I) nor (II) follows.

Hence, the correct answer is "Option 1".

Logic followed here is: (First letter-cluster positional values) - 2 = (Second letter-cluster positional values).

For

1. MQ - KO :

2. KO - IM :

 

3. PS - NR:

  

4. GK - EI :

  

 

Therefore, the letter-cluster pair PS - NR does not belong to the group as it subtracts 1 from the second letter instead of 2.

Hence, the correct answer is "Option 3".