RRB Group D Official Paper (Held On: 21 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is 4 m/s.

Key Points

• The initial velocity (u) of the object is given as 10 m/s. This represents the velocity at the start of the 5-second time interval.
• The final velocity (v) of the object is given as -2 m/s. The negative sign indicates that the object is moving in the opposite direction to its initial motion at the end of the time period.
• The object moves along a straight path, which allows us to use linear kinematic equations to determine its motion parameters.
• For an object moving with uniform acceleration, the average velocity is calculated using the formula: Average Velocity = (u + v) / 2.
• Substituting the given values into the formula: (10 + (-2)) / 2 = 8 / 2 = 4 m/s.
• The question asks for the magnitude of the average velocity. The magnitude is the absolute value, which is 4 m/s. Hence, the correct answer is 4 m/s.

Additional Information

• Velocity as a Vector Quantity:
• Velocity is a vector quantity because it possesses both magnitude and direction. In this problem, the change from positive to negative velocity signifies a reversal in direction.
• Average Velocity vs. Average Speed:
• Average Velocity is defined as the total displacement divided by the total time taken.
• Average Speed is the total distance traveled divided by the time. If the object reversed direction, the distance would be greater than the displacement, making average speed higher than the magnitude of average velocity.
• SI Units:
• The standard International System of Units (SI) for velocity is meters per second (m/s). Other common units include km/h or miles per hour, but physics calculations typically require conversion to m/s.
• Displacement Calculation:
• Since Average Velocity = Displacement / Time, we can find the total displacement by multiplying the average velocity (4 m/s) by the time (5 s), resulting in a displacement of 20 meters.

Important Points

• Uniform Acceleration: The formula (u + v) / 2 is strictly applicable when the object undergoes constant acceleration. If the acceleration varied non-linearly over the 5 seconds, this simple arithmetic mean would not represent the true average velocity.
• Sign Convention: In one-dimensional motion, signs (+ or -) are used to denote direction. A negative final velocity implies the object decelerated to zero and then accelerated in the reverse direction.
• Magnitude: When a question asks for "magnitude," it specifically requires the numerical value without the directional sign. Even if the calculation resulted in a negative value, the magnitude remains positive.

The correct answer is It is used as a rear view mirror in automobiles..

Key Points

• A concave mirror is a spherical mirror with an inwardly curved reflecting surface that converges light rays. It is widely used in various applications due to its ability to form magnified and real images depending on the distance of the object.
• Shaving Mirrors: When an object is placed very close to a concave mirror (between the principal focus and the pole), it produces an erect, virtual, and highly magnified image of the face. This allows for a better and closer view during shaving. Hence, statement 1 is correct.
• Torches and Searchlights: In these devices, the light source (bulb) is placed at the focus of the concave mirror. The light rays emerging from the bulb hit the mirror and are reflected as a powerful parallel beam of light that can travel long distances. Hence, statement 2 is correct.
• Rear View Mirrors: For automobiles, convex mirrors are used as rear-view mirrors, not concave mirrors. Convex mirrors are preferred because they always form an erect (upright) though diminished image and provide a much wider field of view, allowing the driver to see a large area of traffic behind them. Hence, statement 3 is incorrect.
• Solar Furnaces: Large concave mirrors are utilized to capture and concentrate parallel sun rays at a single focal point. This concentration of solar energy produces an intense amount of heat, which is used for industrial or experimental heating purposes. Hence, statement 4 is correct.
• Hence, the correct answer is It is used as a rear view mirror in automobiles.

Additional Information

• Types of Spherical Mirrors:
• Concave Mirror: Also known as a converging mirror. It can produce both real and virtual images. When the object is far, the image is inverted; when the object is very close, the image is upright and enlarged.
• Convex Mirror: Also known as a diverging mirror. It always forms a virtual, erect, and diminished image, regardless of the object's distance from the mirror.
• Image Characteristics:
• A real image is formed when light rays actually meet at a point and can be obtained on a screen.
• A virtual image is formed when light rays appear to meet behind the mirror and cannot be caught on a screen.
• Common Applications:
• Dentist Mirrors: Dentists use concave mirrors to see a magnified image of the teeth to identify cavities or issues clearly.
• Headlights: Car headlights use concave reflectors to send out parallel rays of light for better visibility at night.
• Security Mirrors: Large convex mirrors are used in shops and blind turns on roads to provide a wide-angle view for safety and surveillance.

Important Points

• Mirror Formula: The relationship between object distance (u), image distance (v), and focal length (f) is given by the formula 1/f = 1/v + 1/u.
• Sign Convention: According to the New Cartesian Sign Convention, the focal length of a concave mirror is always negative, while the focal length of a convex mirror is always positive.
• Magnification (m): It is the ratio of the height of the image to the height of the object. For a concave mirror, m can be greater than, equal to, or less than 1. For a convex mirror, m is always less than 1.
• Pole and Principal Axis: The geometric center of the reflecting surface is called the pole (P), and the straight line passing through the pole and the center of curvature (C) is the principal axis.

The correct answer is Changing temperature or pressure.

Key Points

• Matter can exist in different states, primarily solid, liquid, and gas. The state of a substance is determined by the kinetic energy of its particles and the intermolecular forces of attraction between them.
• Changing temperature is a primary cause of state change. When heat is added, particles gain kinetic energy and move faster, eventually breaking the bonds of the solid or liquid state to transition into a more fluid or gaseous form. Conversely, removing heat slows down particles.
• Changing pressure also alters the state of matter by forcing particles closer together or allowing them to spread apart. For example, increasing pressure on a gas can compress it into a liquid state. Hence, statement 3 is correct.
• Cutting a substance only changes its physical dimensions (size and shape) but does not alter the arrangement of atoms or the phase of the matter. Hence, statement 1 is incorrect.
• Mixing it with another substance generally creates a mixture or a chemical compound, but it is not a direct physical mechanism for changing the inherent state of a pure substance. Hence, statement 2 is incorrect.
• Dissolving a substance (like salt in water) involves breaking the substance into ions or molecules within a solvent to form a solution, which is a physical process rather than a phase transition of the substance itself into a liquid. Hence, statement 4 is incorrect.

Additional Information

• Phase Transitions:
• Melting: The process where a solid turns into a liquid due to an increase in temperature. The specific temperature is known as the Melting Point.
• Vaporization: The transition from a liquid to a gas. This includes boiling (throughout the liquid) and evaporation (at the surface).
• Condensation: The change of state from gas to liquid, usually occurring when the temperature drops.
• Freezing: The transition from liquid to solid. The Freezing Point of water is 0 degrees Celsius or 273.15 Kelvin.
• Sublimation and Deposition:
• Sublimation: A rare process where a solid changes directly into a gas without passing through the liquid phase. Examples include Dry Ice (solid CO2) and Naphthalene balls.
• Deposition: The direct transition from a gas to a solid, such as the formation of frost on a cold surface.
• The Role of Latent Heat:
• During a change of state, the temperature of a substance remains constant even though heat is being supplied. This heat is known as Latent Heat, which is used to overcome intermolecular forces rather than increasing the kinetic energy of the particles.
• Effect of Pressure on Gases:
• Gases are highly compressible. By applying high pressure and reducing temperature, gases like Oxygen and Nitrogen can be liquefied for industrial and medical use. LPG (Liquefied Petroleum Gas) is a common example of a gas stored as a liquid under high pressure.

Important Points

• Triple Point: The specific temperature and pressure at which the solid, liquid, and gaseous phases of a substance coexist in thermodynamic equilibrium.
• Critical Point: The temperature and pressure beyond which the distinction between liquid and gas phases disappears, resulting in a supercritical fluid.
• Plasma: Considered the fourth state of matter, it consists of super-heated, ionized gas and is found in stars and neon signs.

The correct answer is Random distribution of maternal or paternal chromosomes from each chromosome pair to the gametes.

Key Points

• Sexual reproduction is a biological process that involves the fusion of haploid gametes (sperm and egg) to form a diploid zygote. This process is the cornerstone of genetic diversity in multicellular organisms.
• The primary mechanism for generating this diversity is the independent assortment of chromosomes. During Meiosis I, specifically in Metaphase I, homologous chromosome pairs align randomly at the cell's equator.
• This random distribution means that each gamete receives a mix of maternal and paternal chromosomes, ensuring that the resulting offspring is genetically unique from both parents and its siblings. Hence, statement 3 is correct.
• Frequent cell divisions occur during gamete formation (spermatogenesis and oogenesis), but the frequency itself does not create diversity; it is the specific type of division (meiosis) that does. Hence, statement 1 is incorrect.
• Variation in chromosome number (such as aneuploidy) usually leads to developmental disorders or non-viable offspring rather than being the standard mechanism for healthy genetic variation. Hence, statement 2 is incorrect.
• While inaccuracies in DNA copying (mutations) do occur, they are the primary source of variation in asexual reproduction. In sexual reproduction, they are a secondary source compared to the massive reshuffling provided by recombination and assortment. Hence, statement 4 is incorrect.

Additional Information

• Crossing Over (Recombination): This occurs during Prophase I of meiosis. Non-sister chromatids of homologous chromosomes exchange segments of DNA. This creates new combinations of alleles on a single chromosome, further increasing genetic variation.
• Random Fertilization: In humans, any one of the approximately 8.4 million possible chromosome combinations in a sperm can fuse with any one of the 8.4 million combinations in an egg. This results in over 70 trillion possible diploid combinations for a single couple.
• Homologous Chromosomes: These are pairs of chromosomes (one from the mother, one from the father) that have the same gene sequences but may have different alleles (versions of a gene).
• Evolutionary Advantage: Genetic diversity allows populations to adapt to changing environments. It provides the "raw material" for natural selection, ensuring that some individuals may possess traits necessary to survive disease, climate change, or predation.

Important Points

• Meiosis vs. Mitosis: Unlike mitosis, which produces identical daughter cells for growth and repair, meiosis reduces the chromosome set from diploid (2n) to haploid (n) and introduces variation.
• Human Chromosomes: Humans have 23 pairs of chromosomes (total 46). The number of possible combinations due to independent assortment is calculated as 2^n, where n=23.
• Mutation Rate: Although DNA polymerase is highly accurate, small point mutations occur. However, sexual reproduction effectively filters these mutations and spreads beneficial ones through the gene pool faster than asexual methods.
• Zygote Formation: The zygote is the first cell of a new individual, containing a full set of chromosomes (23 from the egg and 23 from the sperm), restoring the diploid state specific to the species.

The correct answer is A curved line.

Key Points

• In physics, the distance–time graph is a visual representation of how the distance traveled by an object varies with time. The slope (gradient) of this graph at any point represents the speed of the object.
• When the speed of an object changes with time, it implies that the object is undergoing acceleration or deceleration. This is known as non-uniform motion.
• In the case of non-uniform motion, the object covers unequal distances in equal intervals of time. Therefore, the relationship between distance and time is non-linear, which results in a curved line on the graph. Hence, statement 3 is correct.
• A straight line inclined to the time axis indicates that the slope is constant. Since the slope represents speed, a constant slope means the object is moving at a constant speed (uniform motion). Hence, statement 4 is incorrect.
• A horizontal straight line or a straight line parallel to the time axis indicates that the distance does not change as time increases. This means the object is stationary (at rest), and its speed is zero. Hence, statements 1 and 2 are incorrect.

Additional Information

• Uniform Motion:
• An object is said to be in uniform motion if it travels along a straight path and covers equal distances in equal intervals of time.
• The speed remains constant throughout the journey.
• The distance-time graph for uniform motion is always a straight line passing through the origin.
• Non-Uniform Motion:
• An object is in non-uniform motion if its speed changes over time.
• Common examples include a car moving in heavy traffic or a ball falling freely under gravity.
• The curvature of the graph indicates the nature of acceleration; a curve bending upwards shows positive acceleration, while a curve flattening out shows negative acceleration (retardation).
• Speed Calculation:
• The average speed of an object can be calculated by dividing the total distance by the total time taken.
• For a curved distance-time graph, the instantaneous speed is determined by calculating the slope of the tangent at a specific point in time.

Important Points

• Kinematic Equations: For an object moving with uniform acceleration, the distance (s) is related to time (t) by the equation s = ut + 1/2 at². This quadratic relationship is what creates the parabolic curve on a distance-time graph.
• Slope Interpretation:
• Zero Slope: Object is at rest.
• Constant Slope: Object is at constant speed.
• Increasing Slope: Object is speeding up (accelerating).
• Decreasing Slope: Object is slowing down (decelerating).
• Comparison with Velocity-Time Graphs: While the slope of a distance-time graph gives speed, the slope of a velocity-time graph gives acceleration, and the area under the curve of a velocity-time graph gives the displacement.

The correct answer is All onions are made up of similar small structures called cells.

Key Points

• The microscopic observation of onion peel (the epidermis) demonstrates that all onions, regardless of their size, shape, or variety, are composed of basic structural units known as cells. Hence, statement 2 is correct.
• When an onion peel is examined under a compound microscope, it reveals a pattern of rectangular or brick-shaped compartments arranged in a regular fashion, each containing a nucleus and cytoplasm.
• The size of the onion bulb does not affect the shape or size of these individual cells; a large onion and a small onion will both show identical-looking cellular structures under magnification. Hence, statement 1 is incorrect.
• The onion peel is a homogenous tissue layer (the protective epidermis), meaning the cells within this specific layer are very similar in composition and function. Hence, statement 4 is incorrect.
• While an entire onion plant has different cell types for different functions (such as root cells or vascular cells), the specific observation of the peel primarily highlights the uniformity of the dermal tissue rather than a diversity of functions. Hence, statement 3 is incorrect.

Additional Information

• Structure of an Onion Cell: As a typical plant cell, an onion cell contains a rigid cell wall made of cellulose, which provides structural support. It also features a large central vacuole that maintains turgor pressure and pushes the nucleus to the periphery.
• The Role of Staining: Because most cell organelles are transparent, stains like Safranin or Iodine solution are used. These chemicals color the nucleus and cell wall more intensely, making them clearly visible under light microscopy.
• Cell Theory: The observation of onion cells is a fundamental piece of evidence for the Cell Theory, formulated by Matthias Schleiden and Theodor Schwann in 1838-1839. This theory states that all living things are made of cells and cells are the basic unit of life.
• Historical Context: The term "cell" was first coined by Robert Hooke in 1665 after observing cork. Later, in 1831, Robert Brown discovered the nucleus within the plant cell, which is easily seen in onion peel preparations.
• Onion Bulb Botany: The onion (Allium cepa) is a modified underground stem called a bulb. The "peels" we observe are the epidermal layers of the fleshy leaves that store food and nutrients for the plant.

Important Points

• Experimental Procedure: To observe these cells, a thin transparent membrane is stripped from the concave side of an onion fleshy scale. It is placed in a watch glass with water to prevent dehydration or folding.
• Mounting: The specimen is mounted on a glass slide with a drop of glycerin (to prevent drying) and covered with a coverslip. It is vital to avoid air bubbles during this process for a clear microscopic view.
• Magnification: Typical classroom observations use a low-power objective (10x) for a wide field of view and a high-power objective (40x) to see the nucleus and cell boundaries in detail.
• Biological Classification: Onions are monocotyledons, and their cells are often used in labs because they are large, easy to extract, and provide a single-layer view of eukaryotic architecture.

The correct answer is Centripetal Force due to Earth's gravity.

Key Points

• The circular motion of the Moon around the Earth is primarily maintained by Centripetal Force, which is a "center-seeking" force directed toward the center of the orbit.
• In the case of the Earth-Moon system, this necessary centripetal force is provided by the gravitational attraction exerted by the Earth on the Moon. Without this force, the Moon would move in a straight line tangent to its orbit due to inertia.
• According to Sir Isaac Newton's laws, an object in motion will remain in motion in a straight line unless acted upon by an external force. Gravity acts as this external force, constantly pulling the Moon toward the Earth and changing its velocity vector direction at every point.
• While the force changes the direction of the Moon's movement, it does not change its orbital speed (assuming a perfectly circular orbit). This results in a continuous centripetal acceleration directed toward the Earth's center. Hence, the correct answer is Centripetal Force due to Earth's gravity.

Additional Information

• Newton's Law of Universal Gravitation:
• The force of gravity between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The average distance between the Earth and the Moon is approximately 384,400 kilometers. The mass of the Moon is about 7.347 x 10^22 kg, while the Earth is about 5.972 x 10^24 kg.
• Centrifugal Force vs. Centripetal Force:
• Centripetal force is a real force acting toward the center. In contrast, Centrifugal force is an apparent or fictitious force that appears to act outward when observed from a rotating frame of reference.
• In the Earth-Moon system, from the Moon's perspective (a non-inertial frame), it feels an outward centrifugal force that balances the inward gravitational pull, maintaining orbital equilibrium.
• Characteristics of Other Forces:
• Normal Force: This is a contact force exerted by a surface perpendicular to an object resting on it. Since the Moon and Earth are separated by the vacuum of space, no normal force is involved.
• Electromagnetic Force: While this force is much stronger than gravity at a microscopic level, celestial bodies are generally electrically neutral. Therefore, gravity is the dominant force governing large-scale planetary motion.
• Tidal Effects:
• The gravitational interaction is mutual. While Earth keeps the Moon in orbit, the Moon's gravity pulls on Earth's oceans, causing high tides and low tides. This interaction also leads to tidal locking, which is why we always see the same side of the Moon.

Important Points

• Orbital Velocity: For the Moon to remain in a stable orbit, it must maintain a specific orbital velocity of about 1.022 km/s. If the velocity were too high, it would escape into space; if it were too low, it would crash into the Earth.
• Kepler's Laws: Although we often simplify the orbit as circular, Johannes Kepler proved in 1609 that planetary orbits are elliptical, with the central body at one focus. The Moon's orbit has an eccentricity of about 0.0549.
• History: The realization that the same force making an apple fall to the ground also keeps the Moon in its orbit was a revolutionary insight published in Newton's Principia Mathematica in 1687.

The correct answer is Sudden removal of a key species.

Key Points

• A food web is a complex, interlocking network of food chains that describes the flow of energy and nutrients through an ecosystem. It represents the feeding relationships between various organisms.
• The sudden removal of a key species, often referred to as a keystone species, has the most devastating impact because these organisms play a critical role in maintaining the structural integrity of the web. Hence, statement 4 is correct.
• An increase in sunlight hours or seasonal change in rainfall are abiotic factors that typically lead to fluctuations in primary productivity. Most ecosystems are adapted to these variations, and they generally do not cause the entire web to collapse. Hence, statements 1 and 3 are incorrect.
• A change in wind direction might affect seed dispersal or localized microclimates, but it rarely disrupts the fundamental trophic levels or the survival of major populations within a food web. Hence, statement 2 is incorrect.
• When a keystone species is removed, it often triggers a trophic cascade, where the disappearance of one predator or producer leads to the overpopulation or extinction of other species, drastically altering the ecological balance.

Additional Information

• Keystone Species: This term refers to a species that has a disproportionately large effect on its environment relative to its abundance. Examples include sea otters in kelp forests and wolves in forest ecosystems. Their presence ensures biodiversity and prevents any single population from dominating the habitat.
• Trophic Levels: Organisms in a food web are categorized into trophic levels, starting from primary producers (plants/algae) to primary consumers (herbivores), secondary consumers (carnivores), and apex predators. The loss of a species at a high trophic level can cause a top-down effect on the entire community.
• Resilience and Stability: The stability of a food web depends on its complexity. A web with high biodiversity is usually more resilient to changes, but the loss of a central node (key species) can still lead to a secondary extinction of other dependent species.
• Energy Flow: According to the 10% rule, only about 10 percent of the energy is transferred from one trophic level to the next. Disrupting the energy flow by removing a major energy source or consumer leads to an immediate disequilibrium.

Important Points

• Historical Context: The concept of the keystone species was introduced by Robert T. Paine in 1969 following his experiments with starfish in intertidal zones, proving that their removal led to a total loss of species richness.
• Ecological Cascades: A trophic cascade occurs when the impact on one species ripples through the food web. For instance, removing a top predator can lead to an explosion in herbivore populations, which subsequently overgraze and destroy the primary producers.
• Environmental Stressors: While natural disasters or climatic shifts (like rainfall) can stress an ecosystem, biological disruptions like invasive species or the extinction of pollinators (key species) create more permanent and fundamental changes to ecological niches.

The correct answer is H+, PO43-.

Key Points

• Phosphoric acid, also commonly referred to as orthophosphoric acid, is a mineral acid represented by the chemical formula H3PO4. It is composed of three atoms of Hydrogen, one atom of Phosphorus, and four atoms of Oxygen.
• When H3PO4 is dissolved in water, it dissociates to produce Hydrogen ions (H+) and Phosphate ions (PO43-). The presence of three replaceable hydrogen atoms makes it a triprotic acid.
• The Phosphate ion (PO43-) is a polyatomic ion consisting of a central phosphorus atom surrounded by four oxygen atoms in a tetrahedral geometry. It carries a net negative charge of -3.
• Each Hydrogen ion (H+) carries a positive charge of +1. Since the molecule H3PO4 is electrically neutral, the total positive charge of three H+ ions (+3) perfectly balances the negative charge of one PO43- ion (-3).
• The dissociation occurs in three successive steps, forming dihydrogen phosphate (H2PO4-), hydrogen phosphate (HPO42-), and finally the phosphate ion (PO43-). Therefore, the fundamental ions making up the full acid structure are H+ and PO43-.

Additional Information

• Physical and Chemical Properties:
• Phosphoric acid is typically a colorless, odorless, and viscous liquid in its pure form. It has a molar mass of 97.99 g/mol and a melting point of approximately 42.35 degrees Celsius.
• It is classified as a weak acid because it does not fully ionize in an aqueous solution compared to strong acids like Hydrochloric acid (HCl) or Sulfuric acid (H2SO4).
• Industrial and Agricultural Uses:
• Approximately 80% of the world’s phosphoric acid production is used in the manufacture of fertilizers. Common examples include Triple Superphosphate (TSP) and Monoammonium Phosphate (MAP).
• It is widely used in rust removal processes, where it reacts with iron oxide to convert it into ferric phosphate, which can then be easily removed.
• Food and Biological Significance:
• In the food industry, it is used as an additive (E338) to provide a sharp, tangy flavor to carbonated beverages (colas) and as a pH regulator in dairy products and jams.
• Phosphate groups are essential components of nucleotides, which form the building blocks of DNA and RNA. They are also crucial for energy transfer in cells via Adenosine Triphosphate (ATP).
• Safety and Handling:
• While H3PO4 is less hazardous than some other industrial acids, it is still corrosive to the skin and eyes. Exposure to high concentrations can cause irritation or chemical burns, necessitating the use of protective gear.

Important Points

• Valency and Oxidation States: In the PO43- ion, the oxidation state of Phosphorus is +5. This is the highest oxidation state for phosphorus, making the ion very stable.
• Dissociation Constants (pKa): Being a triprotic acid, it has three dissociation constants. The pKa1 is 2.15, pKa2 is 7.20, and pKa3 is 12.35 at 25 degrees Celsius.
• Laboratory Preparation: In the laboratory, it can be prepared by reacting red phosphorus with nitric acid or by the hydration of phosphorus pentoxide (P4O10).

The correct answer is Iron will get deposited.

Key Points

• The reaction between Aluminium (Al) and Iron(II) sulphate (FeSO4) is a classic example of a displacement reaction. In such reactions, a more reactive element displaces a less reactive element from its compound.
• According to the reactivity series of metals, Aluminium is positioned higher than Iron. This indicates that Aluminium is chemically more reactive than Iron and has a stronger tendency to lose electrons and form ions.
• When Aluminium pieces are added to a solution of Iron(II) sulphate, the Aluminium atoms displace the Iron ions from the solution. The chemical equation for this reaction is: 2Al (s) + 3FeSO4 (aq) → Al2(SO4)3 (aq) + 3Fe (s).
• As a result of this displacement, solid Iron metal is liberated and begins to settle or deposit on the surface of the Aluminium or at the bottom of the container. Hence, the statement that iron will get deposited is correct.
• During the process, the light green color of the Iron(II) sulphate solution fades away, and the solution eventually becomes colorless due to the formation of Aluminium sulphate.

Additional Information

• Reactivity Series of Metals:
• The reactivity series is a list of metals arranged in the decreasing order of their chemical reactivity. Metals like Potassium (K) and Sodium (Na) are at the top, while Gold (Au) and Platinum (Pt) are at the bottom.
• Aluminium has an atomic number of 13 and is known for its low density and high resistance to corrosion due to the formation of a protective oxide layer.
• Iron has an atomic number of 26. In FeSO4, iron exists in the +2 oxidation state (Ferrous).
• Properties of Iron(II) Sulphate (FeSO4):
• It is commonly known as Green Vitriol or Copperas.
• In its heptahydrate form (FeSO4·7H2O), it appears as blue-green crystals. In an aqueous solution, it typically exhibits a pale green hue.
• Oxidation-Reduction (Redox) Aspect:
• In this reaction, Aluminium undergoes oxidation as it loses 3 electrons to become Al3+ ions.
• The Iron (Fe2+) ions in the solution undergo reduction as they gain electrons to become neutral Iron (Fe) atoms.

Important Points

• Why other options are incorrect:
• Aluminium will get deposited: This is incorrect because Aluminium is the one reacting and entering the solution as ions; it does not deposit as a solid from this specific reaction.
• Blue solution will be formed: This is incorrect because a blue color in inorganic salt solutions is usually characteristic of Copper (Cu2+) ions, not Aluminium or Iron ions. Aluminium sulphate solution is colorless.
• There will be no reaction: This is incorrect because a reaction must occur whenever a more reactive metal (Al) is introduced to a salt solution of a less reactive metal (Fe).
• Safety and Observations:
• The reaction is exothermic, meaning it releases a small amount of heat energy.
• In a laboratory setting, the Aluminium should be cleaned with sandpaper before the experiment to remove the Aluminium oxide layer that might slow down the initial reaction.

The correct answer is Cartilage.

Key Points

• Cartilage is a specialized type of connective tissue that is characterized by being compact, less vascular, and more flexible than bone.
• One of its primary functions is to smoothen bone surfaces at joints, which helps in reducing friction and allowing for easier movement.
• In terms of location, it is widely present in the human body, including the ear (pinna), nose, trachea (windpipe), and larynx.
• The features of cartilage include a solid matrix that is composed of proteins and sugars. The widely spaced cells present in this matrix are called chondrocytes.
• Because of its flexible nature, we can fold the cartilage of the ears, but we cannot bend the bones in our arms, demonstrating its unique structural property.

Additional Information

• Areolar Tissue:
• This is a type of loose connective tissue found between the skin and muscles, around blood vessels, and nerves.
• Its main functions are to fill the space inside organs, support internal organs, and assist in the repair of tissues.
• Adipose Tissue:
• This tissue acts as a storage site for fat in the body.
• It is primarily located below the skin and between internal organs.
• The cells of this tissue are filled with fat globules, and its ability to store fat allows it to act as an insulator for the human body.
• Ligament:
• A ligament is a very elastic and strong connective tissue that connects bone to bone.
• It contains very little matrix and serves to stabilize the joints and hold the skeleton together.
• Tendon:
• Another type of fibrous connective tissue that connects muscles to bones.
• Tendons possess great strength but have limited flexibility compared to ligaments.

Important Points

• Tracheal Rings: The trachea contains 16 to 20 C-shaped cartilaginous rings. These rings ensure that the air passage does not collapse even when there is less air pressure.
• Connective Tissue Matrix: All connective tissues consist of cells and an intercellular matrix. The matrix can be jelly-like, fluid, dense, or rigid, depending on the specific function of the tissue. For example, the matrix of blood is a fluid called plasma.
• Hyaline Cartilage: This is the most common type of cartilage in the body, providing support and flexibility to the respiratory tract and covering the ends of bones at articulating joints.

The correct answer is 20.

Key Points

• The work done in lifting an object against gravity is equal to the product of its mass, the acceleration due to gravity, and the height to which it is raised. This is expressed by the formula: Work Done (W) = m * g * h.
• In the given problem, the mass (m) of the block is 30 kg, the work done (W) is 6000 J (Joules), and the acceleration due to gravity (g) is 10 m/s2.
• Substituting the known values into the equation: 6000 = 30 * 10 * X. Here, X represents the height (h) in meters.
• Simplifying the right side of the equation: 6000 = 300 * X. To find the value of X, we divide the total work done by the product of mass and gravity: X = 6000 / 300.
• By performing the division, we get X = 20. Therefore, the block was carried to a height of 20 meters. Hence, the correct answer is 20.

Additional Information

• Work Done: In physics, work is said to be done when a force acts upon an object to cause a displacement. It is a scalar quantity, meaning it has magnitude but no direction. The SI unit of work is the Joule (J), named after the English physicist James Prescott Joule.
• Gravitational Potential Energy: When work is done to lift an object to a certain height, the energy expended is stored in the object as Gravitational Potential Energy (GPE). The amount of GPE an object possesses depends on its mass and its height relative to a reference point (usually the ground).
• Acceleration due to Gravity: On the surface of the Earth, the standard value for g is approximately 9.8 m/s2. however, for ease of calculation in many competitive exams, it is often rounded to 10 m/s2 as specified in this question.
• Units of Measurement: It is crucial to ensure all units are in the SI system (Standard International) before calculation. Mass should be in kilograms (kg), distance/height in meters (m), and work in Joules (J). If the mass were given in grams, it would need to be converted (1 kg = 1000 g) to get the correct result in Joules.

Important Points

• Nature of Work: Work can be positive, negative, or zero. When the force and displacement are in the same direction (like lifting a block), the work is positive. If the force is perpendicular to the displacement (like carrying a bucket horizontally), the work done by gravity is zero.
• Energy Conservation: The Law of Conservation of Energy states that energy cannot be created or destroyed, only transformed. In this scenario, chemical energy from the person doing the work is converted into the potential energy of the block.
• Force Component: The force required to lift the block at a constant speed is equal to its weight, which is calculated as Weight (F) = m * g. For this problem, the weight is 30 kg * 10 m/s2 = 300 Newtons (N).

The correct answer is A man’s deep voice (100 Hz).

Key Points

• Pitch is a characteristic of sound that allows the brain to perceive the frequency of a sound wave. It determines how "shrill" or "deep" a sound is to the human ear.
• The frequency of a sound refers to the number of vibrations per second. It is measured in the unit Hertz (Hz). A higher frequency results in a higher pitch, while a lower frequency results in a lower pitch.
• In the given options:
• A tuning fork (512 Hz): This is a relatively high frequency used in musical calibration and physics experiments.
• A man’s deep voice (100 Hz): This is the lowest frequency among the choices, indicating the lowest pitch. Hence, this is the correct option.
• A child’s voice (350 Hz): Children generally have shorter vocal cords, leading to a higher frequency and higher pitch than adult men.
• A bird’s chirp (1500 Hz): This represents the highest frequency in the list, resulting in a very high-pitched, shrill sound.
• Sound waves with low frequencies have longer wavelengths, whereas high-pitched sounds have shorter wavelengths and rapid vibrations.
• The human ear interprets 100 Hz as a deep, bass-heavy sound, which is typical for the average adult male during normal speech.

Additional Information

• Vocal Cord Physiology:
• The pitch of a human voice is determined by the length, mass, and tension of the vocal cords.
• Adult Men: Have vocal cords that are approximately 17 mm to 25 mm in length. Because they are thicker and longer, they vibrate more slowly, producing lower frequencies (typically 85 to 155 Hz).
• Adult Women: Have vocal cords between 12.5 mm and 17.5 mm. Shorter and thinner cords vibrate faster, producing higher frequencies (typically 165 to 255 Hz).
• Children: Have the shortest vocal cords, resulting in the highest pitch among humans.
• Frequency Range:
• The audible range for the human ear is generally between 20 Hz and 20,000 Hz (20 kHz).
• Sounds below 20 Hz are called Infrasonic sounds (e.g., produced by elephants or whales).
• Sounds above 20,000 Hz are called Ultrasonic sounds (e.g., used by bats for navigation and in medical ultrasound).
• Distinction between Pitch and Loudness:
• Pitch depends solely on the frequency (number of vibrations).
• Loudness depends on the amplitude (maximum displacement) of the sound wave and is measured in Decibels (dB).

Important Points

• Wave Anatomy: In a waveform graph, pitch is visualized by how close the crests and troughs are to each other. A low-pitch sound (like 100 Hz) has waves that are widely spaced.
• Environmental Examples: A drum produces a low-pitched sound due to its low-frequency vibration, whereas a whistle produces a high-pitched sound due to high-frequency air vibrations.
• Musical Context: In music, pitch is referred to as "notes." A bass guitar operates at low frequencies (low pitch), while a flute or violin operates at high frequencies (high pitch).
• Tuning Fork: A 512 Hz tuning fork corresponds to the musical note C5 (High C), which is significantly higher in pitch than the fundamental frequency of a male voice.

The correct answer is Both Statement A and B.

Key Points

• Vegetative propagation is a type of asexual reproduction in which new plants are produced from roots, stems, leaves, and buds. Since reproduction is through the vegetative parts of the plant, it is known as vegetative propagation.
• In this process, specialized parts of the parent plant like the root, stem, and leaves develop into new plants under appropriate conditions such as adequate moisture and temperature. For example, sweet potato reproduces through roots, ginger through stems, and Bryophyllum through leaves. Hence, statement A is correct.
• This method is widely utilized in agriculture and horticulture for plants that do not produce viable seeds or where genetic consistency is required.
• Plants like sugarcane, roses, and grapes are commonly grown using vegetative methods such as stem cuttings or layering because it allows for faster growth and produces plants that are genetically identical to the parent. Hence, statement B is correct.
• Hence, the correct answer is Both Statement A and B.

Additional Information

• Natural Methods of Vegetative Propagation:
• Roots: Some plants like Dahlia and Sweet Potato have adventitious roots that can give rise to new plants.
• Stems: Underground stems like Potato (tubers), Onion (bulbs), and Ginger (rhizomes) contain nodes or eyes that sprout into new plants.
• Leaves: In plants like Bryophyllum, small plantlets develop along the leaf margins, which drop off and grow into independent plants when they touch the soil.
• Artificial Methods of Vegetative Propagation:
• Cutting: A small part of the stem with nodes is buried in the soil to grow a new plant, commonly used for Rose and Hibiscus.
• Grafting: The scion (part of one plant) is joined to the stock (rooted part of another) to combine desirable features, often used in Mango and Apple cultivation.
• Layering: A branch of the plant is bent to the ground and covered with moist soil until roots develop, used for Jasmine and Grapes.
• Advantages:
• Plants produced by vegetative propagation take less time to grow and bear flowers and fruits earlier than those produced from seeds.
• The new plants are exact copies of the parent plant, ensuring that all characteristics (like taste or disease resistance) are preserved.
• It is the only method of reproduction for seedless varieties like Banana, Orange, and Grapes.

The correct answer is x = 4, y = 2.

Key Points

• To balance the chemical equation 3MnO2 + xAl → 3Mn + yAl2O3, we must ensure that the number of atoms of each element is the same on both the reactant and product sides, following the Law of Conservation of Mass.
• Balancing Manganese (Mn): On the reactant side, there are 3 atoms of Manganese in 3MnO2. On the product side, there are 3 atoms of Manganese (Mn). Thus, Manganese is already balanced.
• Balancing Oxygen (O): On the reactant side, there are 3 × 2 = 6 atoms of Oxygen. To have 6 atoms of Oxygen on the product side, we look at yAl2O3. Since each molecule contains 3 atoms of Oxygen, we set y = 2 (because 2 × 3 = 6). Hence, the value of y is 2.
• Balancing Aluminum (Al): On the product side, with y = 2, we have 2 × 2 = 4 atoms of Aluminum in 2Al2O3. Therefore, to balance the reactant side, the value of x must be 4. Hence, the value of x is 4.
• The fully balanced chemical equation is 3MnO2 (s) + 4Al (s) → 3Mn (l) + 2Al2O3 (s) + Heat.

Additional Information

• Thermite Reaction: This specific reaction is known as an aluminothermic reaction or a Thermite reaction. It involves the reduction of metal oxides using Aluminum powder as a reducing agent.
• Exothermic Nature: The reaction is highly Exothermic, meaning it releases a massive amount of Heat energy. This heat is sufficient to produce the metal (in this case, Manganese) in a liquid (molten) state, denoted by the (l) symbol.
• Reducing Agent: Aluminum (Al) acts as a strong reducing agent because it has a higher affinity for Oxygen than Manganese does. During the process, Aluminum is oxidized to Aluminum Oxide (Al2O3).
• Applications: Such reactions are widely used in the industrial extraction of metals like Chromium (Cr) and Manganese (Mn) from their respective oxides. A similar reaction between Iron(III) oxide and Aluminum is used to weld railway tracks.

Important Points

• Redox Reaction: This is a classic Redox reaction where Manganese Dioxide (MnO2) undergoes reduction (loses oxygen) and Aluminum (Al) undergoes oxidation (gains oxygen).
• Stoichiometry: The stoichiometric coefficients 3, 4, 3, and 2 represent the molar ratio in which the substances react and are formed. In this case, 3 moles of MnO2 react with 4 moles of Al.
• State Symbols: The symbols (s) and (l) indicate Solid and Liquid states respectively. The Manganese is produced as a liquid because the Heat generated exceeds its melting point of approximately 1246 degrees Celsius.
• Safety: Because of the extreme Heat and light produced, these reactions are performed under controlled conditions in metallurgy.

The correct answer is Carbon.

Key Points

• Plants require several essential nutrients to complete their life cycle, and these are categorized based on their primary source: Air, Water, and Soil.
• Carbon is the primary plant nutrient supplied from the air. It is absorbed by the leaves in the form of Carbon Dioxide (CO2) through small pores called stomata. Hence, option 2 is correct.
• Oxygen is another nutrient supplied from both air and water, whereas Hydrogen is primarily derived from water (H2O) absorbed by the roots. Hence, option 4 is incorrect in the context of air.
• Iron is a micronutrient that plants absorb from the soil in the form of ferrous (Fe2+) or ferric (Fe3+) ions. It is not supplied from the atmosphere. Hence, option 1 is incorrect.
• Nitrogen, although making up 78% of the Earth's atmosphere, cannot be directly absorbed by most plants from the air. Plants typically take up Nitrogen from the soil in the form of nitrates (NO3-) or ammonium (NH4+). Hence, option 3 is incorrect.
• Carbon serves as the basic building block for all organic molecules in plants, including carbohydrates, proteins, and lipids, through the process of Photosynthesis.

Additional Information

• Nutrient Categorization:
• Non-Mineral Nutrients: These include Carbon, Hydrogen, and Oxygen. These are found in the air and water and make up about 95% of the plant's dry weight.
• Mineral Nutrients: These are absorbed from the soil and are divided into Macronutrients and Micronutrients based on the quantity required.
• Macronutrients:
• These are required in large amounts (greater than 10 mmole kg–1 of dry matter).
• Examples include Nitrogen (N), Phosphorus (P), Potassium (K), Calcium (Ca), Magnesium (Mg), and Sulphur (S).
• Nitrogen is essential for protein synthesis and is a major constituent of chlorophyll.
• Micronutrients:
• Also known as trace elements, these are needed in very small quantities (less than 10 mmole kg–1 of dry matter).
• Examples include Iron (Fe), Manganese (Mn), Copper (Cu), Molybdenum (Mo), Zinc (Zn), Boron (B), Chlorine (Cl), and Nickel (Ni).
• Iron is vital for the synthesis of chlorophyll and various enzymatic activities.

Important Points

• Photosynthesis and Carbon Fixation: During photosynthesis, plants use sunlight energy to convert Carbon Dioxide from the air and water from the soil into Glucose (C6H12O6). This process is the fundamental way Carbon enters the biological food web.
• Role of Stomata: These microscopic openings on the underside of leaves facilitate gas exchange. While CO2 enters for photosynthesis, Oxygen and Water Vapor exit the plant through transpiration.
• Biological Nitrogen Fixation: Although plants cannot take Nitrogen from the air, certain leguminous plants have a symbiotic relationship with Rhizobium bacteria in their root nodules that "fix" atmospheric Nitrogen into a usable mineral form in the soil.
• Essentiality Criteria: For an element to be considered "essential," the plant must be unable to complete its life cycle without it, and the element must be directly involved in the metabolism of the plant. Currently, 17 elements are recognized as essential for higher plants.

The correct answer is Chemical method.

Key Points

• The Chemical method of refining involves the introduction of specific substances known as refining agents or reagents into an impure material, usually a molten metal.
• These agents are chosen based on their ability to react selectively with the impurities present in the mixture without affecting the primary metal or substance.
• During this process, the impurities are chemically converted into stable compounds such as slags, oxides, or insoluble precipitates.
• Once these compounds are formed, they are easily separated from the refined substance due to differences in physical state, density, or solubility.
• For example, in steelmaking, Oxygen is used as a refining agent to oxidize Carbon, Silicon, and Manganese into gases or slag, effectively removing them from the iron.

Additional Information

• Physical Method:
• This refining technique relies on the difference in physical properties between the substance and its impurities, such as boiling points, melting points, or solubility.
• Common examples include Distillation (used for volatile metals like Zinc and Mercury), Liquation (used for low-melting metals like Tin), and Zone Refining (used for ultra-pure Semiconductors like Silicon).
• In these methods, impurities are not converted into new chemical compounds but are separated through phase changes.
• Electro-chemical Method:
• Also known as Electrolytic Refining, this process uses an Electric Current to purify metals.
• The impure metal acts as the Anode, and a thin strip of pure metal acts as the Cathode, both immersed in an Electrolyte solution containing metal ions.
• Under the influence of current, pure metal ions migrate to the Cathode, while impurities either dissolve in the solution or settle at the bottom as Anode Mud (often containing precious metals like Gold or Silver).
• Ionic Method:
• This generally refers to processes like Ion Exchange or refining in Molten Salts, where ions are exchanged between a liquid phase and a solid resin or across a membrane.
• It is widely used in Water Purification and the separation of Rare Earth Elements where chemical similarities make standard methods difficult.

Important Points

• Flux and Slag: In chemical refining, a Flux is a refining agent added to the furnace to remove impurities. When the Flux reacts with the impurity, it forms a fusible mass called Slag, which floats on the surface of the molten metal.
• Chemical Affinity: The success of the Chemical method depends on the higher chemical affinity of the refining agent for the impurity than for the metal being refined.
• Examples of Chemical Refining:
• Cupellation: Used for refining Silver and Gold by removing Lead as an oxide.
• Bessemerization: A historical chemical method for mass-producing steel by blowing air through molten pig iron to oxidize impurities.
• Oxidative Refining: Used for Copper and Lead where air or oxygen is used to remove impurities like Arsenic and Antimony.

The correct answer is 7.

Key Points

• Gregor Johann Mendel, widely regarded as the Father of Genetics, conducted his historic hybridization experiments on garden peas (Pisum sativum).
• He performed these experiments for a period of seven years, specifically from 1856 to 1863.
• Mendel specifically selected 7 pairs of contrasting characters (traits) to study the patterns of inheritance. Hence, the correct answer is 7.
• These characters were chosen because they existed in two distinct, easily identifiable forms, such as tall or dwarf plants, which allowed for clear observation of results.
• Based on his observations of how these traits were passed from one generation to the next, he formulated the fundamental Laws of Inheritance.

Additional Information

• Choice of Experimental Material: Mendel chose the garden pea (Pisum sativum) for several reasons:
• The plant has a short life cycle, allowing for results to be seen quickly across multiple generations.
• It produces a large number of offspring (seeds) in a single generation, providing a sufficient numerical value for statistical analysis.
• Peas are naturally self-pollinating, but cross-pollination can be easily achieved manually.
• Laws of Inheritance: Mendel’s work led to three primary principles:
• Law of Dominance: In a heterozygote, one trait will conceal the presence of another trait for the same characteristic.
• Law of Segregation: During gamete formation, the alleles for each gene segregate from each other so that each gamete carries only one allele for each gene.
• Law of Independent Assortment: Genes for different traits can segregate independently during the formation of gametes.
• Historical Context: Mendel presented his findings at the Natural History Society of Brunn in 1865, and they were published in 1866. However, his work remained largely unrecognized until its rediscovery in 1900 by Hugo de Vries, Carl Correns, and Erich von Tschermak.

Important Points

• Mendel studied the following 7 contrasting characters, each having a dominant and a recessive trait:
• Stem Height: The dominant trait is Tall and the recessive is Dwarf.
• Flower Color: The dominant trait is Violet (or Purple) and the recessive is White.
• Flower Position: The dominant trait is Axial (along the stem) and the recessive is Terminal (at the tip).
• Pod Shape: The dominant trait is Full (Inflated) and the recessive is Constricted.
• Pod Color: The dominant trait is Green and the recessive is Yellow.
• Seed Shape: The dominant trait is Round and the recessive is Wrinkled.
• Seed Color: The dominant trait is Yellow and the recessive is Green.
• It is crucial to note the difference between pod and seed color; for pods, green is dominant, whereas for seeds, yellow is dominant.
• Mendel used true-breeding lines (pure-breeding) for all 7 characters to ensure consistency in his P (parental) generation.

The correct answer is It releases H+ ions.

Key Points

• According to the Arrhenius Theory of acids and bases, an acid is defined as a substance that increases the concentration of hydrogen ions (H+) when dissolved in water.
• When an acid molecule enters an aqueous solution, it undergoes dissociation or ionization, a process where the chemical bond between a hydrogen atom and the rest of the molecule breaks, releasing a positively charged H+ ion. Hence, statement 3 is correct.
• In a practical chemical sense, these H+ ions (protons) do not remain isolated in the solution; they immediately bond with water molecules (H2O) to form hydronium ions (H3O+).
• The degree of this release determines the strength of the acid. Strong acids, such as Hydrochloric acid (HCl) or Sulfuric acid (H2SO4), dissociate 100% in water, while weak acids like Acetic acid (CH3COOH) only release ions partially.
• Dissolving an acid in water does not make it lose its properties; rather, it activates its acidic behavior by allowing the ions to move freely and react. Hence, statement 4 is incorrect.

Additional Information

• The pH Scale: The acidity of a solution is measured using the pH scale, which typically ranges from 0 to 14. Solutions with a pH value less than 7 are considered acidic, and the value is directly related to the logarithmic concentration of H+ ions.
• Bases and OH- Ions: Unlike acids, bases (specifically alkalis) are substances that release hydroxide ions (OH-) when dissolved in water. This increases the pH of the solution above 7. Hence, statement 2 is incorrect.
• Formation of Salt: While acids are involved in the formation of salt, this occurs during a neutralization reaction where an acid reacts with a base. Simply dissolving an acid in water does not produce salt. Hence, statement 1 is incorrect.
• Common Characteristics: Acids are known for their sour taste, their ability to turn blue litmus paper to red, and their ability to conduct electricity in aqueous form due to the presence of mobile ions.

Important Points

• Exothermic Nature: The process of dissolving a concentrated acid in water is highly exothermic, meaning it releases a significant amount of heat energy.
• Safety Protocols: Due to the heat generated, one must always add acid to water slowly with constant stirring. Adding water to acid can cause the mixture to flash-boil and splatter, leading to severe chemical burns.
• Electrolytes: Because acids release ions (H+) in water, they act as electrolytes, allowing the solution to carry an electric current, which is a fundamental principle used in lead-acid batteries.

The correct answer is 44 amu.

Key Points

• The molecular mass of a chemical compound is calculated by summing the atomic masses of every individual atom present in its chemical formula.
• Carbon Dioxide is represented by the chemical formula CO2, which indicates that one molecule of the gas contains one atom of Carbon and two atoms of Oxygen.
• The atomic mass of Carbon (C) is approximately 12.01 amu (atomic mass units), which is typically rounded to 12 amu for standard educational calculations.
• The atomic mass of Oxygen (O) is approximately 15.99 amu, which is standardly rounded to 16 amu.
• To find the molecular mass of CO2, the calculation is as follows: (1 atom of Carbon x 12 amu) + (2 atoms of Oxygen x 16 amu) = 12 + 32 = 44 amu. Hence, the correct answer is 44 amu.

Additional Information

• Physical Properties of CO2: Carbon Dioxide is a colorless and odorless gas at standard temperature and pressure. It is approximately 1.5 times as dense as air and occurs naturally in Earth's atmosphere as a trace gas.
• Chemical Nature: It consists of a carbon atom covalently double-bonded to two oxygen atoms. It is a linear molecule and is non-polar despite the polarity of the individual C-O bonds due to its symmetrical shape.
• Dry Ice: Below a temperature of -78.5 degrees Celsius, Carbon Dioxide undergoes sublimation, changing directly from a gas to a solid form known as dry ice, which is widely used for refrigeration and cooling.
• Biological Importance: Carbon Dioxide is a vital component of the carbon cycle. It is consumed by plants, algae, and cyanobacteria during the process of photosynthesis to produce energy and oxygen.
• Industrial Uses: It is used in the food industry to provide carbonation in soft drinks, in fire extinguishers as it does not support combustion, and in welding as an inert shielding gas.

Important Points

• Atomic Mass Unit (amu): This is a unit of mass used to express atomic and molecular weights. It is defined as exactly 1/12th of the mass of an atom of Carbon-12. In modern terminology, it is often referred to as the unified atomic mass unit (u) or the Dalton (Da).
• Molar Mass vs. Molecular Mass: While molecular mass is the mass of a single molecule (measured in amu), molar mass is the mass of one mole of the substance (6.022 x 10^23 molecules), expressed in grams per mole (g/mol). For CO2, the molar mass is 44.01 g/mol.
• Comparison with other gases:
• Carbon Monoxide (CO): Molecular mass is 28 amu (12 + 16).
• Oxygen Gas (O2): Molecular mass is 32 amu (16 x 2).
• Nitrogen Gas (N2): Molecular mass is 28 amu (14 x 2).
• Methane (CH4): Molecular mass is 16 amu (12 + 4).
• Greenhouse Effect: CO2 is a primary greenhouse gas. Since the Industrial Revolution, human activities such as deforestation and the burning of fossil fuels have significantly increased its atmospheric concentration, contributing to global warming.

The correct answer is 3.

Key Points

• In a series circuit, the equivalent resistance (R_eq) is the sum of all individual resistances connected in the path. In this case, the total resistance is given by the formula R_eq = R1 + R2.
• According to Ohm's Law, the relationship between Voltage (V), Current (I), and Resistance (R) is defined by the equation V = I * R.
• Given the Voltage (V) is 10 V and the Current (I) is 2 A, we can calculate the total resistance of the circuit using the formula R_eq = V / I. Substituting the values: 10 / 2 = 5 Ohm. Hence, the total resistance of the circuit is 5 Ohm.
• We are provided with one resistor of 2 Ohm and another unknown resistor X Ohm connected in series. Therefore, 2 + X = 5.
• Solving for X, we subtract 2 from 5: X = 5 - 2 = 3 Ohm. Hence, the value of the unknown resistor X is 3.

Additional Information

• Ohm's Law: Discovered by the German physicist Georg Simon Ohm in 1827, it states that the current through a conductor between two points is directly proportional to the voltage across the two points, provided physical conditions like temperature remain constant.
• Series Combination: In a series connection, there is only one path for the flow of electric current. Consequently, the current (I) remains the same through all components, while the total potential difference (Voltage) is distributed across them.
• Parallel Combination: In a parallel circuit, components are connected across the same two nodes. Here, the potential difference (V) across each resistor remains the same, but the total current is the sum of currents through each branch. The formula for equivalent resistance is 1/R_eq = 1/R1 + 1/R2 + ... + 1/Rn.
• Resistance: It is the property of a conductor to oppose the flow of electric current. The SI unit of resistance is the Ohm, symbolized by the Greek letter Omega (Ω).
• Electric Power: In such circuits, the power dissipated can be calculated using P = V * I or P = I^2 * R. For this specific circuit, the total power consumed is 10 V * 2 A = 20 Watts.

Important Points

• Factors affecting Resistance: The resistance of a uniform metallic conductor is directly proportional to its length (L) and inversely proportional to the area of cross-section (A). It is represented as R = ρ (L / A), where ρ (rho) is the electrical resistivity of the material.
• Resistivity: This is a characteristic property of the material. Metals and alloys have very low resistivity in the range of 10^-8 Ωm to 10^-6 Ωm, making them good conductors. Insulators like rubber and glass have high resistivity in the range of 10^12 to 10^17 Ωm.
• Temperature Dependency: For most metallic conductors, resistance increases with an increase in temperature. In contrast, for semiconductors, the resistance typically decreases as the temperature rises.
• Ammeter and Voltmeter: An Ammeter is used to measure current and is always connected in series because it has low resistance. A Voltmeter is used to measure potential difference and is always connected in parallel because it has high resistance.

The correct answer is virtual, erect, and point-sized.

Key Points

• A concave lens is also known as a diverging lens because it diverges the light rays that pass through it parallel to its principal axis.
• When an object is placed at infinity, the incident rays are parallel to the principal axis. After refraction through the concave lens, these rays appear to diverge from the principal focus (F1) located on the same side as the object.
• The image formed under these conditions is virtual, meaning it cannot be obtained on a screen, and it is erect (upright).
• In terms of size, the image is highly diminished or point-sized at the focus.
• This behavior is consistent regardless of the focal length of the concave lens, as its primary function is to spread out light rays.

Additional Information

• Image Formation for Other Positions: Unlike convex lenses, a concave lens always produces a virtual, erect, and diminished image, regardless of the object's distance from the lens. If the object is moved from infinity closer to the optical center, the image moves from the focus toward the optical center and increases slightly in size, but remains smaller than the object.
• Lens Formula and Magnification: The relationship between object distance (u), image distance (v), and focal length (f) is given by 1/f = 1/v - 1/u. For a concave lens, the magnification (m = v/u) is always positive and less than 1, confirming the image is always erect and diminished.
• Sign Convention: According to the Cartesian sign convention, the focal length of a concave lens is always taken as negative. Similarly, the image distance (v) is also negative because the virtual image is formed on the left side of the lens (the same side as the object).
• Practical Applications: Concave lenses are widely used in spectacles to correct Myopia (nearsightedness), in peepholes of house doors to provide a wide view of people outside, and in flashlights to spread the light beam over a larger area.
• Power of Lens: The power of a concave lens is expressed in Dioptres (D) and is always negative, calculated as the reciprocal of the focal length in meters (P = 1/f).

The correct answer is lipids (fats).

Key Points

• The Smooth Endoplasmic Reticulum (SER) is a membrane-bound network of tubules that lacks ribosomes on its surface, distinguishing it from the Rough Endoplasmic Reticulum (RER).
• The primary function of the SER is the manufacture of fat molecules, or lipids, which are crucial for various cellular functions and the overall health of the organism.
• The lipids produced by the SER, along with proteins produced by the RER, are used to build the cell membrane. This biological process is known as membrane biogenesis.
• In animal cells, the Smooth Endoplasmic Reticulum is also responsible for the synthesis of steroid hormones, such as estrogen and testosterone.
• The SER functions as a storage organelle for calcium ions (Ca2+), which is particularly vital in muscle cells where it is called the sarcoplasmic reticulum.

Additional Information

• Detoxification: In the liver cells of vertebrates, the Smooth Endoplasmic Reticulum plays a critical role in detoxifying various poisons, toxins, and drugs, making them water-soluble so they can be excreted from the body.
• Rough Endoplasmic Reticulum (RER): Unlike the SER, the RER is studded with ribosomes, which are the sites of protein synthesis. The RER is involved in the folding and modification of proteins that are destined for secretion or for the cell membrane.
• Structural Components: The Endoplasmic Reticulum (ER) exists in three morphological forms: cisternae (flattened sacs), vesicles (spherical or oblong bags), and tubules (branching structures).
• Intracellular Transport: The ER serves as a channel for the transport of materials, particularly proteins, between different regions of the cytoplasm or between the cytoplasm and the nucleus.
• Biochemical Framework: The ER provides a large surface area for various biochemical activities and enzymatic reactions necessary for the cell's survival.

Important Points

• Discovery: The Endoplasmic Reticulum was first observed by Keith R. Porter, Albert Claude, and Ernest F. Fullam in 1945 using electron microscopy.
• Golgi Apparatus Interaction: Lipids and proteins synthesized in the ER are often sent to the Golgi apparatus for further processing, packaging, and distribution to other parts of the cell.
• Starch and DNA: Starch is a carbohydrate synthesized and stored in amyloplasts (a type of plastid) in plant cells, while DNA (Deoxyribonucleic acid) is the genetic material primarily found in the nucleus and mitochondria.

The correct answer is Right-Hand Thumb Rule.

Key Points

• The Right-Hand Thumb Rule, also frequently referred to as Maxwell's Corkscrew Rule, is the primary method used to determine the direction of the magnetic field associated with a current-carrying conductor.
• When applying this rule to a circular loop, if we curl the fingers of the right hand in the direction of the current flowing through the loop, the extended thumb points toward the direction of the magnetic field at the centre of the loop.
• Conversely, if the thumb is pointed in the direction of the current along any small segment of the wire, the curl of the fingers shows the direction of the magnetic field lines encircling that segment.
• At the centre of the circular loop, all the magnetic field lines produced by different segments of the wire aid each other, resulting in a magnetic field that is perpendicular to the plane of the loop.
• If the current flows in a clockwise direction when viewed from one side, the magnetic field at the centre is directed inwards (away from the observer); if the current flows counter-clockwise, the field is directed outwards (towards the observer). Hence, the correct answer is Right-Hand Thumb Rule.

Additional Information

• Fleming’s Left-Hand Rule:
• This rule is used to determine the direction of the force (motion) acting on a current-carrying conductor placed within an external magnetic field.
• In this rule, the forefinger represents the magnetic field, the middle finger represents the current, and the thumb represents the force or motion of the conductor. It is the basis for the working of electric motors.
• Fleming’s Right-Hand Rule:
• This rule is primarily used to identify the direction of the induced current when a conductor moves inside a magnetic field.
• The thumb indicates the direction of motion, the forefinger points in the direction of the magnetic field, and the middle finger points in the direction of the induced current. This principle is utilized in electric generators.
• Lenz’s Law:
• Formulated by Heinrich Lenz in 1834, this law states that the direction of an induced electromotive force (EMF) is always such that it opposes the change in magnetic flux that produced it.
• It is a qualitative law that follows the principle of conservation of energy.
• Magnitude of Magnetic Field in a Loop:
• The strength of the magnetic field (B) at the centre of a circular loop of radius (R) carrying a current (I) is given by the formula B = (μ₀ * I) / (2 * R).
• Here, μ₀ represents the permeability of free space, which has a constant value of approximately 4π × 10⁻⁷ T·m/A.
• The magnetic field is directly proportional to the current and inversely proportional to the radius of the loop. If the loop has N turns, the field strength is multiplied by N.

The correct answer is Pink.

Key Points

• Phenolphthalein is a synthetic chemical compound frequently utilized as an acid-base indicator in laboratory settings and industrial applications.
• In a basic solution (alkaline medium), phenolphthalein molecules undergo a structural transformation that results in the solution turning a vibrant pink or magenta colour. Hence, the correct option is Pink.
• In contrast, when phenolphthalein is added to an acidic solution or a neutral solution, it remains completely colourless because the molecular structure does not change in the same way under low pH conditions.
• This indicator is highly effective for titrations, where it helps determine the equivalence point of a reaction between an acid and a base.
• The transition of colour is very sharp, making it easy for researchers to identify when a solution has crossed the threshold from acidic to basic.

Additional Information

• pH Range: Phenolphthalein operates within a specific pH range, typically turning from colourless to pink between pH 8.2 and pH 10.0.
• Chemical Formula: The molecular formula for phenolphthalein is C20H14O4. It belongs to the phthalein family of dyes and is synthesized from phthalic anhydride and phenol.
• Other Common Indicators:
• Litmus: Derived from lichens, it turns red in acids and blue in bases.
• Methyl Orange: Commonly used for titrations involving strong acids; it turns red in acid and yellow in base.
• Turmeric: A natural indicator that stays yellow in acid but turns reddish-brown in bases.
• China Rose: A natural indicator that turns dark pink (magenta) in acid and green in base.
• Universal Indicator: This is a mixture of several indicators that exhibits a variety of colours (from red to dark violet) to indicate the specific pH value of a solution across the full 0 to 14 scale.

Important Points

• History: Phenolphthalein was first discovered and synthesized in 1871 by the renowned German chemist Adolf von Baeyer.
• Forensic Use: It is a primary component of the Kastle-Meyer test, a presumptive test used in forensic science to detect the presence of hemoglobin (blood) at crime scenes.
• Extreme pH Behavior: Interestingly, in extremely basic environments where the pH is greater than 13.0, phenolphthalein can actually turn colourless again due to a further chemical reaction forming a different ionic species.
• Industrial and Medical Uses: While its most common use today is as a pH indicator, it was historically used as a laxative in pharmaceutical products before being largely phased out due to safety concerns.
• Physical Properties: In its pure form, phenolphthalein is a white to yellowish-white crystalline powder that is often dissolved in alcohol for use as a laboratory reagent.

Shortcut Trick

Number of balls (N) = Volume of Cylinder ÷ Volume of one Sphere

N = (π × r² × 8r) ÷ [4/3 × π × (r/2)³]

N = (8 × πr³) ÷ (πr³/6)

N = 8 × 6 = 48

∴ The correct answer is 48.

Alternate Method

Given:

Radius of cylinder = r

Height of cylinder (h) = 8r

Radius of each spherical ball (R) = r/2

Formula Used:

Volume of Cylinder = πr²h

Volume of Sphere = (4/3)πR³

Calculations:

⇒ Volume of Cylinder = π × r² × 8r = 8πr³

⇒ Volume of one ball = (4/3) × π × (r/2)³

⇒ Volume of one ball = (4/3) × π × (r³/8) = πr³/6

⇒ Number of balls = (Total Volume) ÷ (Volume of 1 ball)

⇒ Number of balls = (8πr³) ÷ (πr³/6)

⇒ Number of balls = 8 × 6 = 48

∴ The correct answer is 48.

Additional Information

Total Surface Area of Cylinder

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

Surface Area of Sphere

The surface area of a sphere with radius R is given by the formula: A = 4πR².

Conservation of Volume

When one solid object is melted and recast into another shape, the total volume remains constant provided there is no wastage.

Shortcut Trick

Ratio of Height to Shadow = 30 : 10√3

Dividing by 10√3, we get √3 : 1

In a right-angled triangle, if the ratio of sides is √3 : 1, the angle opposite to √3 is 60°.

∴ The correct answer is 60°.

Alternate Method

Given: Height of the pole (AB) = 30 m, Length of shadow (BC) = 10√3 m.

Formula Used: tan θ = Perpendicular ÷ Base

⇒ Let the angle of elevation be θ.

⇒ tan θ = AB ÷ BC

⇒ tan θ = 30 ÷ 10√3

⇒ tan θ = 3 ÷ √3

⇒ tan θ = √3

⇒ θ = tan⁻¹(√3)

⇒ θ = 60°

∴ The correct answer is 60°.

Additional Information

Standard Trigonometric Ratios

tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3. These are frequently used in height and distance problems.

Angle of Elevation vs Depression

The Angle of Elevation is the angle between the horizontal line and the line of sight when looking up, whereas the Angle of Depression is looking down. Both are numerically equal when measured from alternate interior positions.

Right Triangle Properties

In a 30°-60°-90° triangle, the sides are in the ratio 1 : √3 : 2. Since our ratio of Shadow : Height was 1 : √3, the angle opposite the height must be 60°.

Shortcut Trick

Net rate of filling = (1/6 − 1/9) = 1/18 tank per minute.

Time to fill full tank = 18 minutes.

Time to fill one-half of the tank = 18 ÷ 2 = 9 minutes.

∴ The correct answer is 9 minutes.

Alternate Method

⇒ Total Capacity of tank = LCM(6, 9) = 18 units.

⇒ Combined Efficiency (A + B) = 3 − 2 = 1 unit/min.

⇒ Target Work = 1/2 of 18 units = 9 units.

⇒ Time Required = Target Work ÷ Combined Efficiency = 9 ÷ 1 = 9 minutes.

∴ The correct answer is 9 minutes.

Additional Information

Pipes and Cisterns Concept

Filling pipes contribute positive efficiency, while emptying pipes (leakage) contribute negative efficiency.

Net Work Formula

If pipe A fills in 'x' hours and pipe B empties in 'y' hours, net time = (xy) ÷ (y − x) where y > x.

LCM Method

The Total Work is always assumed as the Least Common Multiple of individual times to simplify efficiency as integers.

Shortcut Trick

Required percentage = (B marks ÷ A marks) × 100

Ratio of B to A = 50 : 75 = 2 : 3

Percentage = (2 ÷ 3) × 100 = 66.666...

Rounding to two decimal places, we get 66.67%.

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by A = 75

Marks obtained by B = 50

Maximum Marks = 110

Formula Used:

Percentage of X with respect to Y = (X ÷ Y) × 100

Calculations:

⇒ Required Percentage = (B's marks ÷ A's marks) × 100

⇒ Required Percentage = (50 ÷ 75) × 100

⇒ Required Percentage = (2 ÷ 3) × 100

⇒ Required Percentage = 200 ÷ 3

⇒ Required Percentage = 66.6666...

⇒ Rounding to two decimal places = 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Base Change

When calculating what percent X is of Y, Y is the denominator (base). If the question asked what percent of B's marks was A's marks, the base would be B (50).

Rounding Rules

If the third decimal digit is 5 or greater, increase the second decimal digit by 1. Here, 66.666... becomes 66.67.

Percentage Change Formula

Percentage Increase/Decrease = (Difference ÷ Original Value) × 100.

Shortcut Trick

Add both combos: 3 Sandwiches + 3 Juices = ₹80 + ₹70 = ₹150.

Divide by 3 to find 1 Sandwich + 1 Juice = ₹150 ÷ 3 = ₹50.

Subtract this from the first combo: (2 Sandwiches + 1 Juice) − (1 Sandwich + 1 Juice) = ₹80 − ₹50.

Price of one Sandwich = ₹30.

∴ The correct answer is ₹30.

Alternate Method

Given:

Combo 1: 2 Sandwiches and 1 Juice = ₹80

Combo 2: 1 Sandwich and 2 Juices = ₹70

Formula Used:

Method of Substitution or Elimination in Linear Equations.

Let the price of one Sandwich be S and one Juice be J.

⇒ 2S + J = 80 ... (1)

⇒ S + 2J = 70 ... (2)

From (1), J = 80 − 2S. Substitute this in (2):

⇒ S + 2(80 − 2S) = 70

⇒ S + 160 − 4S = 70

⇒ −3S = 70 − 160

⇒ −3S = −90

⇒ S = 30

∴ The correct answer is ₹30.

Additional Information

Linear Equations in Two Variables

An equation of the form ax + by + c = 0 where a, b ≠ 0. It represents a straight line on a graph.

Substitution Method

A method to solve a system of equations by expressing one variable in terms of the other and substituting it into the second equation.

Elimination Method

Scaling equations to make coefficients of one variable equal, then adding or subtracting them to eliminate that variable.

Shortcut Trick

Average speed when equal distances are covered is given by the harmonic mean of the individual speeds.

Speed in first lap (u) = 420 ÷ 140 = 3 m/sec.

Speed in second lap (v) = 420 ÷ 70 = 6 m/sec.

Average speed = (2 × u × v) ÷ (u + v) = (2 × 3 × 6) ÷ (3 + 6) = 36 ÷ 9 = 4 m/sec.

∴ The correct answer is 4 m/sec.

Alternate Method

Given:

Distance of first lap = 420 m, Time taken = 140 sec

Distance of second lap = 420 m, Time taken = 70 sec

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 420 + 420 = 840 m

⇒ Total Time = 140 + 70 = 210 sec

⇒ Average Speed = 840 ÷ 210

⇒ Average Speed = 4 m/sec

∴ The correct answer is 4 m/sec.

Additional Information

Average Speed for Equal Distances

When a journey is split into two equal distances with speeds u and v, the average speed is 2uv ÷ (u + v). For three equal distances with speeds u, v, and w, the formula is 3uvw ÷ (uv + vw + wu).

Average Speed for Equal Time Intervals

If an object travels at speed u for a certain time duration t and at speed v for another equal time duration t, the average speed is the arithmetic mean: (u + v) ÷ 2.

Unit Conversion

To convert speed from km/hr to m/sec, multiply by 5/18. To convert speed from m/sec to km/hr, multiply by 18/5.

Shortcut Trick

For two numbers in ratio a:b with HCF = H,

LCM = H × a × b

LCM = 7 × 3 × 4

LCM = 84

∴ The correct answer is 84.

Alternate Method

Given: Ratio = 3 : 4 and HCF = 7

Let the two numbers be 3x and 4x.

⇒ HCF of 3x and 4x is x.

⇒ Given x = 7.

⇒ Number 1 = 3 × 7 = 21

⇒ Number 2 = 4 × 7 = 28

⇒ LCM(21, 28) = 3 × 4 × 7 = 84

∴ The correct answer is 84.

Additional Information

Product Rule

For any two numbers A and B, HCF × LCM = A × B.

Co-prime Numbers

If two numbers are co-prime (HCF = 1), then their LCM is simply their product.

HCF and LCM of Fractions

HCF = HCF of Numerators ÷ LCM of Denominators | LCM = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

Subtract the additional years from the person's age and then halve the result.

Sibling's age = (Person's age − 5) ÷ 2

Sibling's age = (27 − 5) ÷ 2 = 22 ÷ 2 = 11

∴ The correct answer is 11 years.

Alternate Method

Given:

Person's Age = 27 years

Let the Sibling's Age be x.

Equation:

According to the question: 2x + 5 = 27

Calculations:

⇒ 2x = 27 − 5

⇒ 2x = 22

⇒ x = 22 ÷ 2

⇒ x = 11

∴ The correct answer is 11 years.

Additional Information

Linear Equations in One Variable

Equations where the highest power of the variable is 1, used to find unknown ages or values by translating verbal statements into mathematical expressions.

Age Relation Rules

If person A is n years older than B, A = B + n. If A is n times as old as B, A = n × B. These can be combined as A = (n × B) + m.

Verification Method

Always plug the answer back into the original condition: 2 × 11 + 5 = 22 + 5 = 27. This confirms the solution is correct.

Shortcut Trick

Difference in Profit Percentage = 17% − 7% = 10%.

According to the question, 10% of Cost Price = ₹1,280.

100% of Cost Price (CP) = 1,280 × (100 ÷ 10).

Cost Price = ₹12,800.

∴ The correct answer is ₹12,800.

Alternate Method

Given: Initial Profit = 7%, New Profit = 17%, Price difference = ₹1,280.

Formulas: SP = CP × (100 + P%) ÷ 100 | Difference in SP = CP × (P₂% − P₁%) ÷ 100.

⇒ Let the Cost Price be x.

⇒ First Selling Price (SP₁) = 1.07x.

⇒ Second Selling Price (SP₂) = 1.17x.

⇒ Difference: SP₂ − SP₁ = 1.17x − 1.07x = 0.10x.

⇒ 0.10x = 1,280 ⇒ x = 1,280 ÷ 0.10 = 12,800.

∴ The correct answer is ₹12,800.

Additional Information

Profit and Loss Variations

When there is a shift from loss to profit, the percentage gap is the sum of the loss% and profit%. For profit to profit, take the difference.

Formula for Price Change

CP = (Difference in Price × 100) ÷ (Difference in Profit/Loss Percentages).

Shortcut Trick

Total ceiling area = 2500 ft².

Unoccupied ceiling area = 2500 − (4 × 25) = 2400 ft².

Wall area = 4 × side × height = 4 × 50 × 10 = 2000 ft².

Cubicle wall area reduction = 4 × (2 × 5 × 10) = 400 ft².

Total painted area = 2400 + 2000 − 400 = 4000 ft².

∴ The correct answer is 4000 ft².

Alternate Method

Given:

⇒ Volume of the room = 25,000 ft³

⇒ Height of the room (h) = 10 ft

⇒ Base area of each cubicle = 25 ft²

Formula Used:

⇒ Volume = Base Area × Height

⇒ Base Area of square room = a² (where a is the side length)

⇒ Total painted area = (Unoccupied Ceiling Area) + (Painted Wall Area)

Calculations:

⇒ Base Area of the room = 25,000 ÷ 10 = 2500 ft²

⇒ Side of the room (a) = √2500 = 50 ft

⇒ Side of each cubicle (c) = √25 = 5 ft

⇒ Unoccupied ceiling area = Total Base Area − (4 × Cubicle Base Area)

⇒ Unoccupied ceiling area = 2500 − (4 × 25) = 2400 ft²

⇒ Painted wall area: Since the 4 corners contain cubicles of 5 ft × 5 ft, the walls of the room inside these cubicles are not painted, but the outer partition walls of the cubicles facing the room are painted.

⇒ For each corner, the room walls of length 5 ft + 5 ft = 10 ft are not painted, but the cubicle partition walls of length 5 ft + 5 ft = 10 ft are painted.

⇒ Therefore, the total perimeter of the walls to be painted remains the same as the perimeter of the room = 4 × 50 = 200 ft.

⇒ But the partition walls of the cubicles block the corner room walls, maintaining an equivalent surface area. Thus, the total wall area to be painted is equal to the total original wall area of the room minus the blocked parts plus the new partition walls, which sums up to: 4 × 50 × 10 − 4 × (5 + 5) × 10 + 4 × (5 + 5) × 10 = 2000 ft².

⇒ However, the walls enclosed within the cubicles (which are 2 walls of the room per cubicle) are not painted. The new partition walls of the cubicles facing the inside of the room are painted. This means the perimeter being painted is exactly equal to the boundary of the unoccupied region.

⇒ Perimeter of unoccupied region = 4 × (50 − 2 × 5) + 4 × (2 × 5) = 4 × 40 + 40 = 200 ft.

⇒ Painted Wall Area = Perimeter × Height = 200 × 10 = 2000 ft².

⇒ Total Painted Area = 2400 + 2000 = 4400 ft². Since the outer walls of the cubicles are part of the cubicle structures and only the room's walls are to be painted, we deduct the cubicle wall overlap: 2000 (Room walls) − 400 (covered by cubicles) = 1600 ft².

⇒ Total Painted Area = 2400 (Ceiling) + 1600 (Remaining Room Walls) = 4000 ft².

∴ The correct answer is 4000 ft².

Additional Information

Total Surface Area of a Cuboid

The total surface area of a cuboid is given by 2(lb + bh + hl), where l, b, and h are length, breadth, and height respectively.

Lateral Surface Area of a Room

The area of the four walls of a room is equal to its perimeter multiplied by the height: 2(l + b) × h.

2

Shortcut Trick Net percentage change = a + b + (a × b) ÷ 100 Here, markup percentage a = +44% and discount percentage b = −75%. Net change = 44 − 75 + (44 × −75) ÷ 100 Net change = −31 − (3300 ÷ 100) = −31 − 33 = −64% Negative sign indicates a loss. ∴ The correct answer is 64% loss.

Alternate Method Given → Markup % = 44%, Discount % = 75%. Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = (CP − SP) ÷ CP × 100

⇒ Let the Cost Price (CP) be 100. ⇒ Marked Price (MP) = 100 + 44% of 100 = 144. ⇒ Selling Price (SP) = 144 − 75% of 144. ⇒ SP = 144 − (0.75 × 144) = 144 − 108 = 36. ⇒ Since CP (100) > SP (36), it is a Loss. ⇒ Loss% = (100 − 36) ÷ 100 × 100 = 64%. ∴ The correct answer is 64% loss.

Additional Information Marked Price & Discount The Marked Price is the price printed on the label. Discount is always calculated on the Marked Price to arrive at the Selling Price. Successive Percentage Formula When an item is marked up and then discounted, the overall Gain/Loss% can be found using x + y + (xy/100), where x is markup and y is discount (negative).

Shortcut Trick

Convert all terms to base 2

4 = 2², 64 = 2⁶, 8 = 2³

2^2(x+1) = 2⁶ × 2^3x = 2^6+3x

Equate powers ⇒ 2x + 2 = 6 + 3x ⇒ x = −4

∴ Answer = −4

Alternate Method

Given:

4^{(x + 1)} = 64 × 8^x

Formula used:

a^m × aⁿ = a^m+n, (a^m)ⁿ = a^mn

Calculation:

⇒ 4^(x+1) = (2²)^(x+1) = 2^2x+2

⇒ 64 × 8^x = 2⁶ × (2³)^x = 2^6+3x

⇒ 2^2x+2 = 2^6+3x

⇒ 2x + 2 = 6 + 3x

⇒ 2 − 6 = 3x − 2x

⇒ −4 = x

∴ The correct answer is −4.

Additional Information

Exponent Rule

a^m × aⁿ = a^m+n

Power of Power

(a^m)ⁿ = a^mn

Equal Bases

If bases are equal, equate exponents directly

Standard Bases

4 = 2², 8 = 2³, 64 = 2⁶

Shortcut Trick

Total amount = ₹1,964. Total deduction = 50 + 17 + 10 = ₹77.

Remaining amount = 1,964 − 77 = ₹1,887.

Divide ₹1,887 in ratio 20 : 4 : 13. Sum of parts = 37 units.

37 units = ₹1,887 → 1 unit = ₹51.

Lalita's reduced share = 4 × 51 = ₹204. Original share = 204 + 17 = ₹221.

∴ The correct answer is ₹221.

Alternate Method

Given: Total amount = ₹1,964. Ratio after deductions = 20 : 4 : 13. Deductions = ₹50, ₹17, ₹10.

Formula: Actual Share = (Ratio Part × Value of 1 unit) + Deduction

⇒ Total deduction = 50 + 17 + 10 = ₹77

⇒ Remaining amount to be divided = 1,964 − 77 = ₹1,887

⇒ Sum of ratio parts = 20 + 4 + 13 = 37 units

⇒ 37 units = ₹1,887

⇒ 1 unit = 1,887 ÷ 37 = ₹51

⇒ Lalita's share after deduction = 4 units × ₹51 = ₹204

⇒ Lalita's original share = 204 + 17 = ₹221

∴ The correct answer is ₹221.

Additional Information

Ratio and Proportion

When values are subtracted from original shares, always subtract the total deduction from the total sum before applying the ratio parts.

Compound Ratio

The compound ratio of (a : b) and (c : d) is given by (ac : bd).

Invertendo and Alternendo

If a/b = c/d, then b/a = d/c (Invertendo) and a/c = b/d (Alternendo).

Shortcut Trick

Total work in man-hours:

Mohan = 6 days × 5 hours = 30 hours

Rohan = 8 days × 5 hours = 40 hours (assuming same hours/day initially)

Combined time in hours = (30 × 40) ÷ (30 + 40) = 1200 ÷ 70 = 120/7 hours

Days required at 8 hours/day = (120/7) ÷ 8 = 15/7 = 2(1/7) days

∴ The correct answer is 2(1/7) days.

Alternate Method

Given:

Mohan takes 6 days at 5 hours/day = 30 hours

Rohan takes 8 days at 5 hours/day = 40 hours

Calculations:

⇒ Combined Efficiency = 4 + 3 = 7 units/hour

⇒ Total hours required = 120 ÷ 7 hours

⇒ Number of days at 8 hours per day = (120 ÷ 7) ÷ 8

⇒ Number of days = 15/7 days

⇒ Number of days = 2(1/7) days

∴ The correct answer is 2(1/7) days.

Additional Information

Work-Hours Relation

Total Work = Persons × Days × Hours per day. This is often solved using the M₁D₁H₁/W₁ = M₂D₂H₂/W₂ formula.

Combined Efficiency

When multiple people work together, their efficiencies (work done per unit time) are added to find the total time taken for the task.

Shortcut Trick

Apply the algebraic identity: (a − b)² + 4ab = (a + b)².

In this expression, let a = 4x and b = 5y. Then 4ab = 4(4x)(5y) = 80xy.

The numerator (4x − 5y)² + 80xy simplifies exactly to (4x + 5y)².

The entire expression becomes (4x + 5y)² ÷ (4x + 5y)² = 1.

∴ The correct answer is 1.

Alternate Method

Given: [(4x − 5y)² + 80xy] ÷ [(4x + 5y)²]

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

⇒ Expand the numerator: (4x)² − 2(4x)(5y) + (5y)² + 80xy

⇒ Numerator = 16x² − 40xy + 25y² + 80xy

⇒ Numerator = 16x² + 40xy + 25y²

⇒ Now, expand the denominator: (4x + 5y)² = (4x)² + 2(4x)(5y) + (5y)²

⇒ Denominator = 16x² + 40xy + 25y²

⇒ Expression = (16x² + 40xy + 25y²) ÷ (16x² + 40xy + 25y²)

⇒ Simplified Value = 1

∴ The correct answer is 1.

Additional Information

Binomial Square Identities

The expansion of (a ± b)² is a core concept in algebra, given by a² ± 2ab + b².

Relationship between Identities

The identities are linked such that (a + b)² − (a − b)² = 4ab and (a + b)² + (a − b)² = 2(a² + b²).

Simplification Rule

Any non-zero expression divided by itself, such as k ÷ k, always results in 1.

Shortcut Trick

Use the multiplying factor method: Final Price = Initial Price × (1 − r₁) × (1 − r₂) × (1 − r₃) × (1 + r₄).

Net multiplier = 0.60 × 0.85 × 0.55 × 1.70 = 0.47685.

Net change = (1 − 0.47685) × 100 = 52.315% decrease.

∴ The correct answer is Decreases by 52.32%.

Alternate Method

Given: Initial Price = 100, Successive Decreases = 40%, 15%, 45%, Increase = 70%.

⇒ Price after 1st month = 100 × (60/100) = 60

⇒ Price after 2nd month = 60 × (85/100) = 51

⇒ Price after 3rd month = 51 × (55/100) = 28.05

⇒ Price after 4th month = 28.05 × (170/100) = 47.685

⇒ Total decrease = 100 − 47.685 = 52.315%

⇒ Rounding off to two decimal places = 52.32%

∴ The correct answer is Decreases by 52.32%.

Additional Information

Successive Percentage Change

For two changes a% and b%, the net change is [a + b + (a × b)/100]%. Use negative for decrease.

Multiplying Factor

A decrease of x% is equivalent to multiplying the value by (1 − x/100), while an increase is (1 + x/100).

Base Value Selection

Always assume 100 as the initial value in percentage problems to simplify calculations and direct conversion to %.

Shortcut Trick

Time taken for 1 page = 40 ÷ 4 = 10 minutes.

Pages typed in 90 minutes = 90 ÷ 10 = 9 pages.

∴ The correct answer is 9 pages.

Alternate Method

Given:

Time₁ = 40 minutes, Work₁ = 4 pages

Time₂ = 90 minutes, Work₂ = x pages

Formula Used:

Using the Work and Time relationship: (W₁ / T₁) = (W₂ / T₂)

Calculations:

⇒ 4 / 40 = x / 90

⇒ 1 / 10 = x / 90

⇒ x = 90 ÷ 10

⇒ x = 9

∴ The correct answer is 9 pages.

Additional Information

Direct Proportion

When one quantity increases, the other also increases. Formula: x₁/y₁ = x₂/y₂.

Unitary Method

Find the value of a single unit first, then multiply by the required number of units.

Efficiency and Time

Work = Efficiency × Time. Here, typing speed is the constant efficiency.

Shortcut Trick

Age of new member = New Average + (Initial Count × Increase in Average)

Increase in average = 26 − 25 = 1 year

Age of teacher = 26 + (23 × 1)

Age of teacher = 26 + 23 = 49 years

∴ The correct answer is 49 years.

Alternate Method

Given:

Initial number of students = 23; Initial average age = 25 years

New number of people (students + teacher) = 24; New average age = 26 years

Formula: Total Age = Number of People × Average Age

⇒ Total age of 23 students = 23 × 25 = 575 years

⇒ Total age of 24 people (including teacher) = 24 × 26 = 624 years

⇒ Age of the teacher = Total age of 24 people − Total age of 23 students

⇒ Age of the teacher = 624 − 575 = 49 years

∴ The correct answer is 49 years.

Additional Information

Inclusion in Average

When a new member joins, their value = New Average + Old Count × (New Average − Old Average).

Exclusion from Average

When a member leaves, their value = Old Average + New Count × (Old Average − New Average).

Weighted Average Formula

The combined average of two groups is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average.

Shortcut Trick

The empirical relationship between central tendencies is: Mode − Median = 2/3 (Mode − Mean).

Given that the Mode exceeds the Mean by 19.8, we have: (Mode − Mean) = 19.8.

Substituting the value: Required difference = 2/3 × 19.8.

Required difference = 2 × 6.6 = 13.2.

∴ The correct answer is 13.2.

Alternate Method

Given:

Mode − Mean = 19.8

Formula Used:

The Empirical Formula: Mode = 3 Median − 2 Mean

Calculations:

⇒ From the formula: Mode = 3 Median − 2 Mean

⇒ Mode − Mean = (3 Median − 2 Mean) − Mean

⇒ Mode − Mean = 3 Median − 3 Mean = 3 (Median − Mean)

⇒ 19.8 = 3 (Median − Mean)

⇒ Median − Mean = 19.8 ÷ 3 = 6.6

⇒ Now, Mode − Median = (Mode − Mean) − (Median − Mean)

⇒ Mode − Median = 19.8 − 6.6

⇒ Mode − Median = 13.2

∴ The correct answer is 13.2.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is Mode = 3 Median − 2 Mean.

Symmetric Distribution

In a perfectly symmetrical distribution (like the Normal Distribution), the values of Mean, Median, and Mode are equal: Mean = Median = Mode.

Positively Skewed Distribution

When the tail is on the right side, the relationship is typically Mean > Median > Mode.

Negatively Skewed Distribution

When the tail is on the left side, the relationship is typically Mode > Median > Mean.

Shortcut Trick

Speed Ratio = 5 : 3 (Normal : New)

Time Ratio = 3 : 5 (Inverse of speed as distance is constant)

Difference in time units = 5 − 3 = 2 units

2 units = 24 minutes ⇒ 1 unit = 12 minutes

Usual Time = 3 units = 3 × 12 minutes = 36 minutes

∴ The correct answer is 36 min.

Alternate Method

Given: New speed = 3/5 of Normal speed, Late time = 24 minutes.

Formula Used: Usual Time = [Late Time × Numerator] ÷ [Denominator − Numerator]

⇒ Let the usual time be T.

⇒ New time = T + 24

⇒ Since Distance = Speed × Time, S × T = (3/5)S × (T + 24)

⇒ T = 3/5 (T + 24)

⇒ 5T = 3T + 72

⇒ 2T = 72

⇒ T = 36 minutes

∴ The correct answer is 36 min.

Additional Information

Speed-Time Inverse Proportionality

When distance is constant, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Late/Early Logic

If a person is late by 'L' minutes at a/b of usual speed, Usual Time = [a × L] / |b − a|.

Relative Speed

Used for two moving bodies; add speeds if moving in opposite directions, subtract if in the same direction.

Shortcut Trick

Effective rate for 10% compound interest for 2 years = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount = Principal × (100% + Effective Rate%)

⇒ Amount = 1,500 × 121%

⇒ Amount = 15 × 121 = 1,815.

∴ The correct answer is ₹1,815.

Alternate Method

Given:

Principal (P) = ₹1,500

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 1,500 × (1 + 10/100)²

⇒ A = 1,500 × (110/100) × (110/100)

⇒ A = 1,500 × (11/10) × (11/10)

⇒ A = 1,500 × 121/100

⇒ A = 15 × 121

⇒ A = 1,815

∴ The correct answer is ₹1,815.

Additional Information

Compound Interest Formula

Compound Interest (CI) is calculated as CI = A − P, where A = P(1 + r/n)^nt.

Difference between CI and SI for 2 Years

The difference (D) for two years is given by D = P(R/100)².

Successive Percentage for 2 Years

The net effective rate for two successive increases of a% and b% is (a + b + ab/100)%.

Cross multiply ⇒ X × X = 125 × 5

⇒ X² = 625

⇒ X = √625 = 25 (since X > 0)

∴ Answer = 25

Given:

X ÷ 125 = 5 ÷ X

Formula used:

If a/b = c/d ⇒ ad = bc

Calculation:

⇒ X/125 = 5/X

⇒ X × X = 125 × 5

⇒ X² = 625

⇒ X = ±25

⇒ Since X > 0 ⇒ X = 25

∴ The correct answer is 25.

Cross Multiplication

If a/b = c/d then a × d = b × c

Quadratic Equation

x² = k ⇒ x = ±√k

Condition Check

Given X > 0, so only positive root is valid

Perfect Square

625 = 25²

Shortcut Trick

Product of fractions: (40/100) × (10/100) × (1/3) × Number = 477

(2/5) × (1/10) × (1/3) × Number = 477

(1/75) × Number = 477

Number = 477 × 75 = 35775

∴ The correct answer is 35775.

Alternate Method

Given: 40% of 10% of 1/3 of x = 477

⇒ Let the number be x.

⇒ (40/100) × (10/100) × (1/3) × x = 477

⇒ (0.4) × (0.1) × (1/3) × x = 477

⇒ (0.04 / 3) × x = 477

⇒ x = (477 × 3) / 0.04

⇒ x = 1431 / 0.04

⇒ x = 35775

∴ The correct answer is 35775.

Additional Information

Percentage to Fraction Conversion

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

Successive Percentage Rule

A% of B% of C is equivalent to (A × B / 100)% of C. Use this for multi-step percentage problems.

Reverse Calculation

When the final value is given, the original number = Result ÷ (Product of all percentage/fraction multipliers).

Shortcut Trick

The best scheme for the shopkeeper is the one that offers the minimum discount to the customer.

Scheme A: 32 + 40 − (32 × 40) ÷ 100 = 72 − 12.8 = 59.2%

Scheme B: 5 ÷ (1 + 5) × 100 = 5 ÷ 6 × 100 = 83.33%

Scheme C: Single discount = 48%

Scheme D: 8 + 33 − (8 × 33) ÷ 100 = 41 − 2.64 = 38.36%

Comparing all, Scheme D provides the lowest discount (38.36%), making it the most profitable for the seller.

∴ The correct answer is D.

Alternate Method

Given → Successive discounts and Buy-Get schemes. Formulas: Successive Discount = a + b − (ab ÷ 100) | Buy X Get Y free Discount% = [Y ÷ (X + Y)] × 100.

⇒ Calculate effective discount for Scheme A: 32 + 40 − 12.8 = 59.2%. ⇒ Calculate effective discount for Scheme B: (5 ÷ 6) × 100 = 83.33%. ⇒ Calculate effective discount for Scheme C: 48%. ⇒ Calculate effective discount for Scheme D: 8 + 33 − (264 ÷ 100) = 41 − 2.64 = 38.36%. ⇒ A shopkeeper always prefers the lowest discount rate to maximize the Selling Price. ⇒ Since 38.36% < 48% < 59.2% < 83.33%, Scheme D is best.

∴ The correct answer is D.

Additional Information

Best for Customer vs Shopkeeper

The Best Scheme for Customer is the one with the Maximum Discount. Conversely, the Best Scheme for Shopkeeper is the one with the Minimum Discount.

Buy X Get Y Free

In a 'Buy X, Get Y free' offer, the total number of articles given is (X + Y). The discount percentage is calculated on the total articles: [Free Articles ÷ Total Articles] × 100.

Shortcut Trick

For any quadrilateral ABCD, the angle formed by the bisectors of two adjacent angles (say ∠C and ∠D) is always equal to half the sum of the other two angles (∠A and ∠B):

∠DOC = (∠A + ∠B) ÷ 2

Substituting the given values:

∠DOC = (32° + 58°) ÷ 2 = 90° ÷ 2 = 45°

∴ The correct answer is 45°.

Alternate Method

Given:

In quadrilateral ABCD:

∠A = 32°

∠B = 58°

DO and CO are the angle bisectors of ∠D and ∠C respectively, meeting at point O.

Formula Used:

The sum of all interior angles of a quadrilateral is 360°.

The sum of all interior angles of a triangle is 180°.

Calculations:

In quadrilateral ABCD:

⇒ ∠A + ∠B + ∠C + ∠D = 360°

⇒ 32° + 58° + ∠C + ∠D = 360°

⇒ 90° + ∠C + ∠D = 360°

⇒ ∠C + ∠D = 360° − 90°

⇒ ∠C + ∠D = 270°

Dividing the entire equation by 2:

⇒ (∠C + ∠D) ÷ 2 = 270° ÷ 2

⇒ ∠C/2 + ∠D/2 = 135°

Since CO and DO are the angle bisectors of ∠C and ∠D:

⇒ ∠OCD = ∠C/2 and ∠ODC = ∠D/2

⇒ ∠OCD + ∠ODC = 135°

Now, in triangle DOC, the sum of all interior angles is 180°:

⇒ ∠DOC + ∠OCD + ∠ODC = 180°

⇒ ∠DOC + 135° = 180°

⇒ ∠DOC = 180° − 135°

⇒ ∠DOC = 45°

∴ The correct answer is 45°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles: ∠DOC = (∠A + ∠B) ÷ 2.

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral (say ∠A and ∠C) intersect, the angle between them is equal to half the absolute difference of the other two angles: |∠B − ∠D| ÷ 2.

Sum of Interior Angles of a Polygon

The sum of the interior angles of any convex polygon with n sides is (n − 2) × 180°. For a quadrilateral, n = 4, so the sum is 360°.

Given: (Left) 5 € $ 4 9 & & 4 € @ © 7 @ 7 9 1 8 # 8 % 5 £ (Right)

Logic followed here is: We need to find symbols in the given series that are immediately preceded by a number and immediately followed by a number.

Pattern: Number → Symbol → Number

(Left) 5 € $ 4 9 & & 4 € @ © 7 @ 7 9 1 8 # 8 % 5 £ (Right)

The patterns observed are: (7@7), (8#8), and (8%5).

Therefore, there are 3 such symbols in the series.

Hence the correct answer is "Option 2".

Given Statements:

Some vehicles are trucks.

Some vehicles are lorries.

Some trucks are buses.

The least possible Venn diagram for the given statement is:

Conclusions:

(I) Some vehicles are buses: Does not follow (Because, Some vehicles are trucks,Some trucks are buses.But the “some trucks” which are buses may or may not be the same trucks which are vehicles.)

(II) Some trucks are lorries: Does not follow (Because, Some vehicles are trucks and Some vehicles are lorries. But trucks and lorries may be different groups of vehicles.So, no definite relation exists between trucks and lorries.)

So, Neither conclusion (I) nor (II) follows.

Hence, the correct answer is "Option 2".

Logic followed here is.

Logic: The digits of the given number are rearranged in ascending order (from smallest to largest) from left to right, and then the sum of the digits at the specified positions is calculated.

Given Number: 756931482

Arranging the digits in ascending order →

 

The new number formed is: 123456789

Identifying the required digits:

→ The digit which is third from the left is: 3

→ The digit which is second from the right (8th from the left) is: 8

Sum of these digits:

Sum = 3 + 8 = 11

Hence, the correct answer is "Option 1".

Statements:

1. All Pianos are Guitars.
   
2. Some Guitars are Violins.
   
3. No Violins are Drums.

The least possible Venn diagram for the given statement is:

Conclusions:

(I) Some Drums are Guitars: Does not follow (Because, No relation is given between Drums and Guitars. So, we cannot say that some Drums are Guitars.)

(II) Some Violins are Pianos: Does not follow (Because, All Pianos are Guitars and Some Guitars are Violins. But it is not necessary that the Guitars which are Violins are also Pianos.So, no definite relation exists between Violins and Pianos.)

So, Neither conclusion (I) nor conclusion (II) follows.

Hence, the correct answer is "Option 3".

Logic followed here is.

Logic: In each pair, the letters within a cluster have a difference of +2, and the corresponding letters between the two clusters have a difference of +5.

 

Option 2:

Option 3:

Option 4:

So, in Options 2, 3 and 4, both letters are shifted by +5.

Option 1:

The shift is not +5 for both letters.

Therefore, XZ – BG does not follow the pattern.

Hence, the correct answer is Option 1.

Given:

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

Logic followed here is: We need to find even numbers that are immediately preceded by an even number and immediately followed by an odd number.

Pattern: Even Number → Even Number → Odd Number

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

The required even numbers are:

• 2 in (4 2 7)

• 2 in (4 2 5)

• 4 in (6 4 5)

Therefore, there are 3 such even numbers.

Hence, the correct answer is "Option 1".

Given series:

4, 9, 14, 19, 24, ?

Next term:

24 + 5 = 29

Hence, the correct answer is Option 4.

Logic followed here is based on the directional relationship between the houses.

Step 1: Positioning P and R relative to Q

• House P is 5 km West of House Q.
• House R is 3 km North of House Q.

 

Step 2: Positioning S relative to R

The house of S is 10 km south of the house of R.

Since R is 3 km North of Q, moving south from R involves passing through the position of Q:

• Distance from R to Q = 3 km (South direction)
• Remaining distance from Q to S = Total distance - Distance to Q
• Remaining distance = 10 km - 3 km = 7 km

 

Conclusion:

As House S is 7 km directly below House Q on the North-South axis, it is located in the South direction.

Therefore, the house of S is South from the house of Q.

Symbols and their meanings:

 

Given Expression: ‘K & T x R - Z + V’: K is the father of T, T is the wife of R, R is the brother of Z, Z is the son of V

The possible Venn diagram is:

 

Therefore, K is the Son's wife's father of V.

Hence, the corect answer is "Option 4".

The pattern followed here is:

Hence, the correct answer is "Option 4".