RRB Group D Official Paper (Held On: 14 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is It is equal to the weight of the fluid displaced by the body.

Key Points

• Archimedes' principle states that when a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.
• The upward force mentioned in the principle is known as the buoyant force or upthrust.
• The magnitude of this force depends on the density of the fluid and the volume of the immersed part of the body. Hence, statement 3 is correct.
• While pressure exerted by the fluid increases with depth and contributes to the upward force (due to the pressure difference between the top and bottom of the object), the force itself is not numerically equal to the pressure. Hence, statement 1 is incorrect.
• The magnitude of the force is equal to the weight (a force measured in Newtons), not the volume (measured in cubic meters). Volume determines how much fluid is moved, but the force is the weight of that volume. Hence, statement 2 is incorrect.
• The weight of the immersed body only equals the buoyant force when the object is floating in equilibrium. If the object sinks, its weight is greater than the buoyant force. Hence, statement 4 is incorrect.

Additional Information

• Buoyancy: It is the property of a fluid to exert an upward force on any object immersed in it. This phenomenon is why objects feel lighter when submerged in water.
• Density: It is defined as the mass per unit volume of a substance. The SI unit of density is kg/m³. An object will float if its density is less than the density of the liquid.
• Relative Density: This is the ratio of the density of a substance to the density of water at 4 degrees Celsius. Since it is a ratio of similar quantities, it has no units.
• Applications of Archimedes' Principle:
• Designing Ships and Submarines: Used to ensure they displace enough water to stay afloat or submerge as needed.
• Lactometers: These instruments are used to determine the purity of a sample of milk based on its density.
• Hydrometers: These are used for determining the density of liquids.

Important Points

• Apparent Weight: When an object is in a fluid, its apparent weight is its actual weight minus the buoyant force. This explains why heavy stones are easier to lift while they are still under the water.
• Sinking vs. Floating:
• If the buoyant force is less than the weight of the object, the object will sink.
• If the buoyant force is equal to or greater than the weight of the object, the object will float.
• The principle applies to all fluids, which include both liquids and gases. For example, a hot air balloon rises because the weight of the air it displaces is greater than the weight of the balloon itself.
• The discovery is attributed to Archimedes, a Greek mathematician and inventor from Syracuse, in the 3^rd century BC.

The correct answer is Temperature at which eggs are incubated.

Key Points

• In many reptiles, the sex of the offspring is not determined by sex chromosomes at the time of fertilization, but rather by the environmental temperature during egg incubation. This biological phenomenon is known as Temperature-dependent Sex Determination (TSD).
• This process occurs during a specific window of embryonic development known as the thermosensitive period (TSP). This period typically falls within the middle third of the total incubation time.
• The temperature experienced by the developing embryo during the TSP triggers physiological changes that determine whether the gonads will develop into testes or ovaries. Hence, the statement regarding temperature at which eggs are incubated is correct.
• While most mammals and birds rely on Genotypic Sex Determination (GSD) involving X, Y or Z, W chromosomes, TSD is prevalent among various species of turtles, crocodilians (alligators and crocodiles), tuataras, and some species of lizards.
• Unlike GSD, where the sex ratio is usually 50:50, TSD can lead to highly skewed sex ratios in a population depending on the nesting environment and seasonal temperatures.

Additional Information

• Types of TSD Patterns:
• Pattern Ia (MF): In many turtles, lower temperatures during incubation produce males, while higher temperatures produce females.
• Pattern Ib (FM): In some lizards and crocodilians, the pattern is reversed, where lower temperatures result in females and higher temperatures result in males.
• Pattern II (FMF): In some species like snapping turtles and crocodiles, females are produced at both extreme low and extreme high temperatures, while males are produced at intermediate temperatures.
• Biological Mechanism:
• The conversion of androgens into estrogens is mediated by the enzyme aromatase. In species with TSD, the activity of this enzyme is often temperature-sensitive. High temperatures in certain species increase aromatase activity, leading to ovary development and female offspring.
• Comparison with GSD:
• In Genotypic Sex Determination (GSD), sex is fixed by DNA. For example, in humans, the presence of the Y chromosome (specifically the SRY gene) determines maleness. In some reptiles like snakes, GSD is the primary method, often using the ZZ/ZW system where females are the heterogametic sex (ZW).
• Environmental and Evolutionary Impact:
• Climate change poses a significant threat to species with TSD. Rising global temperatures can lead to an overproduction of one sex (e.g., nearly all female sea turtles), which could eventually lead to population collapse due to a lack of reproductive partners.
• Ecological factors such as nest depth, shading from vegetation, and soil moisture also play critical roles in regulating the incubation temperature within the nest.

Important Points

• The pivotal temperature is defined as the specific incubation temperature that produces an equal ratio (50%) of males and females. Even a small change of 1-2 degrees Celsius can completely flip the sex ratio of a clutch.
• It is important to note that hormonal levels in the mother do not directly determine the genomic or environmental sex of the offspring in the same way incubation temperature does, although they can influence egg quality.
• Reptiles like Green Sea Turtles are classic examples where warming sands due to global warming are currently resulting in extreme female bias in hatchlings.

The correct answer is Boron.

Key Points

• Plants require various essential nutrients for their growth, metabolism, and reproductive cycles, which are categorized based on the quantity required into macronutrients and micronutrients.
• Oxygen is primarily obtained by plants from the atmosphere during respiration and is also derived from water during the process of photosynthesis. Hence, statement 1 is incorrect.
• Hydrogen is derived almost exclusively from water (H2O) absorbed by the roots from the environment. Hence, statement 2 is incorrect.
• Carbon is taken up by plants from the atmosphere in the form of carbon dioxide (CO2) to build organic molecules through carbon fixation. Hence, statement 3 is incorrect.
• Boron is a vital mineral micronutrient that is generally supplied to the plant by the soil. It is absorbed by plant roots primarily in the form of undissociated boric acid. Hence, statement 4 is correct.

Additional Information

• Classification of Nutrients:
• Mineral Nutrients: These are elements like Nitrogen (N), Phosphorus (P), Potassium (K), Calcium (Ca), Magnesium (Mg), and Sulfur (S), along with micronutrients like Iron (Fe) and Boron (B), which are primarily sourced from the soil.
• Non-Mineral Nutrients: These include Carbon, Hydrogen, and Oxygen. They are considered non-mineral because they are supplied by air and water rather than soil minerals.
• Role of Boron in Plants:
• Boron is essential for cell wall formation and maintaining the structural integrity of plasma membranes.
• It plays a critical role in reproductive growth, specifically facilitating pollen germination and pollen tube growth.
• It is also involved in sugar transport and calcium utilization within the plant tissues.
• Micronutrients from Soil:
• Apart from Boron, other micronutrients supplied by the soil include Manganese (Mn), Copper (Cu), Molybdenum (Mo), Zinc (Zn), Iron (Fe), Chlorine (Cl), and Nickel (Ni).
• These elements are required in very small quantities, typically less than 10 mmole per kg of dry matter, but their deficiency can lead to significant physiological disorders.

Important Points

• Nutrient Uptake Mechanism:
• Plants absorb mineral nutrients from the soil solution through their root hairs. This process involves both passive transport (diffusion) and active transport (utilizing ATP energy).
• The availability of Boron in the soil is highly dependent on the soil pH; it is most readily available to plants in acidic to neutral soils (pH 5.0 to 7.0).
• Source Summary:
• Air provides Carbon and Oxygen.
• Water provides Hydrogen and Oxygen.
• Soil provides all other 13 essential mineral nutrients, including Nitrogen, Phosphorus, and Boron.

The correct answer is It has a sweet smell and is volatile.

Key Points

• Ethanol, also known as ethyl alcohol, is a colorless liquid that possesses a distinct, pleasant sweet smell and a characteristic burning taste.
• It is a highly volatile substance, which means it evaporates rapidly at standard temperature and pressure due to its relatively low boiling point of 78.37°C (351.5 K). Hence, statement 1 is correct.
• The formation of scum (insoluble precipitates) is a property of soap reacting with calcium and magnesium ions in hard water. Ethanol is a covalent organic compound and does not react with these ions to form scum. Hence, statement 2 is incorrect.
• Ethanol has a very low melting point of -114.1°C (159 K). Consequently, it exists in a liquid state at room temperature (approx 25°C) and does not solidify under normal environmental conditions. Hence, statement 3 is incorrect.
• Chemically, ethanol is neutral in nature. Although its formula C2H5OH contains a hydroxyl group, it does not ionize in water to release hydroxide ions (OH-), nor does it change the color of litmus paper. Hence, statement 4 is incorrect.
• Hence, the correct answer is It has a sweet smell and is volatile.

Additional Information

• Chemical Nature and Formula: Ethanol is a primary alcohol with the chemical formula C2H5OH. It is the second member of the homologous series of alcohols.
• Solubility: It is highly soluble in water in all proportions. This solubility is due to the ability of the hydroxyl group in ethanol to form hydrogen bonds with water molecules.
• Combustion: It is a flammable liquid that burns with a clean blue flame. The reaction produces carbon dioxide and water vapor, releasing a significant amount of energy.
• Applications as Fuel: Because it is a renewable source of energy (often produced from biomass), it is used as an additive in petrol. In India, the Ethanol Blended Petrol (EBP) Programme aims to increase the blend percentage to 20% (E20) by the year 2025.
• Industrial Uses: It serves as a versatile solvent in the manufacture of paints, varnishes, perfumes, and medicines like tincture of iodine and cough syrups.

Important Points

• Reaction with Sodium: When ethanol reacts with sodium metal, it evolves hydrogen gas and forms sodium ethoxide. This reaction is often used as a chemical test for the presence of the alcoholic group.
• Oxidation: Upon heating with alkaline potassium permanganate or acidified potassium dichromate, ethanol undergoes oxidation to form ethanoic acid (acetic acid).
• Denatured Alcohol: To prevent the illegal consumption of industrial-grade ethanol, it is made unfit for drinking by adding poisonous substances like methanol or pyridine and dyes like copper sulphate to give it a blue color. This is known as denaturation.
• Medical Use: In a concentration of 70%, it is an effective antiseptic used in hand sanitizers to kill microorganisms by denaturing their proteins and dissolving their lipid membranes.

The correct answer is The change in velocity is zero over time interval.

Key Points

• Non-uniform motion is defined as the motion of an object in which the object travels with varied speed and does not cover the same distance in equal time intervals.
• In non-uniform motion, the velocity of the object does not remain constant. Since velocity is a vector quantity, it changes if either the speed (magnitude) or the direction of motion changes.
• The statement "The change in velocity is zero over time interval" implies that the initial velocity and final velocity are identical, which means the acceleration is zero. This is a characteristic of Uniform Motion, not non-uniform motion. Hence, statement 1 is NOT true.
• An object in non-uniform motion covers unequal distances in equal intervals of time. By extension, the relationship between distance and time is not linear, meaning it can move unequal distances in unequal time intervals depending on the rate of acceleration. Hence, statement 2 is correct in the context of non-uniformity.
• Because the rate of motion is constantly fluctuating, the object will possess different speeds at different points or instances along its path. Hence, statement 3 is correct.
• The core definition of non-uniform motion is that the velocity changes with time. This change indicates the presence of acceleration (positive or negative). Hence, statement 4 is correct.

Additional Information

• Acceleration: In non-uniform motion, the acceleration of the body is non-zero. Acceleration is the measure of the rate of change of velocity with respect to time. If the velocity increases, it is called acceleration; if it decreases, it is called retardation or deceleration.
• Types of Non-Uniform Motion:
• Uniformly Accelerated Motion: The velocity changes by equal amounts in equal intervals of time (e.g., a ball rolling down an inclined plane).
• Non-uniformly Accelerated Motion: The velocity changes by unequal amounts in equal intervals of time (e.g., a car moving in heavy city traffic).
• Average Speed: For an object in non-uniform motion, the average speed is calculated by dividing the total distance traveled by the total time taken. Unlike uniform motion, the instantaneous speed at any given point is usually different from the average speed.
• Examples: Common real-world examples include a train pulling out of a station, a person jogging in a crowded park, or a pendulum swinging back and forth.

Important Points

• Graphical Representation:
• In a Distance-Time Graph, non-uniform motion is represented by a curved line (parabola or zig-zag), indicating that the slope (speed) is changing.
• In a Velocity-Time Graph, non-uniform motion is represented by any line that is not horizontal. A horizontal line would indicate constant velocity (uniform motion).
• Uniform Motion vs. Non-Uniform Motion:
• Uniform Motion: Covers equal distance in equal time; zero acceleration; constant velocity.
• Non-Uniform Motion: Covers unequal distance in equal time; variable acceleration; changing velocity.
• Directional Change: It is important to note that even if an object moves at a constant speed in a circle (Uniform Circular Motion), it is considered non-uniform motion because its direction is constantly changing, resulting in a change in velocity.

The correct answer is Tracheids, xylem parenchyma and xylem fibres.

Key Points

• Xylem is defined as a complex permanent tissue in plants, primarily responsible for the conduction of water and dissolved minerals from the roots to the rest of the plant parts.
• It is termed a complex tissue because it consists of more than one type of cell that coordinates to perform a common function. The four main components of xylem in angiosperms are Tracheids, Vessels, Xylem Parenchyma, and Xylem Fibres.
• Tracheids, xylem parenchyma, and xylem fibres are all integral parts of the xylem tissue. While vessels are also a key component in flowering plants, the provided option correctly identifies three valid xylem constituents. Hence, statement 1 is correct.
• Companion cells are a component of Phloem, not xylem. They help in the loading and unloading of sugars into sieve tubes. Hence, statement 2 is incorrect.
• Collenchyma is a simple permanent tissue that provides mechanical support to young stems and petioles; it is not a sub-type of xylem tissue. Xylem has its own supporting tissue known as Xylem Fibres (or sclerenchyma). Hence, statement 3 is incorrect.
• Sieve cells and phloem parenchyma are specialized cells belonging to the Phloem, which is the vascular tissue responsible for transporting organic nutrients (food). Hence, statement 4 is incorrect.

Additional Information

• Components of Xylem:
• Tracheids: These are elongated, tube-like dead cells with thick, lignified walls and tapering ends. They are the primary water-conducting elements in Gymnosperms and lower vascular plants.
• Vessels: Long, cylindrical structures formed by several cells called vessel members. They have perforated end walls and are much more efficient at water transport than tracheids. They are characteristic features of Angiosperms.
• Xylem Parenchyma: These are the only living cells in the xylem tissue. They have thin cellulosic walls and function mainly in the storage of food (starch or fat) and the lateral conduction of water.
• Xylem Fibres: These are dead cells with highly thickened lignified walls and obliterated central lumens. Their primary function is to provide mechanical strength and support to the plant body.
• Direction of Flow: Unlike phloem, which exhibits bidirectional transport, the flow in xylem is strictly unidirectional (from roots upwards to the leaves).
• Wood: In botanical terms, the secondary xylem produced during the secondary growth of a plant is what we commonly refer to as wood.

Important Points

• Evolutionary Fact: In Gymnosperms (like pines and cedars), vessels are absent; water is conducted almost entirely through tracheids.
• Primary Xylem: It is categorized into Protoxylem (first formed) and Metaxylem (later formed). In stems, the protoxylem lies towards the center (Endarch condition), while in roots, it lies towards the periphery (Exarch condition).
• Lignification: The deposition of lignin in the cell walls of tracheids and vessels provides the structural rigidity necessary to withstand the negative pressure generated during transpiration.

The correct answer is In gaseous state, intermolecular distance is very large and intermolecular attraction is very low.

Key Points

• Matter exists in three primary physical states: Solid, Liquid, and Gas. The state of matter is determined by the balance between intermolecular forces and thermal energy.
• In the gaseous state, the particles (atoms or molecules) possess high kinetic energy, which overcomes the forces of attraction between them.
• The intermolecular distance in gases is very large. In fact, the distance between gas molecules is much greater than the actual size of the molecules themselves. Hence, the statement regarding large intermolecular distance is correct.
• The intermolecular attraction (cohesive forces) between gas particles is very low or negligible. This lack of attraction allows gas particles to move independently and expand to fill any container. Hence, the statement regarding low intermolecular attraction is correct.
• Regarding the other options:
• Statement 1 suggests both distance and attraction are very large; this is contradictory as large distance usually results in weak attraction. Hence, statement 1 is incorrect.
• Statement 2 describes the Solid state, where particles are close together (low distance) and strongly bonded (large attraction). Hence, statement 2 is incorrect.
• Statement 4 suggests both distance and attraction are very low; while attraction is low, the distance is actually very high in gases. Hence, statement 4 is incorrect.

Additional Information

• Characteristics of Gases:
• Compressibility: Due to the large intermolecular distances, gases are highly compressible. Applying pressure reduces the empty space between molecules.
• Shape and Volume: Gases have no definite shape and no definite volume. They take the shape and volume of the container they occupy.
• Diffusion: Because of weak intermolecular forces and high speed of particles, gases diffuse very rapidly into each other.
• Density: Gases have very low density compared to solids and liquids because the same mass occupies a much larger volume.
• Kinetic Molecular Theory (KMT):
• This theory explains that gas particles are in constant, random motion.
• The collisions between gas particles and the walls of the container are perfectly elastic, which exerts pressure.
• The average kinetic energy of the particles is directly proportional to the absolute temperature (measured in Kelvin).
• Comparison with other states:
• Solids: Have fixed shape and volume, strongest intermolecular forces, and minimum intermolecular space.
• Liquids: Have fixed volume but no fixed shape, intermediate intermolecular forces, and moderate intermolecular space compared to gases.

Important Points

• Van der Waals Forces: In real gases, there are slight attractive forces known as Van der Waals forces. However, in an Ideal Gas model, these attractions are assumed to be zero.
• Boyle's Law: States that at constant temperature, the volume of a gas is inversely proportional to its pressure ($P \propto 1/V$).
• Charles's Law: States that at constant pressure, the volume of a gas is directly proportional to its absolute temperature ($V \propto T$).
• The transition from liquid to gas is called vaporization or boiling, which occurs when thermal energy is sufficient to break the intermolecular attractions of the liquid phase.

The correct answer is Doubles because resistance is constant.

Key Points

• Ohm’s Law states that the current (I) flowing through a metallic conductor is directly proportional to the potential difference (V) applied across its ends, provided its physical conditions such as temperature remain constant.
• The mathematical expression for this relationship is V = I * R, where R represents the Electrical Resistance of the conductor.
• In this scenario, since the temperature is kept constant, the resistance (R) of the metallic conductor remains unchanged regardless of the change in voltage.
• When the potential difference is increased from V to 2V, the new current (I') can be calculated using the formula I' = (2V) / R. Since I = V / R, it follows that I' = 2 * I.
• The current doubles because it maintains a linear relationship with the potential difference in an ohmic conductor. Hence, statement 2 is correct.
• Statement 1 is incorrect because while current does depend on resistance, it is also a function of the applied potential difference.
• Statement 3 is incorrect because at a constant temperature, the drift velocity of electrons increases proportionally with the electric field (and thus voltage), leading exactly to a doubling of current, not "more than double."
• Statement 4 is incorrect because the question explicitly specifies that the temperature is kept constant, which implies any heating effects (Joule heating) are either neglected or compensated for to keep R stable.

Additional Information

• Electrical Resistance (R): It is the property of a material to oppose the flow of electric current. The SI unit of resistance is the Ohm (represented by the Greek letter Ω).
• Factors Affecting Resistance: The resistance of a conductor depends on its length (directly proportional), area of cross-section (inversely proportional), the nature of the material (resistivity), and temperature.
• Drift Velocity: This is the average velocity attained by charged particles (such as electrons) in a material due to an electric field. The current (I) is defined as I = n * A * e * Vd, where Vd is the drift velocity. Since Vd is proportional to V, current is also proportional to V.
• Ohmic and Non-Ohmic Devices:
• Ohmic Conductors: Materials like Copper, Silver, and Aluminum that follow Ohm's law, showing a straight-line V-I graph passing through the origin.
• Non-Ohmic Conductors: Devices like semiconductors, diodes, and transistors where the resistance changes as the voltage or current changes, resulting in a non-linear V-I characteristic.
• Temperature Coefficient: For pure metals, resistance increases with an increase in temperature. This is because the thermal vibrations of metal ions increase, leading to more frequent collisions with drifting electrons. However, in this specific problem, the constant temperature constraint ensures Ohm's Law is strictly followed.

Important Points

• Ohm's Law Discovery: The law was named after the German physicist Georg Simon Ohm, who published his findings in 1827.
• Power Dissipation: When current flows through a resistor, electrical energy is converted into heat. This is known as Joule Heating, given by P = I^2 * R or P = V^2 / R. In real-world applications, if the voltage doubles, the power dissipation increases four times, which would normally raise the temperature unless cooled.
• Conductance: The reciprocal of resistance is called conductance (G), measured in Siemens (S) or mho. G = 1 / R.

The correct answer is Sodium bicarbonate.

Key Points

• Sodium bicarbonate (NaHCO3), commonly known as baking soda, is a basic salt. It is formed by the neutralization reaction between a strong base, Sodium hydroxide (NaOH), and a weak acid, Carbonic acid (H2CO3). Since the base is stronger than the acid, the resulting salt retains basic characteristics.
• Sodium chloride (NaCl) is a neutral salt. It is produced from the reaction of Sodium hydroxide (NaOH), which is a strong base, and Hydrochloric acid (HCl), which is a strong acid. In an aqueous solution, it typically maintains a pH of around 7.
• Ammonium chloride (NH4Cl) is an example of an acidic salt. It is formed through the reaction of a strong acid, Hydrochloric acid (HCl), and a weak base, Ammonium hydroxide (NH4OH). Due to the dominance of the strong acid, the salt exhibits acidic properties in water.
• Hydrogen chloride (HCl) is not a salt at all; it is a strong mineral acid. In its gaseous form, it is hydrogen chloride, and when dissolved in water, it becomes hydrochloric acid, which is highly corrosive and has a very low pH.
• A basic salt is generally defined as a salt that reacts with water to produce a solution with a pH greater than 7. This occurs because the anion of the weak acid undergoes hydrolysis to produce hydroxide ions (OH-).

Additional Information

• Chemical Nature of Salts:
• Neutral Salts: Formed from a strong acid and a strong base (e.g., KNO3, NaCl).
• Acidic Salts: Formed from a strong acid and a weak base (e.g., (NH4)2SO4, NH4Cl).
• Basic Salts: Formed from a weak acid and a strong base (e.g., CH3COONa, Na2CO3).
• Properties of Sodium Bicarbonate:
• It is a white crystalline solid but often appears as a fine powder.
• It has a slightly salty, alkaline taste.
• The pH of a 0.1 M solution of Sodium bicarbonate is approximately 8.3 at 25 degrees Celsius.
• Applications:
• Baking: It reacts with acidic components in batters to release Carbon dioxide (CO2), causing the dough to rise.
• Medical: Used as an antacid to treat acid indigestion and heartburn by neutralizing excess stomach acid.
• Industrial: Used in soda-acid fire extinguishers and as a mild abrasive in cleaning products.

Important Points

• Salt Hydrolysis: This is the process where the salt reacts with water to give back the acid and base. In the case of Sodium bicarbonate, the bicarbonate ion (HCO3-) reacts with water to form undissociated carbonic acid and hydroxide ions, leading to alkalinity.
• Solubility: Sodium bicarbonate is sparingly soluble in water compared to other sodium salts, but its solubility increases with temperature.
• Safety and Storage: While generally safe, it should be stored in a cool, dry place as moisture can cause it to clump or lose its leavening effectiveness over time.

The correct answer is Graphite.

Key Points

• Graphite is a crystalline allotrope of carbon that is widely recognized for its ability to conduct electricity, which is an exception among non-metals.
• In the structure of graphite, each carbon atom is covalently bonded to three other carbon atoms in the same plane, creating a series of hexagonal rings arranged in layers.
• The carbon atoms in graphite undergo sp2 hybridization. This leaves one valence electron per carbon atom unbonded and "free" within the layers.
• These delocalized electrons are mobile and can move throughout the graphite sheet, allowing it to act as an efficient electrical conductor.
• Due to this property, graphite is extensively used in the production of electrodes in batteries and industrial electrolysis processes.

Additional Information

• Diamond:
• It is another major allotrope of carbon where each atom is bonded to four other carbon atoms in a rigid tetrahedral structure.
• This involves sp3 hybridization, meaning all four valence electrons are tightly held in covalent bonds.
• Because there are no free or delocalized electrons, diamond acts as an electrical insulator.
• Fullerenes (C60):
• Discovered in 1985, these are spherical allotropes of carbon, often called Buckminsterfullerene or buckyballs.
• While they possess pi-electron systems, pure fullerenes are generally insulators or semiconductors under normal conditions rather than conductors.
• Charcoal:
• This is an amorphous allotrope of carbon obtained by the destructive distillation of wood.
• Unlike graphite, its structure is highly disordered and porous, which prevents the effective flow of electrons, making it a poor conductor of electricity.
• Allotropy:
• This is the phenomenon where a single chemical element exists in multiple physical forms that differ in their atomic arrangement.
• Other allotropes of carbon include graphene, carbon nanotubes, and coke.

Important Points

• Lubrication: Beyond conductivity, the weak Van der Waals forces between the hexagonal layers of graphite allow them to slide over each other, making it an excellent dry lubricant.
• Thermal Stability: Graphite has a very high melting point (above 3600 degrees Celsius), which allows it to be used in high-temperature crucibles and nuclear reactors as a neutron moderator.
• Comparison: While diamond is the hardest known natural substance due to its 3D bonding, graphite is soft and slippery due to its 2D layered bonding.

The correct answer is it evaporates rapidly, absorbing heat from the palm.

Key Points

• Evaporation is a surface phenomenon where molecules in the liquid state gain enough energy to enter the gaseous state. This process inherently causes a cooling effect because the escaping high-energy molecules take thermal energy away from the surface.
• Acetone, commonly known as nail polish remover, is a highly volatile liquid. Volatile substances are those that have a high vapor pressure and a low boiling point, meaning they transition from liquid to vapor very easily at room temperature.
• When acetone is poured onto the palm, the molecules use the latent heat of vaporization required for the phase change. They absorb this necessary thermal energy directly from the skin of the palm. Hence, the statement that it evaporates rapidly, absorbing heat from the palm is correct.
• As the palm loses heat to the acetone, the temperature of the skin surface drops. This physical change is detected by thermoreceptors (nerve endings) in the skin, which send signals to the brain, resulting in the sensation of coldness.
• Regarding the other options: Statement 1 is incorrect as the cold sensation is due to heat loss from the hand, not the "release" of cold vapors. Statement 2 is incorrect because acetone does not act as an anesthetic to block nerve signals in this context. Statement 4 is incorrect because the effect is a physical phase change, not a chemical reaction with water.

Additional Information

• Latent Heat of Vaporization: This is the amount of heat energy required to change 1 kg of a liquid into a gas at its boiling point without any change in temperature. For acetone, this value is approximately 518 kJ/kg at its boiling point.
• Boiling Point of Acetone: Acetone has a relatively low boiling point of about 56.05°C (329.2 K) compared to water, which boils at 100°C. This low boiling point is why it feels much colder than water when placed on the skin, as it evaporates much faster.
• Intermolecular Forces: The reason acetone is volatile is due to its weak intermolecular forces. While it has dipole-dipole interactions, it lacks the strong hydrogen bonding present in water, making it easier for acetone molecules to break away from the liquid surface.
• Comparison with Alcohol: Similar to acetone, isopropyl alcohol (rubbing alcohol) and ethanol also produce a cooling sensation. These are also volatile liquids frequently used in sanitizers and medical swabs.
• Factors Affecting Evaporation: The rate at which a liquid like acetone evaporates depends on the surface area (spreading it on the palm increases the rate), temperature, and humidity. Higher temperatures and lower humidity levels accelerate the evaporation process.

Important Points

• Chemical Formula: Acetone is the simplest ketone with the chemical formula (CH3)2CO. It is a colorless, flammable liquid.
• Industrial Uses: Beyond nail polish remover, acetone is used as a solvent for plastics, synthetic fibers, and in the manufacture of methyl methacrylate.
• Biological Context: Small amounts of acetone are produced in the human body through metabolic processes (ketosis). Excessive levels can be found in the breath of individuals with diabetes mellitus.
• Safety: While it provides a cooling sensation, prolonged skin contact with acetone can cause defatting of the skin, leading to irritation or dermatitis, as it dissolves the natural oils present on the skin surface.

The correct answer is 10 m/sec.

Key Points

• The problem involves a car moving with uniform acceleration, which allows the use of the equations of motion derived by Sir Isaac Newton.
• According to the question, the car starts from rest. In physics, when an object starts from rest, its initial velocity (u) is always considered to be 0 m/sec.
• The acceleration (a) provided is 2 m/sec². This means the velocity of the car increases by 2 meters per second every single second.
• The time (t) duration for which the acceleration is applied is 5 seconds.
• To find the final velocity (v), we use the first equation of motion: v = u + at.
• By substituting the given values: v = 0 + (2 × 5). Therefore, v = 10 m/sec.
• The final velocity at the end of 5 seconds is 10 m/sec. Hence, the calculation confirms that the fourth option is correct.

Additional Information

• Uniform Acceleration:
• This occurs when the velocity of an object changes by an equal amount in every equal time interval.
• In a Velocity-Time graph, uniform acceleration is represented by a straight line. The slope of this line gives the value of the acceleration.
• Equations of Motion:
• There are three primary equations for constant acceleration:

1. v = u + at: Relates velocity, acceleration, and time.
2. s = ut + ½at²: Relates displacement (s), initial velocity, time, and acceleration.
3. v² = u² + 2as: Relates final velocity, initial velocity, acceleration, and displacement.

• Scalar and Vector Quantities:
• Velocity is a vector quantity because it has both magnitude and direction. Its S.I. unit is m/s.
• Acceleration is also a vector quantity, representing the rate of change of velocity. Its S.I. unit is m/s².
• Speed is the scalar counterpart of velocity, representing only the magnitude of motion.
• Real-world Application:
• These kinematic equations are fundamental in engineering and automotive safety to calculate braking distances and engine performance. For example, if a car is decelerating (negative acceleration), the same formulas apply by using a negative value for a.

Important Points

• Initial State: Always identify if the object starts from rest (u=0) or comes to a stop (v=0), as these are critical numerical cues in physics problems.
• Units: Ensure all parameters are in S.I. units (meters and seconds) before calculation. If the speed was given in km/hr, it would need to be multiplied by 5/18 to convert to m/sec.
• Direction: Since acceleration is a vector, if the acceleration is in the opposite direction of the velocity, it is termed retardation or deceleration, and its value is taken as negative in the equations.
• Displacement Calculation: If the question asked for the distance traveled in those 5 seconds, we would use s = ut + ½at², resulting in s = 0 + ½(2)(5)² = 25 meters.

The correct answer is Copper.

Key Points

• The ability of one metal to displace another from its salt solution depends on the Reactivity Series of metals. This series ranks metals based on their chemical reactivity, primarily their ability to lose electrons and form positive ions.
• A displacement reaction is a type of chemical reaction where a more reactive element displaces a less reactive element from its compound. In the context of metals, a metal higher in the reactivity series will displace any metal placed below it from its aqueous salt solution.
• Aluminium is significantly more reactive than Lead. In the reactivity series, Aluminium (Al) is positioned much higher than Lead (Pb). Therefore, if Aluminium is added to a lead salt solution, it will displace the lead. Hence, the statement regarding Aluminium is incorrect for this question.
• Calcium and Magnesium are highly reactive metals belonging to the Alkaline Earth Metals group. Both Calcium (Ca) and Magnesium (Mg) sit high up in the reactivity series, well above Lead (Pb). They can easily displace Lead from its salt solution. Hence, the statements regarding Calcium and Magnesium are incorrect for this question.
• Copper (Cu) is placed below Lead (Pb) in the reactivity series. Since Copper is less reactive than Lead, it does not possess the chemical potential to displace Lead ions from a solution. Hence, the statement regarding Copper is correct as it will NOT displace lead.

Additional Information

• The Reactivity Series:
• The series follows a general order: Potassium (K) > Sodium (Na) > Calcium (Ca) > Magnesium (Mg) > Aluminium (Al) > Zinc (Zn) > Iron (Fe) > Tin (Sn) > Lead (Pb) > Hydrogen (H) > Copper (Cu) > Mercury (Hg) > Silver (Ag) > Gold (Au).
• Metals like Potassium and Sodium are so reactive they are often stored in kerosene to prevent reaction with moisture in the air.
• Metals at the bottom, such as Gold and Platinum, are known as noble metals because they are extremely unreactive and do not easily corrode or react with acids.
• Displacement Reaction Mechanism:
• In these reactions, the more reactive metal undergoes oxidation (loses electrons), while the less reactive metal ion in the solution undergoes reduction (gains electrons).
• For example, if Magnesium is put into Lead Nitrate solution, the Magnesium atoms turn into ions, and Lead ions turn into solid metallic Lead.
• Since Copper is lower than Lead, it cannot force its electrons onto Lead ions; therefore, no reaction occurs when Copper is placed in a Lead salt solution.
• Lead (Pb):
• Lead has the atomic number 82. It is a heavy metal that is denser than most common materials.
• It is relatively resistant to corrosion but is placed above Hydrogen in the reactivity series, meaning it can theoretically displace hydrogen from dilute acids, though the reaction is often slowed by the formation of insoluble coatings.
• Copper (Cu):
• Copper has the atomic number 29. It is a transition metal known for high thermal and electrical conductivity.
• It is one of the few metals that occur in nature in a directly usable metallic form (native metals). Because it is below Hydrogen and Lead, it is considered a relatively stable and unreactive metal in many environments.

Important Points

• Electrochemical Context: The reactivity series is fundamentally linked to Standard Reduction Potentials. Metals with a more negative reduction potential are more reactive and act as stronger reducing agents.
• Practical Application: This principle is used in Galvanization, where Zinc (more reactive) is used to coat Iron to prevent rusting. The Zinc displaces the corrosive action from the Iron by acting as a sacrificial anode.
• Metal Extraction: The reactivity of a metal determines the method used to extract it from its ore. Highly reactive metals like Aluminium are extracted via electrolysis, while less reactive metals like Lead or Iron can be extracted by reduction with Carbon.

The correct answer is They always form closed loops.

Key Points

• Magnetic field lines are imaginary lines used to represent the magnetic field around a magnet. They are continuous and form closed loops. Externally, they emerge from the North pole and enter the South pole, while internally, they travel from the South pole back to the North pole. Hence, statement 1 is correct.
• It is a fundamental property that magnetic field lines never intersect or cross each other. If they were to intersect at a certain point, it would imply that a magnetic compass placed at that point would show two different directions for the magnetic field simultaneously, which is physically impossible. Hence, statement 2 is incorrect.
• Magnetic field lines do not start and end at the same pole. They exist as a continuous flow from the North pole to the South pole outside the magnet and complete the circuit inside. This distinguishes them from electric field lines, which can start on a positive charge and end on a negative charge. Hence, statement 3 is incorrect.
• Because magnetic poles always exist in dipoles (North and South), these field lines must return to their source, making them closed curves rather than open curves. There are no isolated magnetic monopoles in nature that would allow for open field lines. Hence, statement 4 is incorrect.
• Hence, the correct answer is they always form closed loops.

Additional Information

• Nature of Magnetic Field:
• The magnetic field is a vector quantity, meaning it possesses both magnitude and direction.
• The SI unit of magnetic field strength is the Tesla (T), named after Nikola Tesla. Another common unit used is the Gauss, where 1 Tesla equals 10,000 Gauss.
• Field Strength and Density:
• The relative strength of the magnetic field is shown by the degree of closeness of the field lines.
• Where the lines are crowded (near the poles), the magnetic field is strongest. As the lines spread apart, the field becomes weaker.
• Directionality:
• The tangent drawn to a magnetic field line at any given point indicates the direction of the magnetic field at that specific location.
• The direction is determined by the way the North pole of a magnetic needle would point when placed in the field.

Important Points

• Gauss's Law for Magnetism: This law states that the net magnetic flux through any closed surface is always zero. This mathematical principle confirms that magnetic monopoles do not exist and that field lines must be closed loops.
• Discovery: The concept of lines of force or field lines was first introduced by the 19^th-century scientist Michael Faraday to visualize electromagnetic induction and magnetic interactions.
• Earth's Magnetic Field: The Earth acts as a giant bar magnet. Its magnetic field lines emerge from the magnetic South Pole (near the geographic North) and enter the magnetic North Pole (near the geographic South). This property is essential for navigation using a magnetic compass.
• Solenoids: In a current-carrying solenoid, the magnetic field pattern is remarkably similar to that of a bar magnet, where the lines form parallel and uniform paths inside the coil, reinforcing the concept of closed loops.

The correct answer is At the centre of curvature, same size.

Key Points

• When an object is placed at the centre of curvature (C) of a concave mirror, the light rays reflecting from the object converge at the same point C.
• According to the laws of reflection, a ray parallel to the principal axis passes through the focus (F) after reflection, and a ray passing through the focus becomes parallel to the principal axis.
• These reflected rays intersect exactly at the centre of curvature, forming the image at that location. Hence, the statement that the image is formed at the centre of curvature is correct.
• The nature of the image formed in this specific case is real and inverted because the light rays actually meet at the point of image formation.
• The size of the image is exactly the same as the size of the object, which is a unique property of this position in concave mirrors. Hence, the correct answer is At the centre of curvature, same size.

Additional Information

• Concave Mirror Characteristics:
• A concave mirror is also known as a converging mirror because it tends to collect light that falls on it, refocusing parallel incoming rays to a single focus.
• It has a reflective surface that is curved inward, resembling a portion of the inner surface of a hollow sphere.
• The uses of concave mirrors include shaving mirrors, dentist's mirrors to see enlarged images of teeth, and reflectors in car headlights and searchlights.
• Other Image Positions in Concave Mirrors:
• At Infinity: If the object is at infinity, the image is formed at the focus (F) and is highly diminished and point-sized.
• Beyond C: When the object is beyond the centre of curvature, the image is formed between F and C, and it is diminished and real.
• Between C and F: If the object is placed between the centre of curvature and the focus, the image is formed beyond C and is magnified.
• At Focus (F): When the object is at the focus, the image is formed at infinity and is highly magnified.
• Between F and Pole (P): This is the only case where a virtual and erect image is formed behind the mirror, which is also magnified.

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.
• Magnification (m): It is defined as the ratio of the height of the image (h') to the height of the object (h), expressed as m = h'/h = -v/u. For an object at C, the magnification is -1, indicating a same-sized, inverted image.
• Radius of Curvature (R): The distance from the pole to the centre of curvature is called the radius of curvature. In spherical mirrors, the focal length is exactly half of the radius of curvature (f = R/2).
• Sign Convention: According to the New Cartesian Sign Convention, the distances measured in the direction of incident light are positive, while those measured against it are negative. For a concave mirror, the focal length and radius of curvature are always negative.

The correct answer is Endothermic energy change.

Key Points

• Decomposition reactions are chemical processes where a single reactant breaks down into two or more simpler products.
• For a decomposition reaction to occur, the chemical bonds within the reactant molecule must be broken. Breaking these bonds requires an input of energy from the external environment.
• Because these reactions involve the absorption of energy (in the form of heat, light, or electricity), they are generally characterized by an Endothermic energy change. Hence, option 1 is correct.
• An Exothermic energy change is the opposite, where energy is released into the surroundings; this is typically seen in combination or combustion reactions rather than decomposition. Hence, option 3 is incorrect.
• The general representation of this reaction is AB + Energy → A + B, indicating that the energy is a necessary "reactant" for the process to proceed.

Additional Information

• Types of Decomposition Reactions:
• Thermal Decomposition: This occurs when heat energy is utilized for the reaction. For example, when Calcium Carbonate (CaCO3) is heated, it decomposes into Calcium Oxide (CaO) and Carbon Dioxide (CO2). This process is vital in the cement industry.
• Electrolytic Decomposition (Electrolysis): This involves passing an electric current through a compound in a liquid or molten state. A well-known example is the electrolysis of water, which decomposes H2O into Hydrogen (H2) and Oxygen (O2) gases.
• Photolytic Decomposition (Photolysis): This is triggered by light energy. For instance, Silver Chloride (AgCl) turns grey in sunlight because it decomposes into Silver metal and Chlorine gas. This principle was historically used in black and white photography.
• Energy Dynamics:
• In an Endothermic reaction, the total energy of the products is higher than the total energy of the reactants because energy has been "stored" in the system during the reaction.
• In contrast, Exothermic reactions (like respiration or burning of natural gas) result in products with lower energy than the reactants, with the excess being released as Heat.

Important Points

• The Law of Conservation of Energy states that energy cannot be created or destroyed, only transformed. In decomposition, kinetic or thermal energy from the environment is transformed into chemical potential energy within the products.
• Activation Energy: Even if a reaction is overall endothermic, it requires a specific amount of activation energy to initiate the bond-breaking process.
• Digestive System: In the human body, the digestion of food is a series of decomposition reactions where complex substances like carbohydrates and proteins are broken down into simpler molecules like glucose and amino acids, often aided by enzymes.

The correct answer is Cooking food.

Key Points

• Ultrasound refers to sound waves with frequencies higher than the upper audible limit of human hearing, which is typically 20,000 Hz (20 kHz).
• Cleaning delicate machinery parts: This is a major application where objects like electronic components, watch parts, or surgical instruments are placed in a cleaning solution. High-frequency ultrasound waves are sent through the liquid, causing high-pressure bubbles to form and collapse (cavitation), which dislodges dust, grease, and dirt from hard-to-reach surfaces. Hence, this is a correct application.
• Detecting cracks and flaws in metal blocks: In large constructions like bridges and machines, ultrasound is used to detect internal defects. The waves are passed through the metal block and detected by sensors; if there is a crack, the waves are reflected back, revealing the flaw. Hence, this is a correct application.
• Imaging of internal organs: Ultrasonography is a diagnostic medical technique used to visualize internal organs such as the liver, kidneys, and uterus. It helps doctors detect abnormalities like stones or tumors. Hence, this is a correct application.
• Cooking food: This is not an application of ultrasound. Cooking generally utilizes thermal energy (conduction, convection, radiation) or Microwaves, which are electromagnetic waves that vibrate water molecules to produce heat. Ultrasound is a mechanical wave and does not possess the thermal properties required for standard cooking. Hence, statement 1 is the correct answer as it is NOT an application.

Additional Information

• Medical Applications (Echocardiography):
• Echocardiography is a specific use of ultrasound to capture images of the heart and its movements, used to diagnose cardiac diseases.
• Another medical use is Lithotripsy, where high-energy ultrasound waves are employed to break kidney stones into small grains that can be passed out through urine.
• Industrial and Marine Uses (SONAR):
• SONAR (Sound Navigation and Ranging) utilizes ultrasound to measure the depth of the ocean and to locate underwater objects like submarines, icebergs, or sunken ships.
• The echo-ranging principle is used here, where the time interval between transmission and reception of the pulse is recorded to calculate distance.
• Biological Use (Echolocation):
• Animals like bats, dolphins, and porpoises emit ultrasound waves to navigate and detect prey in the dark.
• Bats can produce frequencies up to 100 kHz to create a "sound map" of their surroundings.
• Physical Properties:
• Ultrasound waves have a short wavelength, allowing them to penetrate deep into materials and reflect off very small surfaces, making them ideal for non-destructive testing and high-resolution imaging.

The correct answer is The particles of sugar get into the spaces between particles of water.

Key Points

• Everything around us is made of matter, which is composed of extremely tiny particles. These particles are so small that they cannot be seen with the naked eye.
• One of the fundamental characteristics of matter is that there are intermolecular spaces between the particles. In liquids like water, these spaces are larger than in solids but smaller than in gases.
• When sugar (the solute) is added to water (the solvent) and stirred, the sugar crystals break down into individual molecules.
• These sugar molecules then distribute themselves uniformly throughout the liquid by occupying the empty spaces or voids between the water particles. Hence, the statement that particles of sugar get into the spaces between particles of water is correct.
• The sugar particles do not simply float on top because they are denser than water and, once dissolved, they form a uniform mixture. Hence, statement 1 is incorrect.
• A colloidal solution is a mixture where the particle size is between 1 to 1000 nanometers and exhibits the Tyndall effect. Sugar in water forms a true solution, where particles are smaller than 1 nm. Hence, statement 2 is incorrect.
• Sugar is highly soluble in water due to the formation of hydrogen bonds between sugar molecules and water molecules, meaning they mix thoroughly. Hence, statement 4 is incorrect.

Additional Information

• True Solution: A true solution is a homogeneous mixture of two or more substances. In this case, the sugar solution is clear, transparent, and the components cannot be separated by filtration.
• Solute and Solvent: In a solution, the substance that is dissolved is called the solute (e.g., sugar), and the substance in which it is dissolved is the solvent (e.g., water).
• Conservation of Volume: Interestingly, when a small amount of sugar is dissolved in water, the overall volume of the liquid does not increase significantly. This is a classic experiment used in General Science to prove that matter has spaces between its particles.
• Saturation: A solution is called a saturated solution when no more solute can be dissolved in the solvent at a specific temperature. If more sugar is added beyond this point, it will settle at the bottom.
• Kinetic Theory: The particles of matter are in continuous motion. This constant movement helps the sugar particles diffuse and spread evenly throughout the water.

Important Points

• Effect of Temperature: The solubility of sugar in water increases with an increase in temperature. This is because higher temperatures provide more kinetic energy, which increases the vibration of water molecules and expands the intermolecular spaces, allowing more sugar to fit in.
• Physical Change: Dissolving sugar in water is considered a physical change because no new chemical substance is formed; the sugar can be recovered by the process of evaporation or crystallization.
• Particle Size: In a sugar solution, the particles are less than 10⁻⁹ meters (or 1 nm) in diameter, which is why they do not scatter a beam of light.
• Concentration: The amount of sugar present in a given amount of solution is known as its concentration, which can be expressed in terms of molarity, molality, or mass percentage.

The correct answer is It is a gradual process during adolescence involving physical and hormonal changes.

Key Points

• Sexual maturation, commonly known as puberty, is the biological process by which a child's body matures into an adult body capable of sexual reproduction.
• Sexual maturation does not begin after adulthood; rather, it is the bridge that leads to adulthood, typically starting between the ages of 10 and 12 years. Hence, statement 1 is incorrect.
• This process occurs in both males and females. While the specific physical changes differ between the sexes, both undergo significant hormonal and physiological shifts. Hence, statement 2 is incorrect.
• It is characterized as a gradual process that spans several years during the period of adolescence. It involves the complex interaction of the nervous system and the endocrine system to trigger growth and development. Hence, statement 3 is correct.
• Sexual maturation is not a sudden or overnight event. It involves a series of stages (often measured by the Tanner scale) and does not mark the immediate end of body growth; in fact, a significant growth spurt occurs during this phase. Hence, statement 4 is incorrect.
• During this period, the gonads (testes in males and ovaries in females) begin to produce gametes and secrete sex hormones, leading to the development of secondary sexual characteristics.

Additional Information

• Hormonal Regulation:
• The process is initiated by the hypothalamus, which releases Gonadotropin-Releasing Hormone (GnRH).
• This stimulates the pituitary gland to secrete Luteinizing Hormone (LH) and Follicle-Stimulating Hormone (FSH).
• In males, these hormones stimulate the testes to produce testosterone and sperm.
• In females, they stimulate the ovaries to produce estrogen and progesterone, leading to ovulation.
• Changes in Females:
• The first sign is usually breast development (thelarche), followed by the growth of pubic hair.
• Menarche, the first menstrual period, typically occurs about two years after the start of breast development, usually around age 12 to 13.
• There is also a widening of the pelvis and increased deposition of fat in the hips and thighs.
• Changes in Males:
• The first sign is the enlargement of the testes, followed by the growth of pubic hair and facial hair.
• The larynx (voice box) enlarges, causing the voice to deepen.
• Males experience increased muscle mass and a significant increase in height during their growth spurt.

Important Points

• Adolescence: The World Health Organization (WHO) defines adolescence as the period of life between 10 and 19 years of age. It is a critical phase of physical, psychological, and social transition.
• Secondary Sexual Characteristics: These are features that appear during puberty which distinguish the two sexes of a species, but which are not directly part of the reproductive system (e.g., facial hair in men, breasts in women).
• Primary Sexual Characteristics: These refer to the genitals and internal reproductive organs (testes, ovaries, uterus) that are present at birth but mature during sexual maturation.
• Environmental Factors: The timing of sexual maturation can be influenced by genetics, nutrition, and overall health. Improved nutrition globally has led to a historical trend of earlier onset of puberty.

The correct answer is Motor area.

Key Points

• The Motor area, specifically known as the Primary Motor Cortex, is the region of the brain responsible for the direct control and execution of voluntary movements.
• It is located in the frontal lobe of the cerebral cortex, specifically along a ridge called the precentral gyrus.
• This area generates neural impulses that travel down the spinal cord to stimulate skeletal muscles, allowing for conscious actions like walking, writing, or speaking.
• The Motor area is organized somatotopically, meaning different parts of the cortex correspond to different parts of the body; this mapping is often referred to as the Motor Homunculus.
• Control is contralateral, which means the motor area in the left hemisphere of the brain controls the voluntary muscles on the right side of the body, and vice versa.

Additional Information

• Medulla Oblongata:
• This structure is located in the brainstem and is primarily responsible for autonomic (involuntary) functions.
• It regulates vital life processes such as heartbeat, breathing, blood pressure, and reflex actions like sneezing or swallowing.
• Sensory Area:
• Located in the parietal lobe (specifically the postcentral gyrus), the primary somatosensory cortex processes incoming signals from the body.
• It is responsible for perceiving sensations such as touch, pressure, temperature, and pain.
• Association Area:
• These areas are found throughout the cerebral cortex and are not involved in primary motor or sensory functions.
• They function to integrate information from multiple sources, facilitating complex mental processes such as learning, memory, thought, and decision-making.

Important Points

• Voluntary Muscles: These are striated muscles (skeletal muscles) that are under conscious control of the somatic nervous system.
• Role of the Cerebellum: While the motor area initiates the signal for movement, the cerebellum is crucial for coordinating those movements, maintaining balance, and ensuring posture.
• Frontal Lobe Functions: In addition to the motor area, the frontal lobe houses areas responsible for executive function, personality, and speech production (Broca's area).
• Neural Pathway: Motor signals descend from the cerebral cortex through the corticospinal tract, which is the primary pathway for voluntary motor control in humans.

The correct answer is Embryo.

Key Points

• The Embryo is the most vital part of a seed as it is the "baby plant" that contains the potential to grow into a new individual. It consists of the plumule (the embryonic shoot) and the radicle (the embryonic root). Hence, statement 2 is correct.
• Chlorophyll is the green pigment located in the chloroplasts of plant cells, primarily in the leaves and stems. While it is essential for photosynthesis, it is not the structural component within a seed that initiates the growth of a new plant; rather, it is synthesized as the seedling develops. Hence, statement 1 is incorrect.
• A Sepal is a part of the flower whorl (the calyx) that functions to protect the flower bud before it blooms. It is an external floral organ and is not a component found inside a seed. Hence, statement 3 is incorrect.
• Nectar is a sugar-rich liquid secreted by plants in nectaries to attract pollinators such as bees, birds, and bats. It plays a role in pollination and reproduction but is not stored inside the seed for the growth of the new plant. Hence, statement 4 is incorrect.
• A seed essentially consists of three main parts: the seed coat, the embryo, and the endosperm (or cotyledons) which provides nutrition during the initial stages of germination.

Additional Information

• Structure of the Seed Embryo:
• The Radicle is the first part of the embryo to emerge during germination, eventually forming the primary root system to absorb water and nutrients.
• The Plumule develops into the shoot system, producing the first true leaves of the plant.
• The Hypocotyl is the region of the embryo between the radicle and the cotyledons, which helps push the seed upward through the soil.
• Nutritional Support:
• Cotyledons, also known as seed leaves, store food (starch, proteins, and fats) to sustain the embryo until it can perform photosynthesis.
• In monocots (like Maize), there is only one cotyledon, while in dicots (like Beans), there are two cotyledons.
• Seed Coat (Testa):
• The seed coat is the hard outer covering that protects the delicate embryo from mechanical injury, desiccation, and pathogens.

Important Points

• Germination Factors: For the embryo to begin growing, specific environmental conditions must be met, including the presence of moisture (water), oxygen for respiration, and a suitable temperature.
• Dormancy: Many seeds undergo a period of dormancy, a state of metabolic inactivity that prevents the embryo from germinating until conditions are favorable for survival, often triggered by seasonal changes or hormonal levels like Abscisic Acid.
• Seed Dispersal: Plants have evolved various mechanisms to disperse seeds, such as wind (Anemochory), water (Hydrochory), and animals (Zoochory), ensuring the embryo can find a suitable location to grow away from the parent plant.

The correct answer is It converts starch into simple sugars.

Key Points

• Salivary amylase, also known as ptyalin, is a digestive enzyme present in human saliva that initiates the process of chemical digestion in the mouth.
• The primary function of this enzyme is to catalyze the hydrolysis of starch (a complex polysaccharide) into smaller, simpler sugars like maltose, maltotriose, and limit dextrins. Hence, statement 3 is correct.
• Statement 1 suggests the conversion of amino acids into protein. This is a synthetic (anabolic) process known as translation that occurs within the ribosomes of cells, not in the digestive tract. Hence, statement 1 is incorrect.
• Statement 2 refers to the conversion of simple sugars into starch. This is a process of polymerization typically performed by plants to store energy; humans store excess sugar as glycogen in the liver and muscles. Hence, statement 2 is incorrect.
• Statement 4 describes the conversion of proteins into amino acids. This task is performed by proteolytic enzymes such as pepsin in the stomach and trypsin or chymotrypsin in the small intestine. Hence, statement 4 is incorrect.
• About 30% of starch is hydrolyzed in the oral cavity by salivary amylase before the food is swallowed into the esophagus.

Additional Information

• Composition of Saliva: Human saliva is composed of approximately 99.5% water and 0.5% solutes, including electrolytes (sodium, potassium, chloride), mucus, antibacterial agents (lysozyme), and digestive enzymes.
• Salivary Glands: There are three major pairs of salivary glands: the parotid (located near the ears), submandibular (below the jaw), and sublingual (under the tongue). These glands produce between 1 and 1.5 liters of saliva every day.
• Enzymatic Action: Salivary amylase requires chloride ions as activators to function effectively. It specifically targets the alpha-1,4-glycosidic bonds in starch molecules.
• pH Sensitivity: The enzyme works best at an optimum pH of 6.8. When the food (bolus) reaches the stomach, the highly acidic gastric juice (pH 1.5 to 3.5) denatures the salivary amylase, halting carbohydrate digestion until it reaches the small intestine.
• Digestion Stages: Digestion is broadly divided into mechanical digestion (mastication or chewing) and chemical digestion (enzymatic breakdown). Saliva facilitates both by lubricating the food and providing the necessary enzymes.

Important Points

• Pancreatic Amylase: Since food spends very little time in the mouth, most starch digestion is completed in the small intestine by pancreatic amylase, which is secreted by the pancreas into the duodenum.
• Role of Lysozyme: Saliva also contains lysozyme, which acts as an antimicrobial agent by attacking the cell walls of certain bacteria, helping to prevent infections and tooth decay.
• Glucose Absorption: The end product of total carbohydrate digestion is glucose, which is absorbed into the bloodstream through the villi of the small intestine to provide ATP (energy) to the body cells.
• Biological Catalyst: Enzymes like amylase are biological catalysts that speed up chemical reactions without being consumed in the process, allowing digestion to occur rapidly at body temperature (37°C).

The correct answer is white light.

Key Points

• Sir Isaac Newton was the first to use a glass prism to obtain the spectrum of sunlight. He observed that when a narrow beam of sunlight falls on a prism, it splits into a band of seven colors.
• In his experiment, Newton placed a second identical prism in an inverted position with respect to the first prism. He found that the second prism allowed all the colors of the spectrum to pass through it and recombine them.
• The light emerging from the second, inverted prism was a beam of white light. This observation led Newton to the conclusion that sunlight is made up of seven colors.
• The first prism causes dispersion, which is the splitting of white light into its constituent colors (VIBGYOR), while the second prism, being inverted, reverses the effect by refracting the colored rays back together.
• Newton’s experiment was crucial in proving that the prism itself does not "create" colors but rather separates the colors already present in white light.

Additional Information

• Dispersion of Light: This is the phenomenon where white light splits into its component colors when passing through a transparent medium like a glass prism. The band of colored components is called a spectrum.
• VIBGYOR: The acronym stands for Violet, Indigo, Blue, Green, Yellow, Orange, and Red. These represent the sequence of colors in the spectrum, where Violet bends the most and Red bends the least.
• Refraction: The process of recombination and dispersion occurs because different colors of light travel at different speeds in glass. Since each color has a different wavelength, they bend at different angles when entering or exiting the prism.
• Natural Examples: The most common natural phenomenon based on dispersion is the rainbow. It is formed by the dispersion, refraction, and internal reflection of sunlight by tiny water droplets in the atmosphere.
• Wave Nature: The wavelength of Red light is the longest (approximately 700 nm), while the wavelength of Violet light is the shortest (approximately 400 nm) in the visible spectrum.

Important Points

• Historical Context: Newton performed these experiments around 1665-1666. His findings were later published in his famous work titled "Opticks" in 1704.
• Prism Geometry: For successful recombination, the two prisms must be identical and placed parallel to each other but in opposite (inverted) orientations. If the second prism is not inverted, it would simply disperse the light further.
• Visible Spectrum: The visible spectrum is only a small part of the Electromagnetic Spectrum. Human eyes are sensitive to this specific range of frequencies that combine to form white light.
• Angle of Deviation: The total deviation produced by the first prism is canceled out by the second inverted prism, allowing the emergent beam to be parallel to the incident beam, much like light passing through a rectangular glass slab.

The correct answer is Lysosomes.

Key Points

• Lysosomes are membrane-bound vesicular structures formed by the process of packaging in the Golgi apparatus. They contain hydrolytic enzymes but do not possess their own DNA or ribosomes.
• The Nucleus is the most essential part of a eukaryotic cell, acting as the control center. It contains the majority of the cell's genetic material in the form of DNA molecules organized into chromosomes.
• Mitochondria are known as semi-autonomous organelles because they contain their own circular DNA, RNA, and 70S ribosomes, allowing them to synthesize some of their own proteins.
• Plastids, found in all plant cells and euglenoids, also possess their own double-stranded circular DNA and ribosomes. Similar to mitochondria, they can replicate independently of the nucleus.
• Among the given options, only Lysosomes lack a genome, as their primary function is intracellular digestion rather than energy production or genetic storage.

Additional Information

• Lysosomes:
• These are often referred to as the "suicidal bags" of the cell because if the cell gets damaged, lysosomes may burst and the enzymes digest their own cell.
• They are rich in almost all types of hydrolytic enzymes (hydrolases – lipases, proteases, carbohydrases) optimally active at the acidic pH.
• Mitochondria:
• Commonly called the "Powerhouse of the cell," they are the sites of aerobic respiration and produce cellular energy in the form of ATP (Adenosine Triphosphate).
• The presence of mtDNA (Mitochondrial DNA) supports the Endosymbiotic Theory, which suggests they originated from prokaryotic cells.
• Plastids:
• Based on the type of pigments, plastids can be classified into chloroplasts, chromoplasts, and leucoplasts.
• Chloroplasts contain chlorophyll and carotenoid pigments which are responsible for trapping light energy essential for photosynthesis.
• Organelle Autonomy:
• Organelles like the Nucleus, Mitochondria, and Chloroplasts (a type of plastid) are double-membrane bound, whereas Lysosomes are single-membrane bound.
• The DNA found in mitochondria and plastids is circular, resembling bacterial DNA, while nuclear DNA is typically linear.

Important Points

• Genetic Material Location: In a typical animal cell, DNA is strictly confined to the nucleus and the mitochondria. In plant cells, it is found in the nucleus, mitochondria, and plastids.
• Ribosomes: While ribosomes themselves contain RNA, they are not considered organelles that "have" DNA; rather, they are the machinery that translates genetic information.
• Prokaryotic Cells: Unlike eukaryotes, prokaryotes do not have membrane-bound organelles like mitochondria or lysosomes; their DNA is found in a region called the nucleoid.

The correct answer is potential energy and kinetic energy.

Key Points

• Mechanical energy is defined as the energy possessed by an object due to its motion or its stored energy of position. It is a macroscopic form of energy.
• It is mathematically expressed as the sum of kinetic energy and potential energy. Hence, option 1 is correct.
• Kinetic Energy (KE): This is the energy of an object resulting from its motion. The amount of kinetic energy depends on the mass of the object and the square of its velocity (K.E. = 1/2 mv²).
• Potential Energy (PE): This is the stored energy an object has because of its position or state. Common forms include gravitational potential energy (based on height) and elastic potential energy (stored in a stretched spring).
• In the absence of non-conservative forces like friction or air resistance, the total mechanical energy of a system remains constant, which is known as the principle of conservation of mechanical energy.

Additional Information

• Other Forms of Energy:
• Chemical Energy: Energy stored in the bonds of chemical compounds. It is released during a chemical reaction, often in the form of heat.
• Electrical Energy: Energy caused by the movement of electrons through a conductor, such as a wire.
• Heat Energy (Thermal Energy): Energy that comes from the temperature of the matter, related to the vibration of atoms and molecules.
• Light Energy: A form of electromagnetic radiation that is visible to the human eye, traveling in waves.
• Sound Energy: Energy produced by vibrations as they travel through a medium (like air or water).
• While these other forms are important in thermodynamics and electromagnetism, they are not categorized under the mechanical energy of a physical body in classical mechanics.

Important Points

• SI Unit: The standard unit for measuring mechanical energy is the Joule (J), named after the physicist James Prescott Joule. One Joule is equal to 1 kg⋅m²/s².
• Scalar Quantity: Energy, including mechanical energy, is a scalar quantity, meaning it has magnitude but no direction.
• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy.
• Conservative vs. Non-conservative Forces: Forces like gravity and spring forces are conservative because the work done by them depends only on the initial and final positions. Friction is a non-conservative force because it converts mechanical energy into heat, reducing the total mechanical energy available.

Shortcut Trick Use Option Verification method to find the Positive Number quickly. Check Option (4): Number = 8. Increase by 14 ⇒ 8 + 14 = 22. Reciprocal of 8 is 1/8. Now, 176 × (1/8) = 22. Since 22 = 22, the condition is satisfied. ∴ The correct answer is 8.

Alternate Method Given: A positive number increased by 14 equals 176 times its reciprocal. Formula Used: Let the positive number be x. Equation: x + 14 = 176 × (1/x)

Calculations: ⇒ x + 14 = 176/x ⇒ x2 + 14x = 176 ⇒ x2 + 14x − 176 = 0 ⇒ x2 + 22x − 8x − 176 = 0 ⇒ x(x + 22) − 8(x + 22) = 0 ⇒ (x − 8)(x + 22) = 0 ⇒ x = 8 or x = −22 ⇒ Since the number must be positive, x = 8. ∴ The correct answer is 8.

Additional Information Quadratic Equations A quadratic equation of the form ax2 + bx + c = 0 can be solved using the factorization method or the formula x = [−b ± √(b2 − 4ac)] / 2a. Reciprocal Property The reciprocal of a non-zero number n is 1/n. The product of a number and its reciprocal is always 1. Nature of Roots For the equation x2 + 14x − 176 = 0, the discriminant D = b2 − 4ac = 142 − 4(1)(−176) = 196 + 704 = 900. Since D > 0 and a perfect square, roots are real and rational.

Shortcut Trick

Net deviation for the first 9 numbers = 5 × (55 − 58) + 4 × (62 − 58) = 5 × (−3) + 4 × (+4) = −15 + 16 = +1.

To balance the overall average, the sum of the 10^th, 11^th, and 12^th numbers = (3 × 58) − 1 = 174 − 1 = 173.

Let 10^th number = x, then 11^th = x + 10 and 12^th = x − 5.

Sum: x + x + 10 + x − 5 = 173 ⇒ 3x + 5 = 173 ⇒ 3x = 168 ⇒ x = 56.

Average of 10^th (56) and 11^th (66) = (56 + 66) ÷ 2 = 61.

∴ The correct answer is 61.

Alternate Method

Given: Average of 12 numbers = 58; First 5 numbers avg = 55; Next 4 numbers avg = 62.

Relation: 10^th = 11^th − 10; 10^th = 12^th + 5.

⇒ Sum of all 12 numbers = 12 × 58 = 696

⇒ Sum of first 5 numbers = 5 × 55 = 275

⇒ Sum of next 4 numbers = 4 × 62 = 248

⇒ Sum of 10^th, 11^th, and 12^th numbers = 696 − (275 + 248) = 696 − 523 = 173

⇒ Let 10^th number be x. Then 11^th number = x + 10 and 12^th number = x − 5

⇒ x + (x + 10) + (x − 5) = 173

⇒ 3x + 5 = 173 ⇒ 3x = 168 ⇒ x = 56

⇒ 10^th number = 56 and 11^th number = 56 + 10 = 66

⇒ Average of 10^th and 11^th = (56 + 66) ÷ 2 = 122 ÷ 2 = 61

∴ The correct answer is 61.

Additional Information

Sum of Observations

The sum of any number of observations is equal to the product of their average and the total number of observations (Sum = Average × n).

Net Deviation Method

In any data set, the sum of deviations of all observations from the arithmetic mean is always zero (Σ(x_i − Mean) = 0).

Combined Average Formula

If two groups have sizes n₁, n₂ and averages a₁, a₂, their combined average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Shortcut Trick

Identify 9x² − 4y² as (3x)² − (2y)² which factors into (3x − 2y)(3x + 2y).

The term (3x − 2y) cancels out with the first denominator.

The term (3x + 2y) cancels out with the second denominator.

Only the numerator 3x remains as the final result.

∴ The correct answer is 3x.

Alternate Method

Given: ((9x² − 4y²) ÷ (3x − 2y)) × (3x ÷ (3x + 2y))

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

⇒ Factorizing the numerator: 9x² − 4y² = (3x)² − (2y)² = (3x − 2y)(3x + 2y)

⇒ Substituting the factorized form: [(3x − 2y)(3x + 2y) ÷ (3x − 2y)] × [3x ÷ (3x + 2y)]

⇒ Canceling (3x − 2y) from the first part: (3x + 2y) × [3x ÷ (3x + 2y)]

⇒ Canceling (3x + 2y) from the product: 3x

∴ The correct answer is 3x.

Additional Information

Algebraic Identity: Difference of Squares

The identity a² − b² = (a − b)(a + b) is used to simplify rational expressions and factorize polynomials efficiently.

Rational Expression Multiplication

To multiply algebraic fractions, factorize all terms and cancel common factors from the numerator and denominator before multiplying.

Perfect Square Identity

Another related identity is (a + b)² = a² + 2ab + b², which is often used in quadratic simplification.

Shortcut Trick

In right-angled triangle BDC, the angle at C is 90° − 42° = 48°.

For triangle AEC, the exterior angle ∠AEB is equal to the sum of the two opposite interior angles.

∴ ∠AEB = ∠CAE + ∠C = 24° + 48° = 72°.

∴ The correct answer is 72°.

Alternate Method

Given:

In ΔABC, BD ⊥ AC at D, ∠DBC = 42°, and ∠CAE = 24°.

Formula Used:

1. Sum of angles in a triangle = 180°
2. Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Calculations:

In right-angled ΔBDC:

⇒ ∠BDC = 90°

⇒ ∠DBC + ∠BDC + ∠C = 180°

⇒ 42° + 90° + ∠C = 180°

⇒ ∠C = 180° − 132° = 48°

Now, in ΔAEC, E is a point on BC, so ∠AEB is the exterior angle at vertex E:

⇒ ∠AEB = ∠CAE + ∠C

⇒ ∠AEB = 24° + 48°

⇒ ∠AEB = 72°

Option Analysis:

⇒ Option 1 (72°): Correct, as calculated using the exterior angle theorem.

⇒ Option 2 (67°): Incorrect, as it does not satisfy the angle sum property of ΔAEC.

⇒ Option 3 (59°): Incorrect, as it deviates from the geometric properties of the given triangle.

⇒ Option 4 (58°): Incorrect, as it is mathematically inconsistent with the given angle values.

∴ The correct answer is 72°.

Additional Information

Exterior Angle Theorem

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

Angle Sum Property

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

Right-Angled Triangle Properties

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

Shortcut Trick

Ratio of Profit = Ratio of Investment (when time is constant).

Total profit = ₹5,000 | Anmol's share = ₹2,300 | Ayushi's share = 5,000 − 2,300 = ₹2,700.

Ratio of Profit (Anmol : Ayushi) = 2,300 : 2,700 = 23 : 27.

Total ratio parts = 23 + 27 = 50 units → ₹41,400.

1 unit = 41,400 ÷ 50 = ₹828.

Anmol's investment = 23 × 828 = ₹19,044.

∴ The correct answer is ₹19,044.

Alternate Method

Given: Total Investment = ₹41,400, Total Profit = ₹5,000, Anmol's Profit = ₹2,300. Formula: Investment_Anmol = (Profit_Anmol ÷ Total Profit) × Total Investment

⇒ Calculate Anmol's share of profit as a fraction of total profit: 2,300 ÷ 5,000. ⇒ Simplify the fraction: 23 ÷ 50. ⇒ Multiply this fraction by the Total Investment to find Anmol's contribution. ⇒ Anmol's Investment = (23 ÷ 50) × 41,400. ⇒ Anmol's Investment = 23 × 828 = 19,044.

∴ The correct answer is ₹19,044.

Additional Information

Partnership Principles

When the period of investment is the same for all partners, the profit is distributed in the direct ratio of their investments.

Compound Partnership

If investments are for different time periods, Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Shortcut Trick

Since speed is constant, distance is directly proportional to time (D₁/T₁ = D₂/T₂).

180 km → 3 hours

1 km → 3/180 hours = 1/60 hours

420 km → 420 × (1/60) = 7 hours

∴ The correct answer is 7 hr.

Alternate Method

Given:

Initial Distance (D₁) = 180 km

Initial Time (T₁) = 3 hours

New Distance (D₂) = 420 km

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ Speed = 180 ÷ 3 = 60 kmph

⇒ Time required for 420 km = Distance ÷ Speed

⇒ Time = 420 ÷ 60

⇒ Time = 7 hours

∴ The correct answer is 7 hr.

Additional Information

Speed-Distance-Time Relationship

When time is constant, distance is proportional to speed. When distance is constant, speed is inversely proportional to time.

Unit Conversion

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

Average Speed

Average Speed = Total Distance ÷ Total Time. For two equal distances with speeds x and y, Average Speed = 2xy/(x + y).

Shortcut Trick

Total Voters = 25,20,000

Valid Votes = 25,20,000 × 0.98 × 0.95

Losing candidate % = 100% − 65% = 35%

Losing candidate votes = 25,20,000 × 98/100 × 95/100 × 35/100 = 8,21,142

∴ The correct answer is 8,21,142.

Alternate Method

Given: Total voters = 25,20,000 | Voter turnout = 98% | Valid votes = 95% of polled | Winner = 65% of valid.

⇒ Total Polled Votes = 25,20,000 × (98/100) = 24,69,600

⇒ Total Valid Votes = 24,69,600 × (95/100) = 23,46,120

⇒ Losing Candidate Share = 100% − 65% = 35%

⇒ Votes for Loser = 23,46,120 × (35/100)

⇒ Final Calculation = 23,46,120 × 0.35 = 8,21,142

∴ The correct answer is 8,21,142.

Additional Information

Election Base Change

Always calculate the next percentage on the reduced base (e.g., valid votes are a % of polled votes, not total voters).

Valid vs Invalid Votes

Valid Votes = (Total Votes − Non-voters) − Invalid Votes. The winner and loser percentages are usually calculated on Valid Votes.

Successive Percentage Formula

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ...

Shortcut Trick Scheme A discount% = [Free Items ÷ Total Items] × 100 = [5 ÷ (1 + 5)] × 100 = 83.33%. Scheme B discount% = 64%. Scheme C discount% = a + b − ab ÷ 100 = 3 + 37 − (3 × 37) ÷ 100 = 40 − 1.11 = 38.89%. Comparing 38.89% (C) < 64% (B) < 83.33% (A). Ascending order is C, B, A. ∴ The correct answer is C, B, A.

Alternate Method Given → Scheme A: Buy 1 Get 5 Free; Scheme B: 64% Single Discount; Scheme C: Successive 3% and 37%. Formulas: Discount% = (Free ÷ Total) × 100 | Successive = a + b − ab ÷ 100

⇒ Scheme A: Buy 1 get 5 free means customer gets 6 articles for the price of 1. Discount = 5 ÷ 6 × 100 = 83.33%. ⇒ Scheme B: Single discount = 64.00%. ⇒ Scheme C: Net Discount = 3 + 37 − (3 × 37 ÷ 100) = 40 − 1.11 = 38.89%. ⇒ Arranging in Ascending Order: 38.89% < 64% < 83.33%. ∴ The correct answer is C, B, A.

Additional Information Buy X Get Y Free In such offers, the discount percentage is calculated as [Y ÷ (X + Y)] × 100, where Y is the free quantity and X+Y is the total quantity. Successive Discounts Two successive discounts of x% and y% are always less than a single discount of (x+y)% because the second discount is applied to a reduced price.

Shortcut Trick

For an angle of depression of 30°, the ratio of the height of the building to the distance of the car is 1 : √3.

Given height = 45 m, which corresponds to 1 unit.

Therefore, distance = 45 × √3 = 45 × 1.732 = 77.94 m.

∴ The correct answer is 77.94 m.

Alternate Method

Given: Height of building (AB) = 45 m, Angle of depression = 30°.

Formula Used: tan θ = Perpendicular ÷ Base

⇒ In right-angled triangle ABC, let BC be the distance of the car.

⇒ tan 30° = AB ÷ BC

⇒ 1/√3 = 45 ÷ BC

⇒ BC = 45 × √3

⇒ BC = 45 × 1.732

⇒ BC = 77.94 m

∴ The correct answer is 77.94 m.

Additional Information

Angle of Depression

It is the angle between the horizontal line and the line of sight when looking down at an object. It is numerically equal to the angle of elevation due to alternate interior angles.

Standard Trigonometric Ratios

For a 30° angle, tan 30° = 1/√3. For a 60° angle, tan 60° = √3. For a 45° angle, tan 45° = 1.

Value of √3

The approximate value of √3 is 1.732, and √2 is 1.414. These are frequently used in heights and distances problems.

Shortcut Trick

In a rhombus, the relationship between side (a) and diagonals (d₁, d₂) is: 4a² = d₁² + d₂².

Given diagonals are 24 cm and 7 cm.

Side (a) = √(d₁² + d₂²) ÷ 2 = √(24² + 7²) ÷ 2 = √(576 + 49) ÷ 2 = √625 ÷ 2 = 25 ÷ 2 = 12.5 cm.

Perimeter = 4 × side = 4 × 12.5 = 50 cm.

∴ The correct answer is 50 cm.

Alternate Method

Given: Diagonal d₁ = 24 cm and Diagonal d₂ = 7 cm.

Formula Used: In a rhombus, diagonals bisect each other at 90°. Using Pythagoras Theorem in the small right-angled triangle formed by the halves of the diagonals: Side² = (d₁/2)² + (d₂/2)².

Let the diagonals intersect at point O. In right-angled triangle AOB:

⇒ AO = d₁ ÷ 2 = 24 ÷ 2 = 12 cm

⇒ BO = d₂ ÷ 2 = 7 ÷ 2 = 3.5 cm

⇒ AB² = AO² + BO²

⇒ AB² = 12² + 3.5² = 144 + 12.25 = 156.25

⇒ AB = √156.25 = 12.5 cm

⇒ Perimeter = 4 × Side = 4 × 12.5 = 50 cm

∴ The correct answer is 50 cm.

Additional Information

Area of Rhombus

The area is calculated as half the product of its diagonals: Area = 1/2 × d₁ × d₂.

Properties of Rhombus

All four sides are equal, opposite angles are equal, and diagonals bisect each other at 90°.

Diagonal and Side Relation

For a rhombus with side 'a' and diagonals 'p' and 'q', the identity is p² + q² = 4a².

Shortcut Trick

Effective Rate of Interest for 2 years at 5% = x + y + (xy/100)

⇒ 5 + 5 + (5 × 5)/100 = 10.25%

Total Amount = 100% + 10.25% = 110.25% of Principal

⇒ Amount = 6000 × 110.25 / 100 = 60 × 110.25 = ₹6615

∴ The correct answer is ₹6615.

Alternate Method

Given:

Principal (P) = ₹6000

Rate (R) = 5% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 6000 × (1 + 5/100)²

⇒ A = 6000 × (1 + 1/20)²

⇒ A = 6000 × (21/20) × (21/20)

⇒ A = 6000 × (441 / 400)

⇒ A = 15 × 441

⇒ A = ₹6615

∴ The correct answer is ₹6615.

Additional Information

Compound Interest Formula

The interest earned is given by CI = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Difference between CI and SI for 2 years

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

Half-Yearly Compounding

When interest is compounded half-yearly, the rate becomes R/2 and the time period becomes 2n.

Shortcut Trick

Total distance = 165 + 195 = 360 m

Relative speed (opposite direction) = 83 + 79 = 162 kmph

Relative speed in m/s = 162 × 5/18 = 45 m/s

Time = 360 ÷ 45 = 8 seconds

∴ The correct answer is 8 sec.

Alternate Method

Given: Length of first train (L₁) = 165 m, Length of second train (L₂) = 195 m. Speed of first train (S₁) = 83 km/hr, Speed of second train (S₂) = 79 km/hr.

Formula Used: Time = (L₁ + L₂) ÷ (S₁ + S₂)

⇒ Total Length = 165 + 195 = 360 m

⇒ Relative Speed = (83 + 79) × 5/18 m/s

⇒ Relative Speed = 162 × 5/18 = 9 × 5 = 45 m/s

⇒ Time taken = 360 / 45

⇒ Time taken = 8 seconds

∴ The correct answer is 8 sec.

Additional Information

Relative Speed in Opposite Direction

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

Relative Speed in Same Direction

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

Conversion of Speed Units

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

Shortcut Trick

Let the cost price to B be CP.

Successive Profit Multiplier: B sells at 112% and C sells at 112%.

CP × 1.12 × 1.12 = 313.60

CP × 1.2544 = 313.60

CP = 313.60 ÷ 1.2544 = 250

∴ The correct answer is ₹250.

Alternate Method

Given → Profit % for B = 12%, Profit % for C = 12%, Final SP (to D) = ₹313.60. Formulas: SP = CP × (100 + P%) ÷ 100

⇒ Let CP of B = x

⇒ CP of C (which is SP of B) = x × (112 ÷ 100) = 1.12x

⇒ SP of C (which is CP of D) = 1.12x × (112 ÷ 100) = 1.2544x

⇒ 1.2544x = 313.60

⇒ x = 313.60 ÷ 1.2544 = 250

∴ The correct answer is ₹250.

Additional Information

Successive Percentage Change

When a value changes by a% and then by b%, the net percentage change is given by [a + b + (ab ÷ 100)]%.

Cost Price and Selling Price Relationship

The Selling Price (SP) of the seller is always equal to the Cost Price (CP) of the buyer in a transaction chain.

Shortcut Trick Let D's age be D and B's age be B. Given D = B + 10 → B = D − 10. B = 5 × |C − D|. Since we need minimum D and D = B + 10, D must be greater than B. For minimum D, we check the case where D < C: D − 10 = 5(30 − D). D − 10 = 150 − 5D ⇒ 6D = 160 ⇒ D = 80/3. ∴ The correct answer is 80/3 years. Alternate Method Given: C = 30 years, B = 5|C − D|, D = B + 10.

From the problem, we have two possible cases for the absolute difference |C − D|: Case 1: D ≥ C ⇒ B = 5(D − 30) Substitute B = D − 10: ⇒ D − 10 = 5D − 150 ⇒ 4D = 140 ⇒ D = 35 years Case 2: D < C ⇒ B = 5(30 − D) Substitute B = D − 10: ⇒ D − 10 = 150 − 5D ⇒ 6D = 160 ⇒ D = 160/6 = 80/3 years Comparing 35 and 80/3 (26.66), the minimum age is 80/3. ∴ The correct answer is 80/3 years. Additional Information Absolute Difference in Ages The expression |x − y| ensures a positive value regardless of which person is older, leading to two linear equations. Ages involving Ratios If the ratio of ages is a:b, we assume ages as ax and bx to solve for the constant x.

Shortcut Trick

Initial Savings = 5.5% of 34,000 = ₹1,870. Initial Expenditure = 94.5% of 34,000 = ₹32,130.

New Income = 34,000 × 1.17 = ₹39,780.

New Expenditure = 32,130 × 1.50 = ₹48,195.

New Savings = 39,780 − 48,195 = −₹8,415 (Deficit).

Change in Savings = −8,415 − 1,870 = −₹10,285.

∴ The correct answer is decrease by ₹10,285.

Alternate Method

Given: Initial Income = ₹34,000, Savings % = 5.5%, Income increase = 17%, Expenditure increase = 50%.

Formula: Income = Expenditure + Savings

⇒ Initial Savings = (5.5 ÷ 100) × 34,000 = ₹1,870

⇒ Initial Expenditure = 34,000 − 1,870 = ₹32,130

⇒ New Income = 34,000 + (17 ÷ 100 × 34,000) = 34,000 + 5,780 = ₹39,780

⇒ New Expenditure = 32,130 + (50 ÷ 100 × 32,130) = 32,130 + 16,065 = ₹48,195

⇒ New Savings = 39,780 − 48,195 = −₹8,415

⇒ Change in Savings = New Savings − Initial Savings = −8,415 − 1,870 = −₹10,285

∴ The correct answer is decrease by ₹10,285.

Additional Information

Income-Expenditure Relation

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

Percentage Change

New Value = Original Value × (1 ± Change%/100). A negative result in savings indicates a deficit or debt.

Base Value Importance

Always ensure the percentage increase is applied to the correct base component (e.g., expenditure increase is on initial expenditure, not income).

Shortcut Trick

For two numbers a and b, the third proportional is given by b² ÷ a.

Here, a = 16x² and b = 20xy.

Third Proportional = (20xy)² ÷ 16x²

⇒ (400x²y²) ÷ 16x² = 25y².

∴ The correct answer is 25y².

Alternate Method

Given: First term (a) = 16x², Second term (b) = 20xy.

Formula Used: If a, b, c are in continued proportion, then a/b = b/c ⇒ c = b²/a.

⇒ Let the third proportional be c.

⇒ 16x² ÷ 20xy = 20xy ÷ c

⇒ c = (20xy × 20xy) ÷ 16x²

⇒ c = 400x²y² ÷ 16x²

⇒ c = 25y²

∴ The correct answer is 25y².

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is √(ab).

Fourth Proportional

If four quantities a, b, c, d are in proportion (a:b = c:d), then the fourth proportional d = (b × c) ÷ a.

Continued Proportion

Three numbers a, b, c are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third (a:b = b:c).

Shortcut Trick

Observe the relation between numerator and denominator terms:

0.0079 = 0.079 ÷ 10

Every term in the denominator is 1/10^th of the corresponding term in the numerator.

Since the terms are squared, the denominator becomes 1/100^th of the numerator.

Expression = Numerator ÷ (Numerator ÷ 100) = 100

∴ The correct answer is 100.

Alternate Method

Given:

Expression = (0.079² + 0.25² + 3.37²) ÷ (0.0079² + 0.025² + 0.337²)

Calculations:

⇒ Let a = 0.079, b = 0.25, and c = 3.37

⇒ The denominator terms can be written as:

⇒ 0.0079 = a / 10

⇒ 0.025 = b / 10

⇒ 0.337 = c / 10

⇒ Substituting these variables into the given expression:

⇒ Expression = (a² + b² + c²) ÷ [ (a / 10)² + (b / 10)² + (c / 10)² ]

⇒ Expression = (a² + b² + c²) ÷ [ (a² / 100) + (b² / 100) + (c² / 100) ]

⇒ Expression = (a² + b² + c²) ÷ [ (a² + b² + c²) / 100 ]

⇒ Expression = 100 × [ (a² + b² + c²) ÷ (a² + b² + c²) ]

⇒ Expression = 100 × 1 = 100

∴ The correct answer is 100.

Additional Information

Decimal Fractions

When shifting the decimal point one place to the left, the value is divided by 10. Squaring this shifted value effectively divides the original square by 100.

Algebraic Substitution

Replacing complex decimal numbers with variables (like a, b, c) simplifies the expression visually and makes it easier to cancel out common terms.

BODMAS Rule

Always resolve powers (orders) and division before addition to avoid calculation errors in simplification questions.

Shortcut Trick

Teacher's Age = New Average + (Initial Count × Increase in Average)

Increase in average = 19 − 18 = 1 year

Teacher's Age = 19 + (40 × 1)

Teacher's Age = 19 + 40 = 59 years

∴ The correct answer is 59 years.

Alternate Method

Given:

Number of students = 40

Initial average age = 18 years

New average age (including teacher) = 19 years

Formula Used:

Total Age = Average Age × Number of Persons

Calculations:

⇒ Total age of 40 students = 40 × 18 = 720 years

⇒ Total number of people including teacher = 40 + 1 = 41

⇒ Total age of 41 people = 41 × 19 = 779 years

⇒ Age of the teacher = Total age of 41 people − Total age of 40 students

⇒ Teacher's age = 779 − 720 = 59 years

∴ The correct answer is 59 years.

Additional Information

General Average Formula

Average = (Sum of all observations) ÷ (Total number of observations).

Effect of New Entry

If the average increases by 'x' when one person is added, the new person's value = Old Average + (New Total Strength × x).

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Total Work in terms of Men = 5 × 16 = 80 Man-days.

Total Work in terms of Women = 9 × 8 = 72 Woman-days.

Equating Work: 80M = 72W ⇒ 10M = 9W ⇒ 1M = 0.9W.

Required Group: 3W + 2M = 3W + 2(0.9W) = 3W + 1.8W = 4.8W.

Days = (Total Woman-days) ÷ (Efficiency of Group) = 72 ÷ 4.8 = 15 days.

∴ The correct answer is 15 days.

Alternate Method

Given: 5 Men in 16 days, 9 Women in 8 days.

Formula Used: Total Work = Number of People × Efficiency × Time

⇒ Efficiency of 1 Man = 45 ÷ 5 = 9 units/day

⇒ Efficiency of 1 Woman = 90 ÷ 9 = 10 units/day

⇒ Efficiency of (3 Women + 2 Men) = (3 × 10) + (2 × 9) = 30 + 18 = 48 units/day

⇒ Total Work = 5 × 9 × 16 = 720 units

⇒ Days Required = 720 ÷ 48 = 15 days

∴ The correct answer is 15 days.

Additional Information

Work Equivalence Formula

M₁D₁H₁ = M₂D₂H₂ is used when the amount of work remains constant across two different groups.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Man-Day Concept

The total work is often expressed in Man-days or Woman-days to simplify ratio calculations between different genders or groups.

Shortcut Trick

Convert fractions to decimals for easier comparison.

1/20 = 0.05 and 2/100 = 0.02

Percentage = (0.05 ÷ 0.02) × 100 = 5/2 × 100 = 250%

∴ The correct answer is 250%.

Alternate Method

Given:

First number (A) = 1/20

Second number (B) = 2/100

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

⇒ Required Percentage = [(1/20) ÷ (2/100)] × 100

⇒ Required Percentage = [1/20 × 100/2] × 100

⇒ Required Percentage = [100/40] × 100

⇒ Required Percentage = 2.5 × 100

⇒ Required Percentage = 250%

∴ The correct answer is 250%.

Additional Information

Percentage to Fraction

To convert a percentage to a fraction, divide the number by 100 and remove the % sign (e.g., 25% = 25/100 = 1/4).

Fraction to Percentage

To convert a fraction to a percentage, multiply the fraction by 100 and add the % sign (e.g., 1/5 × 100 = 20%).

Base Value Identification

In questions like "A is what % of B", B is always the denominator (base), and A is the numerator.

Shortcut Trick

Evaluate power first: 3² = 9

Expression becomes: 4 × 17 × 9 ÷ 12 + 26 − 41

Cancel: 4/12 = 1/3 ⇒ 17 × 9 ÷ 3 = 17 × 3 = 51

Now: 51 + 26 − 41

∴ Answer = 36

Alternate Method

Given:

4 × (17 × (3²)) ÷ 12 + 26 − 41

Concept Used:

BODMAS rule

Calculation:

⇒ 4 × (17 × 9) ÷ 12 + 26 − 41

⇒ 4 × 153 ÷ 12 + 26 − 41

⇒ 612 ÷ 12 + 26 − 41

⇒ 51 + 26 − 41

⇒ 77 − 41

⇒ 36

∴ The correct answer is 36.

Additional Information

BODMAS Rule

Brackets, Orders (powers), Division and Multiplication, Addition and Subtraction

Order of Operations

Solve powers first, then × and ÷ from left to right, then + and −

Simplification Trick

Cancel common factors early to reduce calculation

Shortcut Trick

Ratio of dimensions = 8 : 5 : 3. Let common factor be x.

Diagonal square ratio: 8² + 5² + 3² = 64 + 25 + 9 = 98 units.

Actual diagonal squared = (21√2)² = 441 × 2 = 882.

Calculating x² = 882 ÷ 98 = 9 ⇒ x = 3.

Total Surface Area (TSA) = 2 × x² × (8×5 + 5×3 + 3×8) = 2 × 9 × 79 = 1,422 m².

∴ The correct answer is 1,422 m².

Alternate Method

Given: Ratio of Length (L), Breadth (B), and Height (H) = 8 : 5 : 3 and Diagonal (D) = 21√2 m.

Formula Used: Diagonal = √(L² + B² + H²) | TSA = 2(LB + BH + HL)

⇒ Let L = 8x, B = 5x, H = 3x.

⇒ (8x)² + (5x)² + (3x)² = (21√2)²

⇒ 64x² + 25x² + 9x² = 441 × 2

⇒ 98x² = 882

⇒ x² = 9 ⇒ x = 3 m.

⇒ L = 8 × 3 = 24 m, B = 5 × 3 = 15 m, H = 3 × 3 = 9 m.

⇒ TSA = 2 × (24 × 15 + 15 × 9 + 24 × 9)

⇒ TSA = 2 × (360 + 135 + 216)

⇒ TSA = 2 × 711

⇒ TSA = 1,422 m².

∴ The correct answer is 1,422 m².

Additional Information

Total Surface Area of Cuboid

It is the sum of the areas of all 6 faces: TSA = 2(lb + bh + hl).

Diagonal of a Cuboid

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

Volume of Cuboid

The space occupied by the cuboid is calculated as Volume = l × b × h.

Lateral Surface Area (LSA)

The area of the four side walls (excluding top and bottom) is LSA = 2h(l + b).

Shortcut Trick

Combined efficiency of A, B, and C = 1/2 + 1/20 + 1/20 = (10 + 1 + 1)/20 = 12/20 = 3/5 unit/day.

Days taken for same work = 1 / (3/5) = 5/3 days.

Days taken for thrice the work = 3 × (5/3) = 5 days.

∴ The correct answer is 5 days.

Alternate Method

Given: Time taken by A = 2 days, B = 20 days, C = 20 days.

Calculations:

⇒ Total Work (LCM of 2, 20, 20) = 20 units

⇒ Total Efficiency of (A + B + C) = 10 + 1 + 1 = 12 units/day

⇒ New Work = 3 × Total Work = 3 × 20 = 60 units

⇒ Time required = New Work ÷ Total Efficiency

⇒ Time = 60 ÷ 12 = 5 days

∴ The correct answer is 5 days.

Additional Information

Efficiency and Time Relation

Time taken is inversely proportional to efficiency. Work = Time × Efficiency.

Work and Days Direct Proportion

If the amount of work increases by a factor of n, the time taken to complete it (at constant efficiency) also increases by factor n.

Shortcut Trick

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

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

Let Discount% be d. Given MP is 41% above CP, so MP/CP = 141/100.

100 / 141 = (100 − d) / (100 + 22)

100 / 141 = (100 − d) / 122 ⇒ 12200 = 14100 − 141d

141d = 1900 ⇒ d = 1900 ÷ 141 ≈ 13.48%.

∴ The correct answer is 13.48%.

Alternate Method

Given: MP = ₹331, MP is 41% above CP, Profit = 22%.

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

⇒ CP = 331 × 100 ÷ 141 = ₹234.7517...

⇒ Selling Price (SP) = 234.7517 × 122 ÷ 100 = ₹286.397...

⇒ Discount = MP − SP = 331 − 286.397 = ₹44.603...

⇒ Discount% = (44.603 ÷ 331) × 100 = 13.475% ≈ 13.48%.

∴ The correct answer is 13.48%.

Additional Information

Marked Price and Markup

Markup is the extra price added to the Cost Price to arrive at the Marked Price. Markup% = (Markup ÷ CP) × 100.

Effective Percentage Change

Successive application of markup and discount results in profit: %Profit = %Markup − %Discount − (%Markup × %Discount) ÷ 100.

Shortcut Trick

For 7n4 to be divisible by 6, it must be divisible by both 2 and 3.

Since the last digit is 4, it is divisible by 2.

For divisibility by 3, the sum of digits (7 + n + 4 = 11 + n) must be a multiple of 3.

The smallest single-digit whole number n making 11 + n divisible by 3 is 1 (since 11 + 1 = 12).

∴ The correct answer is 1.

Alternate Method

Given: Three-digit number is 7n4 which is divisible by 6.

Formula Used: A number is divisible by 6 if it is divisible by both 2 and 3.

⇒ Divisibility by 2: The unit digit is 4, which is even. Thus, 7n4 is divisible by 2 for any integer n.

⇒ Divisibility by 3: Sum of the digits must be divisible by 3.

⇒ Sum of digits = 7 + n + 4 = 11 + n

⇒ For 11 + n to be divisible by 3, the possible values of (11 + n) are 12, 15, 18, etc.

⇒ If 11 + n = 12 ⇒ n = 1

⇒ If 11 + n = 15 ⇒ n = 4

⇒ If 11 + n = 18 ⇒ n = 7

⇒ The least possible value for n is 1.

∴ The correct answer is 1.

Additional Information

Divisibility Rule of 3

A number is completely divisible by 3 if the sum of all its individual digits is a multiple of 3.

Divisibility Rule of 6

A number is divisible by 6 if it is simultaneously divisible by its co-prime factors, 2 and 3.

Divisibility Rule of 9

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

The position and positional values or English Alphabets are:

Given: PCR45, MGO41, JKL37, ?

The logic followed here is:

Thus, GOI33 will replace the question mark (?) in the following series to make it logically complete.

Hence, the correct answer is "Option 1".

Family Chart:

Decoding the Symbols:

 

 

Given: O @ P # T + G - V

It means: O is the brother of P, P is the father of T, T is the sister of G, G is the wife of V.

Thus, O is V's Wife's father's brother.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: RV23, QU32, PT41, OS50, ?

The logic followed here is:

Thus, NR59 will replace the question mark (?) in the given series to make it logically complete.

Hence, the correct answer is "Option 4".

Given: If 2 is added to each odd digit and 2 is subtracted from each even digit in the number 3267451.

Highest digit = 9

Lowest digit = 0

Difference = 9 − 0 = 9

Thus, 9 will be the difference between the highest and lowest digits in the new number thus formed.

Hence, the correct answer is "Option 1".

Given: 604, 895, 469, 742, 307

Highest number = 895

Lowest number = 307

Third digit of highest number 895 is 5.

Second digit of lowest number 307 is 0

Now add them:

5 + 0 = 5

Thus, 5 will be the resultant, if the third digit of the highest number is added to the second digit of the lowest number.

Hence, the correct answer is "Option 1".

The logic followed here is:

KEPT - ETPK - TPEK

SAIL - ALIS - LIAS

Let's check each option:

1. TURN - UTRN - NURT

This triad does not follow the same logic.

2. MOPS - OSPM - SPOM

This triad follows the same logic.

3. HOPE - HPOE - EPOH

This triad does not follow the same logic.

4. PACK - APCK - KACP

This triad does not follow the same logic.

Thus, MOPS - OSPM - SPOM from the given options, the one which follows the same logic.

 

Hence, the correct answer is "Option 2".

The position and positional values or English Alphabets are:

Given: UPNF is related to YLPD in a certain way based on the English alphabetical order.

The logic followed here is:

Similarly,

Thus, KZVX is related to OVXV, following the same logic.

Hence, the correct answer is "Option 3".

Statements:

No father is a mother.

No mother is a son.

The least possible Venn diagram for the given statement is:

Conclusions:

(I) All mothers are fathers. - Does not follow (Because, no father is a mother, so no mothers are fathers.)

(II) No father is a son. - Does not follow (There is no definite relation between fathers and sons, it can be possible but it is not definite.)

Thus, neither follows.

Hence, the correct answer is "Option 1".

The position and positional values or English Alphabets are:

The pattern followed here is:

Each letter-cluster follows the pattern -4, +3.

Let’s check each option:

1. WSV

This letter-cluster follows the pattern -4, +3.

2. UQT

This letter-cluster follows the pattern -4, +3.

3. PLO

This letter-cluster follows the pattern -4, +3.

4. IFH

This letter-cluster does not follow the pattern -4, +3.

Thus, IFH is the odd one out.

Hence, the correct answer is "Option 4".

Given: Pavan ranked 13^th from the top and 3^rd from the bottom in his class.

Pavan's rank from the top = 13^th

Pavan's rank from the bottom = 3^rd

Total students in the class

= Rank from top + Rank from bottom − 1

= 13 + 3 − 1

= 16 - 1

= 15

Thus, 15 students are there in the class.

Hence, the correct answer is "Option 3".

Shortcut Trick

Interchange signs: (+ → −), (− → +), (× → ÷), (÷ → ×)

New Expression → 172 × 4 − 183 ÷ 3 + 164

Step 1 (Div/Mult): 688 − 61 + 164

Step 2 (Final): 627 + 164 = 791

∴ The correct answer is 791.

Alternate Method

Given Equation: 172 ÷ 4 + 183 × 3 − 164 = ?

According to the question, we interchange the signs as follows:

• is interchanged with −

× is interchanged with ÷

⇒ 172 × 4 − 183 ÷ 3 + 164 = ?

Applying BODMAS rule:

⇒ 172 × 4 − 61 + 164 = ?

⇒ 688 − 61 + 164 = ?

⇒ 852 − 61 = ?

⇒ 791

∴ The correct answer is 791.

Additional Information

BODMAS Rule

It specifies the order of operations: Brackets, Orders, Division × Multiplication (left to right), and Addition × Subtraction (left to right).

Sign Interchange Strategy

When signs are interchanged, always rewrite the entire expression clearly before starting calculations to avoid mixing old and new operators.

Arithmetic Accuracy

In interchanged equations, a single error in sign replacement leads to a completely different result; double-check the mapping (+ ↔ −, × ↔ ÷).