RRB Group D Official Paper (Held On: 19 Dec, 2025 Shift 3) (Previous Year Paper)

The correct answer is Sodium.

Key Points

• Sodium is a highly reactive metal belonging to Group 1 (alkali metals) of the Periodic Table with an atomic number of 11.
• It reacts violently and explosively with cold water to produce sodium hydroxide and hydrogen gas. The reaction is represented as 2Na + 2H2O → 2NaOH + H2.
• The reaction is extremely exothermic, meaning it releases a significant amount of heat. This heat is often sufficient to ignite the hydrogen gas produced, leading to a visible flame or explosion. Hence, the statement regarding sodium reacting violently with cold water is correct.
• Due to its high reactivity with both oxygen and moisture (water vapor) in the air, sodium is always stored under kerosene oil to prevent accidental reactions.
• Potassium is another metal that reacts even more violently than sodium with cold water, as reactivity increases down the group in the periodic table.

Additional Information

• Magnesium (Mg):
• Magnesium has the atomic number 12 and belongs to Group 2 (alkaline earth metals).
• It does not react with cold water. However, it reacts slowly with hot water and more vigorously with steam to form magnesium hydroxide and hydrogen. Hence, the statement that magnesium reacts violently with cold water is incorrect.
• Aluminium (Al):
• Aluminium is a metal in Group 13 with the atomic number 13.
• It does not react with cold or hot water under normal conditions due to the formation of a tough, protective oxide layer on its surface. It only reacts with steam at high temperatures. Hence, the statement regarding aluminium is incorrect.
• Copper (Cu):
• Copper is a transition metal with the atomic number 29.
• It is placed very low in the reactivity series, even below hydrogen. Consequently, it does not react with water or steam at any temperature. Hence, the statement regarding copper is incorrect.
• Reactivity Series:
• The reactivity series is a list of metals arranged in the order of their decreasing chemical reactivity.
• Metals like Potassium, Sodium, Lithium, and Calcium are at the top and can react with cold water. Metals like Gold, Silver, and Platinum are at the bottom and are highly unreactive.

Important Points

• Chemical Hazards: Handling Sodium requires strict safety protocols, including the use of forceps and protective gear, as skin contact can cause severe chemical burns due to the reaction with moisture on the skin.
• Density: Sodium is one of the few metals that is less dense than water (density approx 0.97 g/cm3). Because of this, it floats on the surface of the water while it reacts violently, moving around rapidly as the hydrogen gas is evolved.
• Identification: When sodium reacts with water, the resulting solution of sodium hydroxide is a strong base, which can be identified by turning red litmus paper blue or turning phenolphthalein indicator pink.
• Year of Discovery: Sodium was first isolated in 1807 by Sir Humphry Davy through the electrolysis of caustic soda.

The correct answer is dead cells cannot osmoregulate.

Key Points

• Plasmolysis is a biological process that occurs in living plant cells when they are placed in a hypertonic solution (a solution with a higher solute concentration than the cell sap). During this process, water moves out of the cell through osmosis, causing the protoplasm to shrink away from the cell wall.
• When Rhoeo leaf peels are boiled, the high temperature causes the denaturation of proteins and destroys the plasma membrane. Boiling effectively kills the cells, leading to the loss of their selective permeability.
• The process of osmosis and osmoregulation requires a functional, living semi-permeable membrane. Since boiling renders the cells dead, the membrane can no longer regulate the movement of water or solutes. Hence, the statement that dead cells cannot osmoregulate is correct.
• In dead cells, the physical and chemical barriers that facilitate exosmosis are compromised. Therefore, even when a concentrated salt solution is applied, the characteristic shrinkage of the cytoplasm (plasmolysis) is not observed because the biological machinery required for osmotic movement is non-functional.
• The cell wall of a plant cell is fully permeable to water and dissolved substances; it is the plasma membrane that acts as the regulatory barrier. In boiled leaves, the plasma membrane is ruptured, allowing substances to move freely without osmotic control.

Additional Information

• Rhoeo discolor (Tradescantia spathacea):
• This plant is widely used in botany experiments because its lower epidermal cells contain anthocyanin, a purple-colored pigment located in the vacuoles.
• The presence of this pigment makes the protoplast easily visible under a microscope, allowing students to clearly observe the stages of plasmolysis and deplasmolysis.
• Stages of Plasmolysis:
• Limiting Plasmolysis: The initial stage where the osmotic pressure of the cell equals the external solution, and the turgor pressure becomes zero.
• Incipient Plasmolysis: The stage where the protoplast begins to pull away from the cell wall at the corners.
• Evident Plasmolysis: The final stage where the cytoplasm is completely detached from the cell wall and assumes a spherical or irregular shape in the center.
• Deplasmolysis:
• If a plasmolyzed cell is placed in a hypotonic solution (like distilled water), water enters the cell through endosmosis.
• The protoplast swells and returns to its original position against the cell wall. Like plasmolysis, this process can only occur in living cells.
• Turgor Pressure:
• In a hypotonic environment, water enters the plant cell, creating turgor pressure against the rigid cell wall. This pressure is essential for maintaining the structural integrity and upright posture of non-woody plants.

Important Points

• Permeability: It is important to distinguish that the cell wall is completely permeable (allows everything to pass), whereas the plasma membrane is selectively permeable (allows only specific molecules like water to pass freely).
• Osmoregulation in Plants: It is the mechanism by which plants maintain the water potential and solute concentration within their cells. This is a metabolic process that ceases immediately upon the death of the cell.
• Hypertonic vs. Hypotonic: A hypertonic solution has a lower water potential than the cell, leading to water loss, while a hypotonic solution has a higher water potential, leading to water gain.

p data-end="57" data-start="0">The correct answer is They are used by dentists to see enlarged images of teeth.

 

Key Points

• Concave mirrors are widely used by dentists to see large images of the teeth of patients. When the tooth is placed between the focus (F) and the pole (P) of the mirror, it produces an enlarged, virtual, and upright image. Hence, statement 2 is correct.
• Rear-view mirrors in vehicles are generally convex mirrors because they give an erect, though diminished, image and provide a wider field of view to the driver. Hence, statement 1 is incorrect.
• Concave mirrors are capable of producing magnified images. For example, when an object is placed between the center of curvature (C) and the principal focus (F), the resulting image is magnified and real. Hence, statement 3 is incorrect.
• A concave mirror does not always produce virtual images; in fact, it produces real and inverted images for most positions of the object (at infinity, beyond C, at C, and between C and F). It only produces a virtual image when the object is placed between the focus and the mirror. Hence, statement 4 is incorrect.
• Hence, the correct answer is They are used by dentists to see enlarged images of teeth.

Additional Information

• Definition and Nature:
• A concave mirror is a spherical mirror whose reflecting surface is curved inwards, facing the center of the sphere. It is also known as a converging mirror because it converges light rays that strike its surface.
• The focal length of a concave mirror is considered negative according to the Cartesian sign convention.
• Applications of Concave Mirrors:
• Shaving Mirrors: They are used to see a larger image of the face for a better shave.
• Torches and Headlights: Concave mirrors are used in search-lights and vehicle headlights to get powerful parallel beams of light by placing the bulb at the focus.
• Solar Furnaces: Large concave mirrors are used to concentrate sunlight to produce heat in solar furnaces.
• Image Formation Characteristics:
• The nature, position, and size of the image formed by a concave mirror depend on the distance of the object from the pole.
• If the object is at the Center of Curvature (C), the image is real, inverted, and the same size as the object.
• If the object is at infinity, the image is formed at the focus (F) and is highly diminished.

Important Points

• Comparison with Convex Mirrors:
• Unlike concave mirrors, convex mirrors (diverging mirrors) always produce virtual, erect, and diminished images regardless of the object's distance.
• This property makes convex mirrors ideal for blind turns on roads and shop security mirrors.
• Mirror Formula and Magnification:
• The Mirror Formula is given by 1/f = 1/v + 1/u, where f is focal length, v is image distance, and u is object distance.
• Linear Magnification (m) is the ratio of the height of the image to the height of the object, also expressed as m = -v/u. For a concave mirror, magnification can be less than 1, equal to 1, or greater than 1.

The correct answer is Variation makes species prone to extinction.

Key Points

• Variation refers to the differences in DNA sequences and physical traits among individuals within a population or species. These differences are the primary drivers of evolution and biodiversity.
• Variation ensures species survival during disasters: In a changing environment or during natural calamities, individuals with specific variations may have a better chance of survival compared to others. For example, in a bacterial population, some might be resistant to heat; during a sudden temperature increase, only these variants survive. Hence, statement 1 is correct.
• Variation makes species prone to extinction: This is incorrect. In reality, a lack of variation (genetic uniformity) makes a species more vulnerable to extinction. Higher genetic diversity provides a "buffer" against environmental changes, whereas low variation means the entire population could be wiped out by a single event. Hence, statement 2 is incorrect.
• Variation enables adaptation to changing environments: Through the process of Natural Selection, favorable variations that help an organism survive and reproduce in its current environment are passed on to the next generation. This is the basis of adaptation. Hence, statement 3 is correct.
• Provides resilience against diseases: Genetic variation ensures that not all individuals are equally susceptible to the same pathogens. If a new virus or bacteria attacks a population, some individuals may possess a natural immunity due to their unique genetic makeup. Hence, statement 4 is correct.
• Hence, the incorrect statement (and thus the correct answer) is "Variation makes species prone to extinction".

Additional Information

• Sources of Variation:
• Mutation: A random change in the DNA sequence which can create new alleles.
• Recombination: During Meiosis (cell division), genes are shuffled between chromosomes, leading to new combinations of traits in offspring.
• Sexual Reproduction: The combining of genetic material from two different parents significantly increases the genetic pool of the population.
• Importance in Evolution:
• Charles Darwin, in his book "On the Origin of Species" (1859), highlighted that variation is the raw material upon which natural selection acts.
• Without variation, there would be no speciation (the formation of new and distinct species).
• Types of Variation:
• Continuous Variation: Traits like height or skin color that show a range of small differences.
• Discontinuous Variation: Traits like blood groups or the ability to roll the tongue, which fall into distinct categories with no intermediates.

Important Points

• DNA Copying: Variation often arises during DNA replication before cell division. While the process is highly accurate, occasional "errors" or biochemical inaccuracies occur, leading to variations in the progeny.
• Survival of the Fittest: This concept, coined by Herbert Spencer and utilized by Darwin, explains that individuals with the most useful variations for their specific environment are the ones most likely to survive and reproduce.
• Environmental Factors: While many variations are heritable (passed via genes), some are acquired during an organism's lifetime due to environmental influences (e.g., muscle building or scarring) and are generally not passed to the next generation.
• Population Stability: Variation helps maintain the stability of an ecosystem. A diverse range of species and genetic traits within those species ensures that the ecosystem can recover from stresses such as climate change or habitat loss.

The correct answer is Breakdown of carbohydrates to release energy.

Key Points

• Photosynthesis is a complex anabolic process by which green plants and certain other organisms transform light energy into chemical energy.
• Absorption of light energy by chlorophyll: This is the primary step where chlorophyll pigments, located in the chloroplasts, capture photons from sunlight. Hence, statement 3 is a direct event that occurs during photosynthesis.
• Conversion of light energy into chemical energy: During the light-dependent phase, absorbed solar energy is converted into ATP (Adenosine Triphosphate) and NADPH. This phase also involves the photolysis of water, where water molecules are split into hydrogen and oxygen. Hence, statement 2 is a direct event that occurs during photosynthesis.
• Reduction of carbon dioxide to carbohydrates: This occurs during the Calvin Cycle (light-independent reaction) in the stroma of the chloroplast. The energy from ATP and NADPH is used to fix CO2 into glucose. Hence, statement 1 is a direct event that occurs during photosynthesis.
• Breakdown of carbohydrates to release energy: This process is known as cellular respiration. It is a catabolic process that takes place in the mitochondria to provide energy for the cell's metabolic activities. While plants do respire, this specific action is the functional opposite of photosynthesis. Hence, statement 4 does NOT occur during the process of photosynthesis.

Additional Information

• Site of Photosynthesis:
• Photosynthesis takes place in specialized organelles called chloroplasts. These organelles contain a green pigment called chlorophyll.
• The grana (stacks of thylakoids) are the site for light reactions, where light is trapped.
• The stroma is the site for dark reactions, where the chemical synthesis of carbohydrates occurs.
• Chemical Equation:
• The overall reaction is represented as: 6CO2 + 6H2O + Light Energy → C6H12O6 + 6O2.
• This equation shows that carbon dioxide and water are the raw materials used to produce glucose and oxygen as a byproduct.
• Factors Influencing Photosynthesis:
• Light Intensity: Increased light typically increases the rate of photosynthesis until a saturation point is reached.
• Carbon Dioxide Concentration: As a primary reactant, higher CO2 levels can increase the rate of sugar production.
• Temperature: Since the process involves enzymes (like RuBisCO), it is sensitive to temperature changes, with an optimum range usually between 25°C to 35°C.

Important Points

• Autotrophs: Green plants are considered autotrophs because they can synthesize their own food using inorganic substances and solar energy.
• Stomata: These are tiny pores found mainly on the epidermis of leaves. They facilitate the exchange of gases, allowing CO2 to enter and O2 to exit the plant body.
• Product Storage: The glucose produced during photosynthesis is often converted into starch for long-term storage in different parts of the plant, such as roots, tubers, and seeds.
• Energy Currency: While ATP is produced during the light reaction of photosynthesis, it is immediately consumed in the dark reaction to build carbohydrates. The ATP used for general cellular work in plants is actually generated through respiration.

The correct answer is Substitution reaction.

Key Points

• Saturated hydrocarbons, also known as alkanes, are organic compounds consisting entirely of single covalent bonds between carbon atoms. Their general chemical formula is CnH2n+2.
• Because all the valencies of carbon atoms are fully satisfied by single bonds, alkanes are relatively unreactive under normal conditions. They do not have double or triple bonds to "open up" for new atoms.
• The characteristic reaction of alkanes is the Substitution reaction. In this process, one or more hydrogen atoms of the hydrocarbon are replaced by other atoms or groups of atoms, such as halogens. Hence, option 3 is correct.
• A classic example is the chlorination of methane. In the presence of ultraviolet (UV) light or diffused sunlight, a chlorine atom replaces a hydrogen atom in Methane (CH4) to form Chloromethane (CH3Cl) and Hydrogen Chloride (HCl).
• This reaction can continue until all the hydrogen atoms in the alkane are substituted. For methane, the sequence ends with Tetrachloromethane (CCl4).

Additional Information

• Addition Reaction:
• This reaction is characteristic of unsaturated hydrocarbons like alkenes (double bonds) and alkynes (triple bonds).
• In an addition reaction, molecules like Hydrogen (H2) or Bromine (Br2) are added across the multiple bonds to turn them into saturated compounds. Hence, option 2 is incorrect for alkanes.
• Neutralisation:
• This is a chemical reaction between an acid and a base to produce salt and water.
• Hydrocarbons are generally neutral and do not participate in standard acid-base neutralisation. Hence, option 1 is incorrect.
• Polymerisation:
• This involves the combination of many small molecules called monomers to form a large molecule called a polymer.
• While unsaturated hydrocarbons like ethene undergo polymerisation to form polyethene, saturated alkanes do not typically exhibit this as a characteristic reaction. Hence, option 4 is incorrect.
• Combustion:
• Alkanes react readily with Oxygen (O2) upon heating. Complete combustion produces Carbon dioxide (CO2), Water (H2O), and a large amount of heat and light, making them excellent fuels.

Important Points

• Paraffins: Alkanes were traditionally called paraffins (from Latin parum = little, affinis = affinity), highlighting their low chemical reactivity toward common reagents like acids, bases, or oxidising agents at room temperature.
• Homologous Series: Alkanes form a series where each member differs from the next by a -CH2- unit and has a mass difference of 14 units.
• Isomerism: Alkanes with four or more carbon atoms (starting from Butane) can show structural isomerism, where they have the same molecular formula but different arrangements of atoms.
• Physical State: At room temperature, the first four alkanes (Methane, Ethane, Propane, and Butane) are gases, while those from C5 to C17 are liquids, and higher alkanes are solids.

The correct answer is A compound breaks down into two or more simpler substances.

Key Points

• A Decomposition reaction is a fundamental type of chemical process where a single reactant breaks down into two or more simpler products. This is the exact definition provided in the first option. Hence, statement 1 is correct.
• In these reactions, the chemical bonds in the reactant molecule are broken, requiring an input of energy. The general chemical equation for this process can be represented as AB → A + B.
• Statement 2 describes a Double Displacement reaction, where two compounds exchange ions to form two new compounds, often resulting in a precipitate or water. Hence, statement 2 is incorrect.
• Statement 3 and Statement 4 describe a Combination reaction (also known as a synthesis reaction), where two or more substances combine to form a single product. This is the opposite of decomposition. Hence, statement 3 and 4 are incorrect.
• A classic example of decomposition is the thermal decomposition of Calcium Carbonate (CaCO3), which, when heated, yields Calcium Oxide (CaO) and Carbon Dioxide (CO2).

Additional Information

• Types of Decomposition Reactions:
• Thermal Decomposition: This occurs when heat energy is used to break down a substance. For instance, heating Lead Nitrate crystals leads to the emission of brown fumes of Nitrogen Dioxide and the formation of Lead Oxide.
• Electrolytic Decomposition (Electrolysis): This process uses electrical energy to drive the decomposition. A primary example is the electrolysis of water, where water (H2O) decomposes into Hydrogen gas and Oxygen gas when an electric current is passed through it.
• Photolytic Decomposition (Photolysis): This breakdown is triggered by light energy (photons). For example, Silver Chloride (AgCl) turns grey in sunlight because it decomposes into Silver metal and Chlorine gas.
• Combination Reactions:
• These are reactions where a single substance is formed from multiple reactants. A common example is the burning of magnesium ribbon in air to form Magnesium Oxide or the burning of coal (Carbon + Oxygen → Carbon Dioxide).
• Double Displacement Reactions:
• These involve the exchange of ions between reactants. A well-known example is the reaction between Sodium Sulphate and Barium Chloride, which produces a white precipitate of Barium Sulphate and Sodium Chloride.

Important Points

• Energy Nature: Most decomposition reactions are endothermic in nature because they involve the absorption of energy (heat, light, or electricity) to break existing chemical bonds.
• Biological Application: Decomposition reactions are vital in the human body. During digestion, complex food molecules like starch and proteins undergo decomposition reactions to break down into simpler substances like glucose and amino acids, which the body can then absorb.
• Industrial Use: Decomposition of Calcium Carbonate is a key step in the manufacturing of cement. The Calcium Oxide (Quick Lime) produced is a vital raw material in various industries.
• Photography: The photolytic decomposition of Silver Bromide (AgBr) was traditionally used in Black and White Photography to capture images on film.

The correct answer is 20 m; opposite to the direction of the force.

Key Points

• Work Done is defined as the product of the magnitude of the force acting on a body and the displacement caused by that force in the direction of the force. The formula is W = F × d × cos(θ), where θ is the angle between the force and displacement.
• A retarding force is a force that acts in the opposite direction to the motion (displacement) of an object, such as friction or brakes applied to a vehicle. Because the force and displacement are in opposite directions, the angle θ is 180 degrees.
• In this problem, the force (F) is 10 N and the work done (W) is -200 J. The negative sign of the work done mathematically confirms that the force and displacement are in opposite directions, as cos(180°) = -1.
• To find the magnitude of displacement (d), we rearrange the formula: -200 = 10 × d × (-1). This simplifies to -200 = -10d. By dividing both sides by -10, we get d = 20 meters.
• The magnitude of the displacement is 20 m, and since it is a retarding force causing negative work, the direction of displacement must be opposite to the direction of the force.

Additional Information

• Negative Work: This occurs when the force and displacement are in opposite directions (90° < θ ≤ 180°). A classic example is the work done by friction on a sliding box or the work done by gravity on an object thrown upwards.
• Positive Work: This occurs when the force and displacement are in the same direction (0° ≤ θ < 90°). For instance, when a person pulls a trolley forward, the work done is positive because the force helps the motion.
• Zero Work: Work is zero when the force is perpendicular to the displacement (θ = 90°) or if there is no displacement (d = 0). An example is a person standing still holding a heavy suitcase; no work is done on the suitcase because there is no displacement.
• Units of Measurement: In the SI system (International System of Units), Work is measured in Joules (J), Force in Newtons (N), and Displacement in meters (m). One Joule is equal to the work done by a force of one Newton moving an object one meter.
• Nature of Force: A retarding force results in negative acceleration (deceleration). According to Newton's Second Law (F = ma), the direction of acceleration is always the same as the direction of the net force. Therefore, if the force is retarding, the acceleration is in the opposite direction of the initial velocity.

The correct answer is 2 m.

Key Points

• The relationship between the focal length (f) and the radius of curvature (R) of a spherical mirror (both convex and concave) is given by the formula f = R / 2.
• In this specific problem, the radius of curvature (R) of the convex mirror is given as 4.00 m.
• By substituting the value into the formula: f = 4.00 m / 2 = 2.00 m. Hence, the focal length of the mirror is 2 m.
• For a convex mirror, the principal focus (F) and the center of curvature (C) lie behind the mirror; therefore, according to the sign convention, the focal length is taken as positive.
• The principal focus of a spherical mirror is a point on the principal axis where light rays traveling parallel to the axis appear to diverge from (in the case of a convex mirror) or actually meet (in the case of a concave mirror).

Additional Information

• Convex Mirror Characteristics:
• A convex mirror is also known as a diverging mirror because it reflects parallel light rays away from each other.
• It always forms a virtual, erect, and diminished (smaller than the object) image, regardless of the distance of the object from the mirror.
• The focal length of a convex mirror is always positive according to the New Cartesian Sign Convention.
• Uses in Automobiles:
• Convex mirrors are preferred as rear-view mirrors in vehicles because they provide a much wider field of view compared to plane mirrors.
• Since the images formed are diminished, they allow the driver to see a larger area of traffic behind the vehicle.
• The fact that the image is always erect (upright) is critical for safety while driving.
• Comparison with Concave Mirrors:
• A concave mirror (converging mirror) has a negative focal length because its focus is in front of the reflecting surface.
• Unlike convex mirrors, concave mirrors can produce both real and virtual images depending on the object's position.

Important Points

• New Cartesian Sign Convention:
• The pole (P) of the mirror is taken as the origin.
• Objects are always placed to the left of the mirror, meaning light travels from left to right.
• Distances measured in the direction of incident light (to the right of the pole) are positive.
• Distances measured against the direction of incident light (to the left of the pole) are negative.
• Mirror Formula and Magnification:
• The relationship between object distance (u), image distance (v), and focal length (f) is 1/f = 1/v + 1/u.
• Magnification (m) is the ratio of the height of the image to the height of the object, expressed as m = h'/h = -v/u.
• For a convex mirror, the magnification m is always less than 1 and positive.
• Center of Curvature (C): This is the center of the hollow glass sphere of which the mirror is a part. The distance PC is the radius of curvature (R), which is exactly twice the focal length.

The correct answer is Reduction of DNA in gametes.

Key Points

• The process that ensures the DNA amount does not double in every generation is the reduction of DNA in gametes, which occurs through a specialized cell division called meiosis.
• In sexually reproducing organisms, the body cells (somatic cells) are typically diploid, meaning they contain two sets of chromosomes (one from each parent).
• During the formation of gametes (sperm and egg), meiosis reduces the chromosome number by half, resulting in haploid cells that contain only one set of DNA.
• When fertilization occurs, the male gamete and female gamete fuse to form a zygote, thereby restoring the original diploid (2n) number of chromosomes characteristic of the species.
• Without this reduction division, the DNA content would double with every successive generation, leading to polyploidy and biological instability.

Additional Information

• Meiosis vs. Mitosis:
• Mitosis is the process of somatic cell division where the DNA amount remains constant, producing two identical daughter cells. It is used for growth and repair.
• Meiosis involves two rounds of division (Meiosis I and Meiosis II) but only one round of DNA replication, which is how the reduction is achieved.
• DNA and Chromosomes:
• DNA (Deoxyribonucleic acid) is the hereditary material located in the nucleus. In humans, the diploid number is 46 chromosomes (23 pairs), while the haploid gametes contain 23 chromosomes.
• The concept of germ plasm and the necessity of reduction division was notably proposed by August Weismann in 1887.
• Biological Significance:
• Beyond maintaining ploidy, meiosis facilitates genetic variation through crossing over during prophase I, where homologous chromosomes exchange genetic material.
• Fertilization (or syngamy) is the crucial counterpart to meiosis that ensures the continuity of the species with a stable genome size across generations.
• Other Options Context:
• Division of neurons: Neurons are post-mitotic cells in adults, meaning they generally do not divide, and their division would not affect DNA inheritance across generations.
• Division of body cells: Also known as mitosis, this maintains the DNA amount within an individual but does not handle the generational transfer of DNA.
• Consistency of DNA in gametes: While DNA must be consistent in terms of quality, if the quantity remained the same as somatic cells, the DNA would double upon fusion.

The correct answer is Diamond.

Key Points

• Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. It is one of the most famous allotropes of carbon. Hence, option 3 is correct.
• Quartz is a hard, crystalline mineral composed of silicon and oxygen atoms, chemically known as silicon dioxide (SiO2). It is not an allotropic form of carbon. Hence, option 1 is incorrect.
• Limestone is a sedimentary rock, composed mainly of skeletal fragments of marine organisms such as coral and mollusks. Its major materials are the minerals calcite and aragonite, which are different crystal forms of calcium carbonate (CaCO3). Hence, option 2 is incorrect.
• Alum is a type of chemical compound, usually a hydrated double sulfate salt of aluminum with the general formula XAl(SO4)2·12H2O, where X is a monovalent cation such as potassium or ammonium. Hence, option 4 is incorrect.
• In a Diamond, each carbon atom is connected to four other carbon atoms by strong covalent bonds, forming a rigid three-dimensional tetrahedral structure. This structure accounts for its extreme hardness and durability.

Additional Information

• Allotropy:
• It is the property of some chemical elements to exist in two or more different forms, in the same physical state, known as allotropes of the elements.
• Allotropes typically exhibit different physical properties (like hardness and conductivity) but identical chemical properties.
• Other Allotropes of Carbon:
• Graphite: Carbon atoms are arranged in hexagonal layers. It is soft, slippery, and used as a lubricant and in pencil leads. It is a good conductor of electricity due to delocalized electrons.
• Fullerenes: Discovered in 1985, these are molecules composed entirely of carbon in the form of a hollow sphere, ellipsoid, or tube. Buckminsterfullerene (C60) is the most common form, resembling a soccer ball.
• Graphene: A single layer of carbon atoms arranged in a two-dimensional honeycomb lattice. It is incredibly strong, light, and an excellent conductor of heat and electricity.
• Uses of Diamond:
• Because of its extreme hardness, it is widely used in industrial applications such as cutting, drilling, and grinding.
• Due to its high dispersion of light and brilliance, it is highly valued as a gemstone in jewelry.

Important Points

• Physical Constants: Diamond has a very high melting point (about 3843 Kelvin) and a density of approximately 3.51 g/cm³. Its refractive index is exceptionally high at 2.417, leading to its characteristic sparkle.
• Natural Formation: Natural diamonds are formed at high temperatures and pressures at depths of 150 to 200 kilometers within the Earth's mantle.
• Chemical Symbolism: Carbon has the atomic number 6 and belongs to Group 14 of the periodic table. It displays tetravalency and catenation, which allows it to form a vast variety of compounds and structures.
• Measurement: The weight of a diamond is measured in carats. One carat is equal to 200 milligrams.

The correct answer is It becomes more concentrated.

Key Points

• Concentration refers to the amount of solute that has been dissolved in a given amount of solvent or solution. It is a measure of how "crowded" the solute particles are within the solvent.
• When more solute is added to the same amount of solvent, the ratio of solute to solvent increases. This results in the solution becoming more concentrated. Hence, the correct answer is option 2.
• From a mathematical perspective, concentration (C) is often expressed as C = mass of solute / volume of solvent. If the mass of the solute increases while the volume remains constant, the value of C must increase.
• A concentrated solution contains a relatively large amount of dissolved solute, whereas a dilute solution contains a relatively small amount of solute. Adding solute moves the state of the solution from dilute toward concentrated.
• This process continues until the solution reaches its saturation point, at which no more solute can be dissolved at a given temperature and pressure.

Additional Information

• Solute and Solvent: In a solution, the solute is the substance that is being dissolved (e.g., salt or sugar), and the solvent is the medium in which it dissolves (e.g., water).
• Saturation: An unsaturated solution can still dissolve more solute. A saturated solution is one where the maximum amount of solute has been dissolved; adding more solute after this point will result in the extra solute settling at the bottom as a precipitate.
• Suspension: A suspension is a heterogeneous mixture containing large particles that settle out upon standing. Simply adding more solute to a solution does not automatically make it a suspension unless the particles fail to dissolve and remain large and visible.
• Units of Concentration:
• Molarity (M): Defined as the number of moles of solute per liter of solution.
• Mass Percentage: The mass of the solute divided by the total mass of the solution, multiplied by 100.
• Molality (m): The moles of solute per kilogram of solvent.
• Dilution: This is the opposite of concentration. Dilution occurs when more solvent is added to a solution, which decreases the concentration of the solute.

Important Points

• Factors Affecting Solubility: The ability of a solute to dissolve in a solvent is influenced by temperature (solubility of solids usually increases with heat) and nature of the solute/solvent (the "like dissolves like" principle).
• Homogeneous vs. Heterogeneous: A true solution is always homogeneous, meaning its composition is uniform throughout. Adding solute keeps it a solution until the saturation limit is reached.
• Physical Change: Dissolving a solute in a solvent is generally considered a physical change, as the solute can often be recovered through processes like evaporation or distillation.

The correct answer is Using bulbs or fluorescent tubes.

Key Points

• Judicious energy use involves the efficient and responsible consumption of electricity and other resources to minimize waste and reduce the environmental impact.
• Using bulbs or fluorescent tubes for illumination is a standard and necessary practice; however, choosing Fluorescent tubes or CFLs over traditional incandescent bulbs is a mark of energy efficiency as they consume significantly less wattage for the same amount of light. Hence, statement 4 is correct.
• Keeping appliances on standby mode contributes to "vampire power" or phantom load, where devices continue to draw electricity even when not in active use. This leads to cumulative energy waste over time. Hence, statement 1 is incorrect.
• Heating an empty room continuously represents a total loss of thermal energy, as the heat generated does not benefit any occupant. This practice increases utility costs and carbon footprints unnecessarily. Hence, statement 2 is incorrect.
• Overusing air-conditioners, especially at very low temperature settings or when not needed, places an extreme load on the power grid and consumes high amounts of electrical energy due to the compressor's operation. Hence, statement 3 is incorrect.

Additional Information

• Energy Conservation Act, 2001:
• In India, this Act provides the legal framework for energy efficiency and conservation.
• It led to the establishment of the Bureau of Energy Efficiency (BEE), which regulates energy standards for appliances.
• BEE Star Rating System:
• The Star Labeling program was launched in 2006 to help consumers identify energy-efficient appliances.
• Appliances like Refrigerators, Air Conditioners, and Geysers are rated from 1 to 5 stars; a 5-star rating indicates the highest energy savings.
• Types of Lighting:
• Incandescent Bulbs: Convert only about 10% of energy into light, while 90% is wasted as heat.
• Fluorescent Tubes/CFLs: Use about 75% less energy than incandescent bulbs and last much longer.
• LED (Light Emitting Diode): The most modern and judicious choice, offering the highest durability and lowest energy consumption.
• Sustainable Development Goals (SDG):
• Goal 7 aims to ensure affordable, reliable, sustainable, and modern energy for all by the year 2030.
• Practicing judicious energy use at home directly contributes to reducing Greenhouse Gas (GHG) emissions and combating Climate Change.

Important Points

• Tips for Energy Saving:
• Always switch off lights and fans when leaving a room to ensure zero wastage.
• Use natural sunlight during the day to reduce the dependency on artificial lighting.
• Regular maintenance of electrical appliances improves their operational efficiency and reduces power consumption.

The correct answer is implantation.

Key Points

• Implantation is defined as the process by which the developing embryo (at the blastocyst stage) attaches itself to the endometrium, which is the internal lining of the uterus.
• This critical biological event typically occurs 6 to 12 days after fertilisation has taken place in the fallopian tube.
• During this process, the outer layer of the blastocyst, known as the trophoblast, secretes enzymes that allow it to penetrate the uterine wall, establishing a connection for nutrient exchange between the mother and the developing embryo.
• Once implantation is successful, the embryo begins to receive oxygen and nutrients from the mother's blood, and the production of human Chorionic Gonadotropin (hCG) starts, which is the hormone detected in pregnancy tests. Hence, the correct answer is implantation.

Additional Information

• Fertilisation:
• This is the process of fusion of a male gamete (sperm) with a female gamete (ovum) to form a diploid zygote.
• In humans, fertilisation usually occurs in the ampullary-isthmic junction of the fallopian tube.
• Embryogenesis:
• It is the process by which the embryo forms and develops. It starts with the fertilisation of the ovum and continues through the stages of cleavage, blastulation, gastrulation, and organogenesis.
• During embryogenesis, the single-celled zygote transforms into a multi-cellular complex organism.
• Abortion:
• This refers to the termination of a pregnancy by the removal or expulsion of an embryo or fetus before it can survive outside the uterus.
• It can occur spontaneously (commonly called a miscarriage) or can be purposely induced through medical or surgical means.

Important Points

• Blastocyst Stage: The embryo reaches the uterus about 3 to 4 days after fertilisation and floats in the uterine cavity for another 2 to 3 days before implantation begins.
• Endometrium Role: The hormone progesterone, secreted by the corpus luteum, is essential for maintaining the endometrium in a receptive state for implantation.
• Ectopic Pregnancy: If implantation occurs outside the uterus, most commonly in the fallopian tubes, it is called an ectopic pregnancy, which is a medical emergency.
• Placenta Formation: Following implantation, the trophoblast cells and the uterine tissue together form the placenta, a structural and functional unit between the developing fetus and the maternal body.

The correct answer is Supporting the formation of lysosomes.

Key Points

• The Golgi body, also known as the Golgi apparatus or Golgi complex, is a membrane-bound organelle found in most eukaryotic cells, including plant cells.
• One of the most critical functions of the Golgi apparatus is the modification, sorting, and packaging of proteins and lipids that are synthesized in the cell.
• It plays a vital role in supporting the formation of lysosomes. Lysosomes are formed by the budding off of vesicles from the trans-face of the Golgi apparatus, which contains processed hydrolytic enzymes. Hence, statement 3 is correct.
• Providing protection to the cell is primarily the function of the cell wall (in plants) and the plasma membrane. Hence, statement 1 is incorrect.
• Maintaining the shape of the cell is the responsibility of the rigid cell wall and the internal cytoskeleton (microtubules and microfilaments). Hence, statement 2 is incorrect.
• The synthesis of proteins is carried out by ribosomes, which may be free-floating in the cytoplasm or attached to the Rough Endoplasmic Reticulum (RER). Hence, statement 4 is incorrect.

Additional Information

• Discovery: The organelle was first identified by the Italian physician and biologist Camillo Golgi in 1898. He used a staining technique called the black reaction to visualize it.
• Structure: It consists of a series of flattened, stacked pouches called cisternae. A typical Golgi stack contains 4 to 8 cisternae.
• Dictyosomes: In plant cells, the Golgi bodies are often distributed as separate units throughout the cytoplasm and are specifically referred to as dictyosomes.
• Cell Wall Synthesis: In plants, the Golgi body is especially important because it synthesizes complex polysaccharides such as pectins and hemicelluloses, which are essential components of the plant cell wall.
• Secretion: It acts like a post office or dispatch center where materials are received from the Endoplasmic Reticulum (ER), modified (e.g., glycosylation), and then sent to their final destinations.

Important Points

• Lysosomes: These are often called the suicidal bags of the cell because they contain digestive enzymes that can break down waste materials, cellular debris, or even the cell itself if it becomes damaged.
• Polarity: The Golgi apparatus has a distinct polarity, with a cis face (receiving side near the ER) and a trans face (shipping side towards the plasma membrane).
• Vesicle Transport: Transport vesicles from the Endoplasmic Reticulum fuse with the cis cisternae, and after processing, secretory vesicles bud off from the trans cisternae.
• Storage: Apart from packaging, the Golgi body also serves as a site for the temporary storage of cellular products before they are secreted.

The correct answer is The amount of solute present in a given amount of solution.

Key Points

• The concentration of a solution refers to the amount of solute dissolved in a particular amount (mass or volume) of the solution or solvent.
• It is a quantitative measure that helps determine how "strong" or "weak" a solution is based on the proportion of its components.
• Statement 1: This statement correctly defines concentration as the ratio of the solute to the total solution. This is the standard way to express concentration in chemistry. Hence, statement 1 is correct.
• Statement 2: This statement suggests that concentration is the amount of solution in a solute, which is logically incorrect as the solute is a component of the solution, not vice versa. Hence, statement 2 is incorrect.
• Statement 3: This statement describes the amount of solvent in a solution; while this indicates the degree of dilution, the term "concentration" specifically tracks the solute. Hence, statement 3 is incorrect.
• Statement 4: This statement defines the amount of solvent in a given amount of solute, which is essentially the inverse of the solubility relationship and does not represent the definition of concentration. Hence, statement 4 is incorrect.
• Common mathematical expressions for concentration include Mass by mass percentage, Mass by volume percentage, and Molarity.

Additional Information

• Solute and Solvent:
• The Solute is the substance that is dissolved in another substance (usually present in a smaller quantity), such as salt or sugar.
• The Solvent is the substance in which the solute is dissolved (usually present in a larger quantity), such as water or alcohol.
• Types of Solutions based on Concentration:
• Dilute Solution: A solution containing a very small amount of solute relative to the amount of solvent.
• Concentrated Solution: A solution containing a large amount of solute dissolved in the solvent.
• Saturated Solution: A solution in which no more solute can be dissolved at a specific temperature.
• Unsaturated Solution: A solution where the amount of solute is less than the saturation level, meaning more can still be dissolved.
• Solubility:
• The maximum amount of solute that can be dissolved in 100 grams of solvent at a specific temperature is known as its solubility.
• Factors such as temperature and pressure can significantly affect the solubility of solids and gases in liquids.

Important Points

• Mass by Mass Percentage of a solution: Calculated as (Mass of solute / Mass of solution) × 100.
• Mass by Volume Percentage of a solution: Calculated as (Mass of solute / Volume of solution) × 100.
• Molarity (M): It is defined as the number of moles of solute dissolved in 1 liter of solution. It is the most commonly used unit of concentration in laboratories.
• Molality (m): It is defined as the moles of solute per kilogram of solvent. Unlike molarity, molality does not change with temperature changes because mass is constant.
• Parts Per Million (ppm): Used for expressing very low concentrations, such as the amount of pollutants in the atmosphere or water.

The correct answer is 40K19 and 40Ar18.

Key Points

• Isobars are defined as atoms of different chemical elements that have the same mass number (A) but different atomic numbers (Z).
• In the pair 40K19 and 40Ar18, both species possess a mass number of 40. However, Potassium (K) has an atomic number of 19, while Argon (Ar) has an atomic number of 18. Hence, this pair represents Isobars and is the correct answer.
• The mass number is the total count of protons and neutrons in the nucleus. For 40K19, there are 19 protons and 21 neutrons (40 - 19 = 21). For 40Ar18, there are 18 protons and 22 neutrons (40 - 18 = 22).
• Because Isobars have different atomic numbers, they have different numbers of electrons and thus exhibit completely different chemical properties and occupy different positions in the Modern Periodic Table.
• Other examples of Isobars include Carbon-14 (14C6) and Nitrogen-14 (14N7), where both have a mass number of 14 but different atomic numbers.

Additional Information

• Isotopes:
• These are atoms of the same element that have the same atomic number but different mass numbers.
• Example: 15O8 and 16O8 are isotopes of Oxygen. Similarly, 235U92 and 238U92 are isotopes of Uranium. Hence, options 2 and 4 are incorrect.
• Isotopes have the same chemical properties because they have the same number of electrons.
• Isotones:
• These are atoms of different elements that have the same number of neutrons.
• In the pair 15N7 and 16O8, Nitrogen has 8 neutrons (15 - 7 = 8) and Oxygen also has 8 neutrons (16 - 8 = 8). Hence, option 3 represents isotones and is incorrect.
• Atomic Number (Z): This value represents the number of protons present in the nucleus of an atom. It defines the identity of the element.
• Mass Number (A): This represents the sum of the number of protons and neutrons (collectively called nucleons) in the nucleus.

Important Points

• The term Isobar was originally suggested by Alfred Walter Stewart in the year 1918. It is derived from the Greek words 'isos' meaning equal and 'baros' meaning weight.
• While Isobars have different chemical properties, they may have similar physical properties that depend on the mass, such as density (though they are still distinct due to different atomic structures).
• Potassium-40 is a significant radioactive isotope used in geochronology (Potassium-Argon dating) to determine the age of ancient rocks and minerals.
• In nuclear physics, isobars are important for studying beta decay, where a nucleus transforms into its isobar by changing a neutron into a proton (or vice versa) to reach a more stable state.

The correct answer is Frequency < 20 Hz.

Key Points

• Infrasonic sound, often referred to as infrasound, describes sound waves with a frequency less than 20 Hz. These frequencies are below the lower limit of human audibility. Hence, statement 4 is correct.
• Sound waves are mechanical longitudinal waves that require a medium to propagate. Infrasonic sound can travel through air, liquids, and solids. It is often used to monitor large-scale atmospheric and geological events. Hence, statement 1 is incorrect.
• The range of frequency between 20 Hz and 20,000 Hz (20 kHz) is known as the audible range for the average human ear. Sound within this spectrum is what we typically perceive as speech, music, or environmental noise. Hence, statement 2 is incorrect.
• Frequencies greater than 20 kHz are classified as ultrasonic sound or ultrasound. While humans cannot hear these high-pitched waves, they are widely used in medical imaging (sonography) and industrial cleaning. Hence, statement 3 is incorrect.
• Many large animals, such as elephants, whales, and hippopotamuses, use infrasound for long-distance communication because low-frequency waves undergo less attenuation and can travel further through the environment.

Additional Information

• Characteristics of Infrasound:
• Infrasonic waves possess very long wavelengths, which allow them to bend around obstacles and travel thousands of kilometers without losing much energy.
• Because of their low frequency, they are not easily absorbed by the atmosphere or the ground, making them ideal for detecting remote events.
• Natural Sources:
• Major geophysical events like earthquakes, volcanic eruptions, avalanches, and lightning generate intense infrasonic waves.
• Ocean waves and severe weather systems like hurricanes and tornadoes also produce distinct infrasonic signatures that can be tracked by meteorologists.
• Detection and Monitoring:
• Scientists use specialized instruments called microbarometers to detect minute changes in atmospheric pressure caused by infrasonic waves.
• The International Monitoring System (IMS), established under the Comprehensive Nuclear-Test-Ban Treaty (CTBT), uses a global network of infrasound stations to detect clandestine underground or atmospheric nuclear explosions.
• Human Impact:
• Although humans cannot "hear" infrasound in the traditional sense, exposure to high-intensity low-frequency sound can cause physical sensations like chest wall vibration, nausea, headaches, or feelings of anxiety.
• Man-made sources include wind turbines, heavy machinery, aircraft engines, and large cooling fans in industrial settings.

Important Points

• Speed of Sound: The speed at which infrasound travels depends on the medium. In dry air at 20 degrees Celsius, the speed is approximately 343 meters per second.
• Wave Nature: Like all sound, infrasound is a mechanical wave that consists of periodic compressions and rarefactions of the medium through which it travels.
• Seismology Connection: Infrasonic waves are closely related to seismic waves; while seismic waves travel through the Earth's crust, infrasonic waves travel through the atmosphere, often coupled with the same geological trigger.

The correct answer is 80 N.

Key Points

• According to Newton’s Third Law of Motion, for every action, there is always an equal and opposite reaction.
• In this scenario, when a person pushes a stationary wall with a force of 80 N, this is considered the action force.
• The wall simultaneously exerts a reaction force back onto the person. According to the law, the magnitude of this reaction force must be exactly the same as the action force, which is 80 N.
• While the magnitudes are equal, the directions are opposite; if the push is toward the wall, the reaction is away from the wall toward the person.
• It is a fundamental property of forces that they always exist in pairs. A single isolated force is physically impossible. Hence, the correct answer is 80 N.

Additional Information

• Newton's First Law of Motion:
• Also known as the Law of Inertia, it states that an object will remain at rest or move at a constant velocity unless acted upon by an external unbalanced force.
• Inertia is the inherent property of an object to resist changes in its state of motion.
• Newton's Second Law of Motion:
• This law defines force quantitatively as the product of mass and acceleration (F = ma).
• The SI unit of force is the Newton (N), which is defined as 1 kg·m/s².
• It implies that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• Nature of Force:
• Force is a vector quantity, meaning it possesses both magnitude and direction.
• Common types of forces include gravitational force, frictional force, normal force, and tension.

Important Points

• Action-Reaction Pairs:
• One crucial aspect of Newton’s Third Law is that the action and reaction forces act on two different objects.
• Because they act on different bodies, these forces do not cancel each other out. For example, the 80 N force acts on the wall, while the reaction force of 80 N acts on the person's hands.
• Stationary Wall Mechanics:
• The wall remains stationary despite the 80 N push because it is attached to the earth, and the static friction or structural integrity of the wall provides a counter-force that results in a net force of zero on the wall itself.
• Real-world Examples:
• Walking: As you step, your foot pushes backward on the ground (action), and the ground pushes your foot forward (reaction).
• Swimming: A swimmer pushes the water backward to move their body forward.
• Rocket Propulsion: The rocket engine expels gas downward at high velocity, and the escaping gas exerts an upward force on the rocket.

The correct answer is 10√2 m/s.

Key Points

• When a ball is dropped from a certain height (h), its initial velocity (u) is always taken as 0 m/s because it starts from a state of rest.
• The motion of the ball is a case of uniformly accelerated motion under the influence of gravity (g), where the value of g is provided as 10 m/s^2.
• To find the final velocity (v) or the speed with which the ball strikes the ground, we use the third equation of motion: v^2 = u^2 + 2gh.
• Given values for the calculation:
• Initial velocity (u) = 0 m/s
• Height (h) = 10 m
• Acceleration due to gravity (g) = 10 m/s^2
• Substituting the values into the formula: v^2 = (0)^2 + 2 × 10 × 10. This simplifies to v^2 = 200.
• Taking the square root of both sides, v = √200, which can be further simplified as v = √(100 × 2) = 10√2 m/s. Hence, the speed of the ball is 10√2 m/s.

Additional Information

• Equations of Motion: These equations describe the behavior of a physical system in terms of its motion as a function of time. The three primary equations for constant acceleration are v = u + at, s = ut + 1/2 at^2, and v^2 = u^2 + 2as.
• Free Fall: An object is said to be in free fall when it falls towards the Earth solely under the influence of the gravitational force. In the absence of air resistance, all objects, regardless of their mass, accelerate toward the center of the Earth at the same rate.
• Acceleration due to Gravity (g): This is the acceleration gained by an object due to gravitational force. Its standard value on the surface of the Earth is approximately 9.8 m/s^2, but it is often rounded to 10 m/s^2 in physics problems for ease of calculation.
• Kinetic and Potential Energy: This problem can also be solved using the Law of Conservation of Energy. At the height of 10 m, the ball possesses Potential Energy (PE = mgh). Just before striking the ground, all this PE is converted into Kinetic Energy (KE = 1/2 mv^2). Equating mgh = 1/2 mv^2 leads to the formula v = √(2gh).

Important Points

• Mass Independence: Notice that the mass (m) of the ball does not appear in the final formula for velocity. This confirms that whether a heavy ball or a light ball is dropped, they will strike the ground with the same speed, provided there is no air resistance.
• Direction of Motion: In these calculations, since the ball is moving downwards in the direction of the gravitational force, the value of g is taken as positive (+10 m/s^2). If the ball were thrown upwards, g would be taken as negative (-10 m/s^2).
• Impact of Height: The final speed is directly proportional to the square root of the height. If the height were quadrupled to 40 m, the striking speed would double.

The correct answer is Both A and R are true, and R is the correct explanation of A.

Key Points

• Assertion (A) states that Insulin helps lower blood glucose levels. Insulin is an essential hormone produced by the pancreas that allows your body to use sugar (glucose) from carbohydrates in the food that you eat for energy or to store glucose for future use. Hence, Assertion (A) is correct.
• Reason (R) explains that Insulin converts glucose into glycogen for storage in the liver. When blood glucose levels are high, the pancreas secretes insulin, which signals the liver, muscles, and fat cells to take up the glucose from the blood. In the liver and muscles, this glucose is converted into glycogen through a process called glycogenesis. Hence, Reason (R) is correct.
• The primary mechanism by which insulin lowers the concentration of glucose in the bloodstream is by facilitating its uptake into cells and its conversion into storage forms like glycogen.
• Since the conversion and storage of glucose as glycogen directly results in the reduction of circulating blood sugar, the Reason (R) serves as a scientifically accurate explanation for the Assertion (A).
• Hence, the correct answer is Both A and R are true, and R is the correct explanation of A.

Additional Information

• The Pancreas and Insulin Secretion:
• The pancreas contains clusters of cells known as the Islets of Langerhans.
• Beta cells within these clusters are responsible for the synthesis and secretion of insulin.
• Insulin is considered an anabolic hormone because it promotes the storage of glucose, amino acids, and fats.
• Glucagon - The Antagonist:
• While insulin lowers blood sugar, another hormone called glucagon (secreted by Alpha cells of the pancreas) raises it.
• Glucagon triggers glycogenolysis, which is the breakdown of glycogen back into glucose when blood sugar is low.
• Diabetes Mellitus:
• A deficiency in insulin production or a decrease in the body's responsiveness to it leads to Diabetes Mellitus.
• Type 1 Diabetes occurs when the pancreas produces little or no insulin.
• Type 2 Diabetes occurs when the body's cells become resistant to insulin, leading to hyperglycemia (high blood sugar).

Important Points

• Homeostasis: The balance between insulin and glucagon ensures glucose homeostasis, maintaining blood sugar within a narrow range (typically 70-100 mg/dL during fasting).
• Discovery: Insulin was first isolated in 1921 by Sir Frederick G. Banting and Charles H. Best at the University of Toronto.
• Target Tissues: Beyond the liver, insulin significantly affects skeletal muscle (where it also stores glycogen) and adipose tissue (where it promotes the storage of lipids).
• Glycogenesis vs Glycogenolysis: Glycogenesis is the formation of glycogen from glucose (stimulated by insulin), whereas glycogenolysis is the breakdown of glycogen into glucose (stimulated by glucagon or epinephrine).

The correct answer is It depends on the length of the wire.

Key Points

• Resistivity is a fundamental and intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. Hence, statement 2 is correct.
• It is mathematically defined as the resistance of a conductor having a unit length and a unit area of cross-section. While resistance changes with dimensions, resistivity remains constant for a specific material at a given temperature.
• The resistivity of a material does not depend on its geometrical dimensions such as length or thickness (area). It only depends on the nature of the material and its temperature. Hence, statement 3 is NOT correct.
• The resistivity of metallic conductors typically increases with an increase in temperature because the thermal vibrations of atoms interfere with the flow of electrons. Hence, statement 1 is correct.
• The SI unit of resistivity is the ohm-meter (written as Ω ⋅ m). This unit is derived from the formula for resistance R = ρ (L/A), where ρ is resistivity. Hence, statement 4 is correct.

Additional Information

• Difference between Resistance and Resistivity:
• Resistance (R) is the property of a specific object or component and depends on its length, area, and material. Its unit is the Ohm (Ω).
• Resistivity (ρ) is the property of the substance itself. For example, a 1-meter copper wire and a 10-meter copper wire will have different resistances but the same resistivity.
• Classification of Materials:
• Conductors: Materials like Silver and Copper have very low resistivity, in the range of 10^-8 Ω⋅m to 10^-6 Ω⋅m. Silver is the best conductor with a resistivity of approximately 1.59 × 10^-8 Ω⋅m at 20°C.
• Insulators: Materials like Glass, Rubber, and Ebonite have very high resistivity, ranging from 10^12 Ω⋅m to 10^17 Ω⋅m.
• Alloys: Substances like Nichrome and Constantan have higher resistivity than their constituent metals and are used in heating elements because they do not oxidize (burn) easily at high temperatures.
• Conductivity:
• The reciprocal of resistivity is known as electrical conductivity (σ). It measures how easily a material allows current to flow. Its SI unit is siemens per meter (S/m) or (ohm-meter)^-1.

Important Points

• Factors Affecting Resistance: Resistance of a uniform conductor is directly proportional to its length (L) and inversely proportional to its area of cross-section (A).
• Temperature Coefficient: For most metals, the temperature coefficient of resistivity is positive. However, for semiconductors and insulators, resistivity typically decreases as temperature increases (negative temperature coefficient).
• Usage in Appliances: The high resistivity of alloys like Manganin and Constantan makes them ideal for making standard resistors because their resistivity changes very little with small changes in temperature.

The correct answer is Both (ii) and (iii).

Key Points

• The magnetic field (B) produced by a straight current-carrying conductor at a given point is directly proportional to the magnitude of the current (I) passing through the conductor. If the current increases, the magnetic field strength also increases. Hence, statement (iii) is correct.
• The magnetic field (B) is inversely proportional to the distance (r) from the conductor. As we move further away from the wire, the strength of the magnetic field weakens. Hence, statement (ii) is correct.
• According to the inverse relationship, the field strength would decrease, not increase, as the distance from the source increases. Hence, statement (i) is incorrect.
• Since the magnetic field is linearly proportional to the current, any increase in current (I) will result in an immediate and proportional increase in the magnetic field (B), without any initial decrease. Hence, statement (iv) is incorrect.
• The relationship is mathematically expressed by the formula B = (μ₀I) / (2πr), where μ₀ is the permeability of free space.

Additional Information

• Right-Hand Thumb Rule: This rule is used to find the direction of the magnetic field. If you hold the straight conductor in your right hand such that the thumb points in the direction of the current, then your fingers will curl around the conductor in the direction of the magnetic field lines.
• Magnetic Field Lines: For a straight conductor, the magnetic field lines form concentric circles around the wire. The center of these circles lies on the wire, and the plane of the circles is perpendicular to the conductor.
• Discovery: The relationship between electricity and magnetism was first discovered by Hans Christian Oersted in 1820, when he noticed a compass needle deflect near a current-carrying wire.
• SI Unit: The SI unit of magnetic field induction is the Tesla (T). Another commonly used unit is the Gauss (G), where 1 Tesla = 10,000 Gauss.

Important Points

• Factors affecting Magnetic Field: The strength of the field depends on the medium surrounding the conductor, the magnitude of the current, and the radial distance from the wire.
• Vector Quantity: The magnetic field is a vector quantity, meaning it has both magnitude and direction. At any point, the field is tangent to the magnetic field line at that point.
• Permeability: The value of μ₀ (permeability of free space) is approximately 4π × 10⁻⁷ T·m/A. This constant defines how easily magnetic flux can pass through a vacuum.
• Applications: This principle is fundamental in the design of electromagnets, electric motors, generators, and transformers where controlled magnetic fields are required.

p data-end="57" data-start="0">The correct answer is Rusting of iron.

 

Key Points

• Corrosion is a natural process that converts a refined metal into a more chemically stable form such as oxide, hydroxide, or sulfide. It is the gradual destruction of materials (usually metals) by chemical and/or electrochemical reaction with their environment.
• Rusting of iron is a specific type of corrosion that occurs when iron objects are exposed to oxygen and moisture (water) for a long time. The reddish-brown flaky substance formed is called rust (hydrated iron oxide). Hence, statement 1 is correct.
• Burning of LPG (Liquefied Petroleum Gas) is a combustion reaction. While it is a chemical change, it involves the rapid oxidation of fuel to produce heat and light, rather than the slow deterioration of a metal surface. Hence, statement 2 is incorrect.
• Boiling of water is a physical change where water transitions from a liquid state to a gaseous state (steam). No new chemical substance is formed during this process. Hence, statement 3 is incorrect.
• Dissolving sugar in water is also a physical change. The sugar molecules disperse throughout the water, but their chemical identity remains unchanged, and the sugar can be recovered by evaporating the water. Hence, statement 4 is incorrect.
• Hence, the correct answer is Rusting of iron.

Additional Information

• Chemical Nature of Rusting:
• The process of rusting can be represented by the following equation: 4Fe + 3O2 + 6H2O → 4Fe(OH)3. The final product eventually dehydrates to form Fe2O3·nH2O.
• Both oxygen and water (or moisture) are essential for rusting to occur. In coastal areas, rusting is faster because the presence of salt in water speeds up the process of oxidation.
• Prevention of Corrosion:
• Galvanization: It is the process of depositing a thin layer of zinc on iron or steel to protect it from corrosion. Since zinc is more reactive than iron, it oxidizes first, protecting the underlying metal.
• Alloying: Metals are mixed with other elements to enhance their properties. For example, Stainless steel is made by mixing iron with carbon, chromium, and nickel. It does not rust easily.
• Anodizing: This involves forming a thick oxide layer of aluminum to make it resistant to further corrosion.
• Other Methods: Painting, oiling, greasing, or chrome plating are common ways to prevent the metal surface from coming into contact with air and moisture.

Important Points

• Corrosion in Other Metals:
• Silver: It reacts with sulfur in the air (from hydrogen sulfide) to form a black coating of silver sulfide (Ag2S). This is often called tarnishing.
• Copper: When exposed to moist air, copper reacts with carbon dioxide and water to form a green coating of basic copper carbonate [CuCO3·Cu(OH)2].
• Economic Impact:
• Corrosion causes enormous damage to car bodies, bridges, iron railings, ships, and all objects made of metals, especially those made of iron.
• It is a significant economic burden globally, leading to the loss of millions of tons of steel every year, necessitating constant replacement and maintenance of infrastructure.

The correct answer is also doubles.

Key Points

• According to Newton’s Second Law of Motion, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• The relationship is mathematically expressed by the formula F = ma, where F represents the resultant force, m represents the mass of the object, and a represents the acceleration.
• If the force on an object is doubled (changed from F to 2F) while the mass (m) remains constant, the acceleration must adjust to maintain the equality of the equation.
• Using the formula a = F / m, the new acceleration (a') becomes a' = (2F) / m. Since the original acceleration was F / m, it follows that a' = 2a.
• This demonstrates that when the net force is multiplied by a factor of two, the acceleration also multiplied by the same factor. Hence, the correct answer is also doubles.

Additional Information

• Newton’s Second Law of Motion:
• This law defines the quantitative relationship between force and motion. It allows for the calculation of how much an object will speed up or slow down under a specific load.
• The SI unit of force is the Newton (N), which is the amount of force required to accelerate a mass of 1 kilogram (kg) at a rate of 1 meter per second squared (m/s²).
• Acceleration is a vector quantity, which means it has both magnitude and direction. The acceleration always occurs in the direction of the unbalanced force.
• Inertia and Mass:
• Mass is a measure of an object's inertia, which is its resistance to changes in its state of motion.
• If the mass of an object were to double while the force remained the same, the acceleration would be halved because acceleration is inversely proportional to mass.
• In classical mechanics, mass is typically considered constant unless the object is traveling at speeds approaching the speed of light.

Important Points

• Sir Isaac Newton first published these laws of motion in his monumental work, Philosophiæ Naturalis Principia Mathematica, in the year 1687.
• Net Force: It is important to note that the force mentioned in the second law is the net force (the vector sum of all forces). If multiple forces act on an object but their sum is zero, the acceleration remains zero.
• Real-world applications: This principle is used in automotive engineering to determine engine power, in aerospace to calculate rocket thrust, and in sports science to analyze athlete performance.
• Units: Always ensure mass is in kilograms and acceleration is in m/s² to obtain the force in Newtons. 1 Newton = 1 kg·m/s².

Shortcut Trick

Total age 4 years ago = 44 − (2 × 4) = 36 years.

Ratio of Priya to Rohan 4 years ago = 2 : 1.

Sum of ratio parts (2 + 1) = 3 units → 36 years ⇒ 1 unit = 12 years.

Priya age 4 years ago = 2 × 12 = 24 years.

Present age of Priya = 24 + 4 = 28 years.

∴ The correct answer is 28 years.

Alternate Method

Given: Sum of Priya (P) and Rohan (R) = 44 years. Condition: (P − 4) = 2 × (R − 4).

⇒ P + R = 44 ⇒ R = 44 − P

⇒ P − 4 = 2 × ((44 − P) − 4)

⇒ P − 4 = 2 × (40 − P)

⇒ P − 4 = 80 − 2P

⇒ 3P = 84

⇒ P = 28

∴ The correct answer is 28 years.

Additional Information

Age Difference Rule

The difference between the ages of two persons remains constant throughout their lives, regardless of the number of years passed.

Sum of Ages Change

If the sum of ages of 'n' people is S, then 't' years ago, the sum was S − (n × t), and 't' years hence, it will be S + (n × t).

Shortcut Trick

Given equation: 1/(1 − tanθ) + 1/(1 + tanθ) = 3

Simplify: 2/(1 − tan²θ) = 3 ⇒ 1 − tan²θ = 2/3 ⇒ tan²θ = 1/3

For acute angle θ, tanθ = 1/√3 ⇒ θ = 30°

Therefore, cosecθ = cosec 30° = 2.

∴ The correct answer is 2.

Alternate Method

Given:

1/(1 − tanθ) + 1/(1 + tanθ) = 3

Formula Used:

(a − b)(a + b) = a² − b²

1 + tan²θ = sec²θ

cosecθ = 1/sinθ

Calculations:

⇒ [(1 + tanθ) + (1 − tanθ)] / [(1 − tanθ)(1 + tanθ)] = 3

⇒ 2 / (1 − tan²θ) = 3

⇒ 2 = 3 − 3 tan²θ

⇒ 3 tan²θ = 1

⇒ tan²θ = 1/3

⇒ tanθ = 1/√3 (since θ is an acute angle, tanθ must be positive)

⇒ θ = 30°

⇒ cosecθ = cosec 30° = 2

∴ The correct answer is 2.

Additional Information

Trigonometric Identities

Recall that sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Standard Values for 30°

For θ = 30°, sinθ = 1/2, cosθ = √3/2, tanθ = 1/√3, and cosecθ = 2.

Shortcut Trick

Let the Marked Price be 100%.

Given Discount = 50%, so Selling Price = 100% − 50% = 50%.

The extra amount without discount represents the discount value itself.

Since 50% discount = ₹8,453, then the amount paid (50%) is also ₹8,453.

∴ The correct answer is ₹8,453.

Alternate Method

Given: Discount % = 50%, Extra amount if no discount = ₹8,453.

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

⇒ Let the Marked Price (MP) be x.

⇒ If no discount is given, she pays the Marked Price. The extra amount = MP − SP = Discount.

⇒ Discount = 50% of x = ₹8,453.

⇒ 0.50 × x = 8,453 ⇒ x = 8,453 ÷ 0.50 = ₹16,906.

⇒ Amount paid (SP) = MP − Discount = 16,906 − 8,453 = ₹8,453.

∴ The correct answer is ₹8,453.

Additional Information

Marked Price & Discount

The Marked Price is the price listed on the label. Discount is always calculated on the Marked Price unless stated otherwise.

The 50% Rule

When a 50% discount is applied, the Selling Price and the Discount Value are always equal, as the price is split exactly in half.

Shortcut Trick

Rate of 5% in fraction = 1/20. For 2 years, the ratio of Principal to Amount is (20)² : (21)².

Ratio = 400 : 441

Given, 400 units = ₹4,400 ⇒ 1 unit = ₹11

Amount = 441 units × 11 = ₹4,851

∴ The correct answer is ₹4,851.

Alternate Method

Given:

Principal (P) = ₹4,400

Rate (R) = 5% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4400 × (1 + 5/100)²

⇒ A = 4400 × (1 + 1/20)²

⇒ A = 4400 × (21/20) × (21/20)

⇒ A = 4400 × (441/400)

⇒ A = 11 × 441

⇒ A = ₹4,851

∴ The correct answer is ₹4,851.

Additional Information

Effective Interest Rate for 2 Years

For a rate R% compounded annually for 2 years, the effective rate is R + R + (R × R)/100.

Difference between CI and SI

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

Compounding Half-Yearly

When interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2n).

Shortcut Trick

Wages are distributed in the ratio of the efficiencies of the workers when they work for the same duration.

Efficiency ratio A : B : C = 1/17 : 1/51 : 1/68 = 12 : 4 : 3

Total ratio parts = 12 + 4 + 3 = 19 units

B's share = (4 ÷ 19) × 5,985 = 4 × 315 = ₹1,260

∴ The correct answer is ₹1,260.

Alternate Method

Given: Time taken by A = 17 days, B = 51 days, C = 68 days. Total wage = ₹5,985.

Formula Used: Total Work = LCM of time; Efficiency = Total Work ÷ Time; Wage ∝ Efficiency.

⇒ Total efficiency = 12 + 4 + 3 = 19 units

⇒ 19 units = ₹5,985

⇒ 1 unit = 5,985 ÷ 19 = ₹315

⇒ B's wage = 4 units = 4 × 315 = ₹1,260

∴ The correct answer is ₹1,260.

Additional Information

Wage Distribution Rule

If workers work for the same amount of time, the wage is distributed in the ratio of their efficiencies. If time periods differ, wage ratio = (E₁×T₁) : (E₂×T₂).

Efficiency-Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (Efficiency = 1/Time) when work is constant.

Work and Wages

Total Wage = (Individual Work Done ÷ Total Work) × Total Amount.

Shortcut Trick

Let Total Voters be 100%.

Voters who cast votes = 100% − 20% = 80%.

Winner gets 70% of 80%, Loser gets 30% of 80%.

Difference = (70% − 30%) of 80% = 40% of 80% = 32%.

32% = 23,200 ⇒ 1% = 725 ⇒ 100% = 72,500.

∴ The correct answer is 72,500.

Alternate Method

Given:

Non-voters = 20%

Winner share of cast votes = 70%

Winning margin = 23,200

Calculations:

⇒ Let total number of voters = x

⇒ Votes cast = 80% of x = 0.8x

⇒ Winner's votes = 70% of 0.8x = 0.56x

⇒ Loser's votes = (100% − 70%) of 0.8x = 30% of 0.8x = 0.24x

⇒ Margin = 0.56x − 0.24x = 0.32x

⇒ 0.32x = 23,200

⇒ x = 23,200 ÷ 0.32

⇒ x = 72,500

∴ The correct answer is 72,500.

Additional Information

Invalid Votes Concept

If some votes are invalid, subtract them from the cast votes before calculating the winner's share: Valid Votes = Cast Votes − Invalid Votes.

Percentage Change

Margin percentage based on total voters is (Winner% − Loser%) × (Cast Vote%).

Total Population vs Voters

Voters are usually a subset of the total population; questions may involve multiple stages of reduction from Total Population → Voter List → Casted Votes.

Shortcut Trick

Speed is distance per unit time.

Direct division: 120 ÷ 2 = 60 km/h.

∴ The correct answer is 60 km/h.

Alternate Method

Given:

Distance = 120 km

Time = 2 hours

Formula Used:

Speed = Distance ÷ Time

Calculations:

⇒ Speed = 120 km ÷ 2 hours

⇒ Speed = 60 km/h

∴ The correct answer is 60 km/h.

Additional Information

Unit Conversion

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

Average Speed

Total distance traveled divided by the total time taken. For two equal distances at speeds x and y, Average Speed = 2xy ÷ (x + y).

Shortcut Trick

Let Cost Price (CP) be 100.

Marked Price (MP) = 100 + 25% of 100 = 125.

Selling Price (SP) = 100 + 15% of 100 = 115.

Discount = MP − SP = 125 − 115 = 10.

Discount % = (Discount ÷ MP) × 100 = (10 ÷ 125) × 100 = 8%.

∴ The correct answer is 8%.

Alternate Method

Given: Markup% = 25%, Profit% = 15%. Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = CP × (100 + Profit%) ÷ 100 | Discount% = (MP − SP) ÷ MP × 100.

⇒ Assume CP = 100 units. ⇒ MP = 100 × (125 ÷ 100) = 125 units. ⇒ SP = 100 × (115 ÷ 100) = 115 units. ⇒ Discount Amount = MP − SP = 125 − 115 = 10 units. ⇒ Discount% = (10 ÷ 125) × 100 = 8%.

∴ The correct answer is 8%.

Additional Information

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Formula: SP = MP × (100 − Discount%) ÷ 100.

Relation between CP and MP

The direct ratio is MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Base circumference (2πr) = 176 cm ⇒ Radius (r) = 176 ÷ (2 × 22/7) = 28 cm.

Total Surface Area (TSA) = 2πr(h + r) = (Circumference) × (h + r).

TSA = 176 × (56 + 28) = 176 × 84.

Calculation: 176 × 84 = 14,784 cm².

∴ The correct answer is 14,784 cm².

Alternate Method

Given: Height (h) = 56 cm, Circumference = 176 cm.

Formula Used: Circumference = 2πr | TSA = 2πr(h + r).

⇒ 2 × (22/7) × r = 176

⇒ (44/7) × r = 176

⇒ r = 176 × (7/44) = 4 × 7 = 28 cm

⇒ TSA = 2 × (22/7) × 28 × (56 + 28)

⇒ TSA = 176 × 84

⇒ TSA = 14,784 cm²

∴ The correct answer is 14,784 cm².

Additional Information

Curved Surface Area of Cylinder

The area of the side surface only, given by CSA = 2πrh.

Volume of Cylinder

The total space occupied by the cylinder, calculated as V = πr²h.

Base Area of Cylinder

The area of one of the circular ends, given by Area = πr².

Shortcut Trick

For an equilateral triangle, the sum of perpendicular distances (p₁, p₂, p₃) from an interior point to the sides is equal to the height (h).

h = p₁ + p₂ + p₃ = 12.7 + 14.8 + 11.5 = 39 cm.

Area = h² / √3 = 39² / √3 = 1521 / √3 = 507√3 cm².

∴ The correct answer is 507√3 cm².

Alternate Method

Given: Perpendiculars p₁ = 12.7 cm, p₂ = 14.8 cm, p₃ = 11.5 cm.

Formula: Sum of perpendiculars (p₁+p₂+p₃) = (√3/2) × side (a).

⇒ Let the side of the equilateral triangle be 'a'.

⇒ Area of triangle = Sum of areas of three smaller triangles formed with the interior point.

⇒ (√3/4)a² = (1/2)×a×p₁ + (1/2)×a×p₂ + (1/2)×a×p₃

⇒ (√3/4)a² = (1/2)a (p₁ + p₂ + p₃)

⇒ (√3/2)a = 12.7 + 14.8 + 11.5 = 39

⇒ a = 78 / √3 = 26√3 cm

⇒ Area = (√3/4) × (26√3)²

⇒ Area = (√3/4) × 676 × 3 = 169 × 3 × √3 = 507√3 cm²

∴ The correct answer is 507√3 cm².

Additional Information

Viviani's Theorem

It states that the sum of the distances from any interior point to the sides of an equilateral triangle is constant and equal to the altitude (h) of the triangle.

Equilateral Triangle Formulas

Height (h) = (√3/2)a; Area = (√3/4)a²; In-radius (r) = a/(2√3); Circum-radius (R) = a/√3.

Area in terms of Height

If the height 'h' is known, the area of an equilateral triangle can be directly calculated as Area = h² / √3.

Shortcut Trick

4.2 / 6.3 = 2/3

x/2.1 + 2/3 = 4

x/2.1 = 4 − 2/3 = 10/3

x = (10/3) × 2.1 = (10/3) × (21/10) = 7

∴ Answer = 7

Alternate Method

Given:

x/2.1 + 4.2/6.3 = 4

Formula used:

Simplification of fractions

Calculation:

⇒ 4.2/6.3 = 42/63 = 2/3

⇒ Equation becomes:

x/2.1 + 2/3 = 4

⇒ x/2.1 = 4 − 2/3

⇒ = (12 − 2)/3 = 10/3

⇒ x = (10/3) × 2.1

⇒ = (10/3) × (21/10)

⇒ = 7

∴ The correct answer is 7.

Additional Information

Fraction Conversion

Convert decimals to fractions for easy solving

Equation Solving

Isolate variable step-by-step

Simplification

Reduce fractions before multiplying

Key Insight

Convert all terms into common format

Shortcut Trick

Difference in Distance = 15 − 10 = 5 km

Difference in Fare = ₹330 − ₹230 = ₹100

Per-kilometre Charge = ₹100 ÷ 5 = ₹20 per km

Base Fare = Total Fare − (Distance × Per-km Charge) = 230 − (10 × 20) = ₹30

∴ The correct answer is Base fare = ₹30; Per-kilometre Charge = ₹20.

Alternate Method

Given:

Let Base Fare be 'B' and Rate per km be 'R'.

⇒ Case 1: B + 10R = 230 — (i)

⇒ Case 2: B + 15R = 330 — (ii)

Calculations:

Subtracting equation (i) from (ii):

⇒ (B + 15R) − (B + 10R) = 330 − 230

⇒ 5R = 100

⇒ R = 20

Substitute R = 20 in equation (i):

⇒ B + 10(20) = 230

⇒ B + 200 = 230

⇒ B = 30

∴ The correct answer is Base fare = ₹30; Per-kilometre Charge = ₹20.

Additional Information

Linear Equation Modeling

This problem follows the format y = mx + c, where 'm' is the variable rate, 'c' is the fixed constant (base fare), and 'y' is the total cost.

Simultaneous Equations

To find two unknowns, we require at least two independent linear equations. The elimination method is usually the fastest way to solve for the variable rate.

Unitary Application

In fixed plus variable cost problems, the entire difference in total cost is attributed to the difference in variable quantity only.

Shortcut Trick

Combined efficiency of A, B and C = (Total Work ÷ 3) + (Total Work ÷ 10) + (Total Work ÷ 15)

Using LCM (3, 10, 15) = 30 as base work.

Total efficiency = 10 + 3 + 2 = 15 units/day.

Twice the work = 30 × 2 = 60 units.

Time taken = 60 ÷ 15 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Time taken by A = 3 days, B = 10 days, C = 15 days.

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

⇒ Combined efficiency of (A + B + C) = 10 + 3 + 2 = 15 units/day.

⇒ Original work = 30 units.

⇒ Twice the work = 30 × 2 = 60 units.

⇒ Required time = 60 ÷ 15 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken to complete a task. If time ratio is x : y, then efficiency ratio is y : x.

Pipes and Cisterns

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

Work and Wages

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

Shortcut Trick For any distribution following the Empirical Relationship, if the Mode and Median are equal, the Mean must also be equal to that same value. Given: Mode = 46 and Median = 46. Observation: Since Mode = Median, the distribution is considered symmetrical at this point, leading to Mean = 46. ∴ The correct answer is 46. Alternate Method Given: ⇒ Mode = 46 ⇒ Median = 46 Formula Used: The Empirical Relationship between the three measures of central tendency is: ⇒ Mode = 3 × Median − 2 × Mean

Calculations: ⇒ 46 = 3 × 46 − 2 × Mean ⇒ 46 = 138 − 2 × Mean ⇒ 2 × Mean = 138 − 46 ⇒ 2 × Mean = 92 ⇒ Mean = 92 ÷ 2 ⇒ Mean = 46 ∴ The correct answer is 46. Additional Information Empirical Formula Karl Pearson's formula relates the three averages as: Mode = 3 Median − 2 Mean. It is applicable for moderately skewed distributions. Symmetrical Distribution In a perfectly Symmetrical Distribution (like the Normal Distribution), the values of Mean, Median, and Mode are all identical. Skewed Distributions For Positively Skewed data: Mean > Median > Mode. For Negatively Skewed data: Mode > Median > Mean.

Shortcut Trick

Apply the algebraic identity (a − b)² = a² − 2ab + b².

Square of the first term: (3m)² = 9m².

Square of the second term: (5n)² = 25n².

The middle term is −2 × (3m) × (5n) = −30mn.

Combine all parts: 9m² − 30mn + 25n².

∴ The correct answer is 9m² − 30mn + 25n².

Alternate Method

Given: (3m − 5n)²

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

⇒ Identifying values: a = 3m and b = 5n.

⇒ Expansion: (3m)² − 2 × (3m) × (5n) + (5n)²

⇒ (3 × 3 × m × m) − (2 × 3 × 5 × m × n) + (5 × 5 × n × n)

⇒ 9m² − 30mn + 25n²

⇒ This expression corresponds to Option 4.

∴ The correct answer is 9m² − 30mn + 25n².

Additional Information

Square of a Sum Identity

(a + b)² = a² + 2ab + b². The only difference is the sign of the middle term.

Difference of Two Squares

(a − b)(a + b) = a² − b². This is used for factorization and simplifying products.

Cubic expansion of Difference

(a − b)³ = a³ − 3a²b + 3ab² − b³. Note the alternating signs in the result.

Shortcut Trick

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles:

⇒ ∠STR = (∠P + ∠Q) ÷ 2

⇒ 17° = (∠P + ∠Q) ÷ 2 ⇒ ∠P + ∠Q = 34°

Given ratio ∠P : ∠Q = 8 : 9, the sum of ratio parts is 8 + 9 = 17 units.

Difference between ∠Q and ∠P = 9 − 8 = 1 unit = 34° × (1 ÷ 17) = 2°.

∴ The correct answer is 2°.

Alternate Method

Given:

⇒ In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at point T.

⇒ ∠STR = 17°

⇒ Ratio of ∠P to ∠Q = 8 : 9

Formula Used:

⇒ Sum of interior angles of a quadrilateral = 360°

⇒ Sum of angles in a triangle = 180°

Calculations:

In triangle STR:

⇒ ∠STR + ∠TSR + ∠TRS = 180°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2

⇒ ∠TRS = ∠R ÷ 2

Substituting these values into the triangle angle sum equation:

⇒ 17° + (∠S ÷ 2) + (∠R ÷ 2) = 180°

⇒ (∠S + ∠R) ÷ 2 = 180° − 17°

⇒ (∠S + ∠R) ÷ 2 = 163°

⇒ ∠S + ∠R = 326°

In quadrilateral PQRS, the sum of all interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 326° = 360°

⇒ ∠P + ∠Q = 360° − 326°

⇒ ∠P + ∠Q = 34°

Given that the ratio of ∠P to ∠Q is 8 : 9, let ∠P = 8x and ∠Q = 9x:

⇒ 8x + 9x = 34°

⇒ 17x = 34°

⇒ x = 2°

The difference between the measures of ∠Q and ∠P is:

⇒ ∠Q − ∠P = 9x − 8x = x

⇒ x = 2°

∴ The correct answer is 2°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is always equal to half the sum of the remaining two opposite angles.

Sum of Interior Angles of a Polygon

For any convex polygon with n sides, the sum of all interior angles is given by the formula (n − 2) × 180°.

Shortcut Trick

When distance is constant, Speed is inversely proportional to Time.

Speed Ratio (Usual : New) = 7 : 4 → Time Ratio (Usual : New) = 4 : 7.

Difference in time units = 7 − 4 = 3 units.

Given, 3 units = 12 minutes ⇒ 1 unit = 4 minutes.

Usual Time = 4 units × 4 minutes = 16 minutes.

∴ The correct answer is 16 min.

Alternate Method

Given: New Speed = 4/7 of Normal Speed, Delay = 12 minutes.

Formula Used: Usual Time = [Late time × Numerator] ÷ [Difference between Numerator and Denominator]

⇒ Let Usual Speed be S and Usual Time be T.

⇒ Distance D = S × T

⇒ New Speed = 4/7 S, New Time = T + 12

⇒ D = (4/7 S) × (T + 12)

⇒ S × T = 4/7 S × (T + 12)

⇒ 7T = 4T + 48

⇒ 3T = 48

⇒ T = 16 minutes.

∴ The correct answer is 16 min.

Additional Information

Proportionality Rule

If Speed becomes a/b of its original value, the Time taken becomes b/a of the original time.

Average Speed

Average speed for equal distances at different speeds x and y is given by 2xy ÷ (x + y).

Relative Speed

When two objects move in the same direction, relative speed is the difference of their speeds.

Shortcut Trick

3125 = 5⁵ ⇒ x − y = 5

2187 = 3⁷ ⇒ 3x + y = 7

Solve ⇒ x = 3, y = −2

19x − 18y = 19×3 − 18(−2)

= 57 + 36 = 93

∴ Answer = 93

Alternate Method

Given:

5^{x − y} = 3125

3^{3x + y} = 2187

Formula used:

a^m = aⁿ ⇒ m = n

Calculation:

⇒ 3125 = 5⁵

⇒ x − y = 5 ...(1)

⇒ 2187 = 3⁷

⇒ 3x + y = 7 ...(2)

⇒ Add (1) and (2)

⇒ 4x = 12 ⇒ x = 3

⇒ Substitute in (1)

⇒ 3 − y = 5 ⇒ y = −2

⇒ 19x − 18y = 19×3 − 18(−2)

⇒ = 57 + 36

⇒ = 93

∴ The correct answer is 93.

Additional Information

Exponent Rule

If bases are same, equate powers

Simultaneous Equations

Solve linear equations step-by-step

Substitution Method

Find one variable and substitute

Shortcut Trick

Apply the successive percentage change formula: Net Change = a + b + (a × b)/100

Given: a = +20% (increase) and b = −25% (decrease)

Net Change = 20 − 25 + (20 × (−25))/100

Net Change = −5 − 5 = −10%

∴ The correct answer is 10% decrease.

Alternate Method

Given: First increase = 20%, Second decrease = 25%

Formula Used: Final Price = Initial Price × (100 + %change₁)/100 × (100 + %change₂)/100

⇒ Let the initial price be 100 units.

⇒ Price after 20% increase = 100 + 20% of 100 = 120 units.

⇒ Price after 25% decrease = 120 − 25% of 120

⇒ 120 − (25/100 × 120) = 120 − 30 = 90 units.

⇒ Net change = 100 − 90 = 10 units.

⇒ Net percentage change = (10/100) × 100 = 10%.

∴ The correct answer is 10% decrease.

Additional Information

Successive Percentage Change

When a value is changed by x% and then by y%, the effective change is (x + y + xy/100)%. Use positive for increase and negative for decrease.

Fractional Conversion

20% increase is a multiplier of 6/5 and 25% decrease is a multiplier of 3/4. Net multiplier = 6/5 × 3/4 = 18/20 = 9/10, which indicates a 1/10 or 10% decrease.

Product Stability

If an item price increases by x%, the consumption must decrease by [x/(100+x)] × 100 to keep the expenditure constant.

Shortcut Trick

Expenditure ratio is the product of Price ratio and Quantity ratio.

Expenditure (Jowar : Gram) = (1 × 4) : (5 × 3)

Expenditure (Jowar : Gram) = 4 : 15

∴ The correct answer is 4 : 15.

Alternate Method

Given:

Ratio of Market Prices (Jowar : Gram) = 1 : 5

Ratio of Quantities Consumed (Jowar : Gram) = 4 : 3

Formula Used:

Expenditure = Price × Quantity

Calculations:

⇒ Let the prices be 1x and 5x respectively.

⇒ Let the quantities be 4y and 3y respectively.

⇒ Expenditure on Jowar = 1x × 4y = 4xy

⇒ Expenditure on Gram = 5x × 3y = 15xy

⇒ Ratio of expenditure = 4xy : 15xy

⇒ Ratio = 4 : 15

∴ The correct answer is 4 : 15.

Additional Information

Ratio Multiplication

When two independent variables contribute to a third variable (like Price and Quantity to Expenditure), their respective ratios are multiplied component-wise.

Inverse Relationship

If expenditure is constant, price and quantity are inversely proportional (Price₁/Price₂ = Quantity₂/Quantity₁).

Percentage Change in Expenditure

Net % change in expenditure = a + b + (a × b)/100, where a and b are percentage changes in price and quantity.

Shortcut Trick

Total Cost Price (CP) of 6 notebooks = ₹30.

Total Selling Price (SP) of 6 notebooks = ₹36.

Multiplier for SP = CP × (100 + P%) ÷ 100.

Profit% = (SP − CP) ÷ CP × 100 = (36 − 30) ÷ 30 × 100.

Profit% = 6 ÷ 30 × 100 = 1 ÷ 5 × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: CP of 6 notebooks = ₹30; SP of 6 notebooks = ₹36.

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

⇒ Cost Price (CP) of 1 notebook = 30 ÷ 6 = ₹5.

⇒ Selling Price (SP) of 1 notebook = 36 ÷ 6 = ₹6.

⇒ Profit per notebook = SP − CP = 6 − 5 = ₹1.

⇒ Profit Percentage = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price is greater than Cost Price. It is always calculated on the Cost Price unless specified otherwise.

Unitary Method

To find profit percentage, you can compare the total CP and total SP for the same quantity of items, or find the price per single unit.

Shortcut Trick

Total amount = ₹8,516.

Total deduction = 88 + 31 + 90 = ₹209.

Remaining amount = 8516 − 209 = ₹8,307.

Ratio after deduction = 18 : 17 : 4 (Total parts = 18 + 17 + 4 = 39).

Value of (B − C) ratio difference = (17 − 4) = 13 units.

Difference in original shares = (Difference in ratio parts × Value per unit) + (B deduction − C deduction) = (13 × 8307 ÷ 39) + (31 − 90) = 2769 − 59 = ₹2,710.

∴ The correct answer is ₹2,710.

Alternate Method

Given: Total = ₹8,516 | Deductions: A = ₹88, B = ₹31, C = ₹90 | Ratio (A−88):(B−31):(C−90) = 18:17:4.

Formula: Share = (Ratio Part ÷ Total Parts) × Remaining Sum + Deduction Amount

⇒ Sum after deduction = 8516 − (88 + 31 + 90) = 8516 − 209 = 8307

⇒ Total ratio parts = 18 + 17 + 4 = 39 units

⇒ 39 units = 8307

⇒ 1 unit = 8307 ÷ 39 = 213

⇒ B's share = (17 × 213) + 31 = 3621 + 31 = 3652

⇒ C's share = (4 × 213) + 90 = 852 + 90 = 942

⇒ Difference = 3652 − 942 = 2710

∴ The correct answer is ₹2,710.

Additional Information

Ratio and Proportion Basics

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

Effective Sum Adjustment

When fixed amounts are added or subtracted from shares, always adjust the Total Sum first to align with the given ratio.

Compound Ratio

The compound ratio of a:b and c:d is (a×c):(b×d). This is useful in Partnership and Mixture problems.

Shortcut Trick

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

Given: Old Average = 25 years | New Average = 26 years

Old Count = 43 students | Increase = 26 − 25 = 1 year

⇒ Teacher's Age = 26 + (43 × 1) = 26 + 43 = 69 years

∴ The correct answer is 69 years.

Alternate Method

Given: Number of students = 43, Average age = 25 years, Average with teacher included = 26 years.

Formula Used: Total Sum = Average × Number of Observations

⇒ Total age of 43 students = 43 × 25 = 1075 years

⇒ Total persons (including teacher) = 43 + 1 = 44

⇒ Total age of 44 persons = 44 × 26 = 1144 years

⇒ Teacher's age = (Total age of 44 persons) − (Total age of 43 students)

⇒ Teacher's age = 1144 − 1075 = 69 years

∴ The correct answer is 69 years.

Additional Information

General Inclusion Formula

Value of new entry = (New Number of items × New Average) − (Old Number of items × Old Average).

Net Deviation Method

Sum of deviations from the average is always zero. If the average increases, the new value must be greater than the old average by (Total New Count × Increase).

Average of Groups

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average of each group.

Shortcut Trick

Apply the inverse proportionality rule: Total Work (Man-Days) remains constant.

M₁ × D₁ = M₂ × D₂

1209 × 20 = (1209 + 403) × D₂

1209 × 20 = 1612 × D₂

D₂ = (1209 × 20) ÷ 1612 = 15 days.

∴ The correct answer is 15 days.

Alternate Method

Given: Initial men (M₁) = 1209, Initial days (D₁) = 20, Men added = 403.

Formula Used: M₁D₁ = M₂D₂

⇒ Total food capacity = 1209 × 20 units

⇒ New number of men (M₂) = 1209 + 403 = 1612

⇒ Let the new number of days be D₂.

⇒ 1209 × 20 = 1612 × D₂

⇒ D₂ = (1209 × 20) ÷ 1612

⇒ D₂ = 24180 ÷ 1612

⇒ D₂ = 15

∴ The correct answer is 15 days.

Additional Information

Chain Rule (Inverse Proportion)

When the number of men increases, the number of days for which provisions last decreases proportionally. M₁D₁ = M₂D₂.

Total Work Concept

The product of men and time gives the total volume of work or resources (like food units). Constant Resource = Men × Days.

Man-Hours Formula

If hours per day are also given, use: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Shortcut Trick

Use the multiplying factor method for successive changes.

Net multiplier = (1 − 0.45) × (1 − 0.15) × (1 − 0.50) × (1 + 0.70)

Net multiplier = 0.55 × 0.85 × 0.50 × 1.70 = 0.397375

Decrease % = (1 − 0.397375) × 100 = 60.2625%

∴ The correct answer is Decreases by 60.26%.

Alternate Method

Given:

Initial price of fuel = 100 units

Successive changes: 45% decrease, 15% decrease, 50% decrease, and 70% increase.

Calculations:

⇒ Price after 1^st month = 100 × (55/100) = 55

⇒ Price after 2^nd month = 55 × (85/100) = 46.75

⇒ Price after 3^rd month = 46.75 × (50/100) = 23.375

⇒ Price after 4^th month = 23.375 × (170/100) = 39.7375

⇒ Net Change = Final Price − Original Price = 39.7375 − 100 = −60.2625

⇒ Percentage Change = (−60.2625 / 100) × 100 = −60.2625%

∴ The correct answer is Decreases by 60.26%.

Additional Information

Successive Percentage Change

For two changes x% and y%, the net change is given by the formula: [x + y + (xy/100)]%.

Base Value Change

Percentage increase/decrease is always calculated on the preceding value in a successive series, not the original 100 throughout.

Multiplying Factor

An x% decrease corresponds to a multiplier of (1 − x/100), while an x% increase corresponds to (1 + x/100).

The logic followed here is:

So, the missing term = TRS 74

Hence, the correct answer is Option 4.

The logic followed here is:

Given numbers: 742, 912, 853, 657, 154

Arranging in descending order: 912, 853, 742, 657, 154

Here,

→ 657 remains in the 4^th position

→ 154 remains in the 5^th position

So, total numbers with unchanged positions = 2

Hence, the correct answer is Option 1.

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

Now,

We need to find the total count of prime numbers in the series.

Prime Numbers = 2, 3, 5, 7

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

Total prime numbers = 3 + 3 + 4 + 4 = 14

So, total prime numbers = 14

Hence, the correct answer is Option 4.

The logic followed here is:

thus, "966" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

For, WEAR → EAWR → RAEW

For, SINK → INSK → KNIS

Now, check the options:

Option 1) BITE → BTIE → ETIB

Option 2) YOUR → OYUR → ROUY

Option 3) FANS → ANFS → ANSF

Option 4) MONK → ONMK → KNOM

Hence, the correct answer is Option 4.

Given: Mr. Zzz is 5^th from the right in a row of 63 people. Mr. Ggg is 12^th from the left.

Mr. Zzz’s position from right = 5

Mr. Ggg’s position from left = 12

Total number of people = 63

Number of people between = 63 − (12 + 5)

= 63 − 17

= 46

Hence, the correct answer is Option 4.

The logic followed here is:

Given number: 7593562

Logic:

Add 1 to each even digit

Subtract 2 from each odd digit

Original number: 7 5 9 3 5 6 2

New number: 5 3 7 1 3 7 3

Now, second digit from the left = 3

Hence, the correct answer is Option 3.

The logic followed here is:

Given:

Car A is to the right of Car B

Car D is to the left of Car B

Car F is behind Car D

So,F lies in the South-West direction with respect to B.

Hence, the correct answer is Option 2.

The logic followed here is:

So, the missing term = VLO

Hence, the correct answer is Option 1.