RRB Group D Official Paper (Held On: 10 Feb, 2026 Shift 1) (Previous Year Paper)

The correct answer is Uniform composition.

Key Points

• A mixture consists of two or more different substances that are physically combined but not chemically bonded. The primary way to classify these is based on how the substances are distributed within the sample.
• A homogeneous mixture is characterized by a uniform composition throughout its mass. This means that any sample taken from the mixture will have the exact same proportions of its components. Examples include saltwater or clean air.
• A heterogeneous mixture possesses a non-uniform composition, meaning its components are not evenly distributed. Different regions of the mixture may have different properties and compositions. Examples include a salad or oil and water.
• The term "phase" is often used to distinguish them: a homogeneous mixture consists of a single phase (solid, liquid, or gas), while a heterogeneous mixture consists of two or more phases that remain physically separate.
• Regarding the other options: Chemical reactions are not a property of mixtures, as mixtures are formed through physical changes. A fixed boiling point is a characteristic of pure substances (elements or compounds), not mixtures. High density is a physical property of specific materials but does not determine whether a mixture is homogeneous or heterogeneous.

Additional Information

• Homogeneous Mixtures (Solutions):
• These are often referred to as solutions. A solution consists of a solute (the substance being dissolved) and a solvent (the medium in which it is dissolved).
• In a liquid solution like sugar-water, the sugar particles are so small (typically less than 1 nanometer) that they cannot be seen even with a microscope and do not settle over time.
• Alloys, such as brass (copper and zinc) or steel (iron and carbon), are considered solid homogeneous mixtures.
• Heterogeneous Mixtures (Suspensions and Colloids):
• Suspensions are heterogeneous mixtures where the particles are large enough to be seen and will eventually settle out due to gravity (e.g., sand in water).
• Colloids occupy a middle ground; they may appear homogeneous to the naked eye but are heterogeneous at a microscopic level (e.g., milk or fog). They exhibit the Tyndall Effect, which is the scattering of light by particles.
• Separation Techniques:
• Because mixtures are physical combinations, they can be separated by physical methods.
• Homogeneous mixtures usually require methods that involve phase changes, such as distillation or evaporation, to separate the components.
• Heterogeneous mixtures can often be separated by simpler mechanical means such as filtration, decantation, or using a centrifuge.
• Comparison with Pure Substances:
• A pure substance has constant physical and chemical properties. For instance, distilled water always boils at 100 degrees Celsius at sea level.
• In contrast, mixtures do not have a fixed boiling or melting point; they generally boil or melt over a range of temperatures depending on the concentration of the components.

The correct answer is Salt and water.

Key Points

• A neutralization reaction is a specific type of chemical reaction where an acid and a base react with each other to produce a salt and water.
• When a strong acid (like Hydrochloric acid) reacts with a strong base (like Sodium hydroxide), the hydrogen ions (H+) from the acid and the hydroxide ions (OH-) from the base combine to form neutral water (H2O) molecules.
• The anion (negative ion) from the acid and the cation (positive ion) from the base combine to form an ionic compound called a salt (e.g., Sodium chloride). Hence, statement 4 is correct.
• Neutralization reactions between strong acids and strong bases typically result in a solution with a pH of 7 at 25 degrees Celsius, indicating a neutral solution.
• These reactions are inherently exothermic, meaning they release thermal energy or heat during the process.
• The formation of hydrogen gas typically occurs during the reaction of an acid with a metal, not during a standard acid-base neutralization. Hence, statement 1 is incorrect.
• While water is always formed, it is never the only product; the counter-ions must form a salt. Hence, statement 2 is incorrect.
• The production of carbon dioxide alongside salt and water occurs only when the base is a carbonate or bicarbonate, rather than a simple strong base like a metal hydroxide. Hence, statement 3 is incorrect.

Additional Information

• Strong Acids:
• These are substances that completely ionize in an aqueous solution, meaning they release all their available hydrogen ions.
• Examples include Hydrochloric acid (HCl), Sulfuric acid (H2SO4), and Nitric acid (HNO3).
• They have a very low pH value, usually between 0 and 3.
• Strong Bases:
• These are substances that completely dissociate in water to release hydroxide ions (OH-).
• Common examples include Sodium hydroxide (NaOH), also known as caustic soda, and Potassium hydroxide (KOH).
• They have a high pH value, typically ranging from 12 to 14.
• The pH Scale:
• The pH scale is a logarithmic scale used to measure the acidity or alkalinity of a solution.
• It ranges from 0 to 14, where 7 is neutral, values below 7 are acidic, and values above 7 are basic.
• Titration:
• This is a common laboratory method where a neutralization reaction is used to determine the unknown concentration of an acid or a base.
• An indicator (like phenolphthalein) is often used to signal the equivalence point, where the moles of acid equal the moles of base.

Important Points

• Heat of Neutralization: For any strong acid and strong base reaction, the enthalpy change is approximately constant at -57.3 kJ/mol because the essential reaction is always the formation of water from H+ and OH- ions.
• Applications:
• Antacids: Used to treat acidity in the stomach by neutralizing excess HCl with a weak base like Magnesium hydroxide.
• Soil Treatment: If the soil is too acidic, farmers add quicklime (calcium oxide) or slaked lime (calcium hydroxide) to neutralize it for better crop growth.
• Wastewater Treatment: Industrial effluents containing acids are neutralized with lime before being discharged into water bodies to prevent environmental damage.
• Ionic Equation: The net ionic equation for a strong acid-strong base neutralization is always: H+ (aq) + OH- (aq) → H2O (l).

The correct answer is Four.

Key Points

• Birds and mammals are warm-blooded (homeothermic) animals, meaning they maintain a constant body temperature regardless of the external environment.
• To maintain this internal temperature and support high levels of activity, these animals have high energy needs, which require a very efficient supply of oxygen to the body cells.
• A four-chambered heart consists of two atria and two ventricles, which ensures the complete physical separation of oxygenated blood (rich in oxygen) and deoxygenated blood (rich in carbon dioxide).
• This separation prevents the mixing of blood, allowing for a highly efficient double circulation system where oxygen is delivered to the tissues at a high pressure and rate. Hence, the statement regarding four-chambered hearts is correct.
• The left side of the heart handles oxygenated blood coming from the lungs, while the right side manages deoxygenated blood returning from the body tissues.

Additional Information

• Two-Chambered Heart:
• This type of heart is typically found in fishes.
• It consists of one atrium and one ventricle.
• The blood is pumped to the gills, where it is oxygenated, and then flows directly to the rest of the body. This is known as single circulation.
• Three-Chambered Heart:
• Found in amphibians (like frogs) and most reptiles (like lizards and snakes).
• It consists of two atria and one single ventricle.
• In the single ventricle, there is some mixing of oxygenated and deoxygenated blood, which is less efficient but sufficient for these cold-blooded (poikilothermic) animals with lower metabolic rates.
• Exceptional Cases in Reptiles:
• While most reptiles have three chambers, crocodiles and alligators are notable exceptions as they possess a four-chambered heart, similar to birds and mammals.
• Double Circulation:
• This refers to the blood passing through the heart twice during one complete cycle of the body.
• It includes pulmonary circulation (between heart and lungs) and systemic circulation (between heart and the rest of the body).
• This mechanism is vital for maintaining the high metabolic rate and homeostasis in mammals and birds.

Important Points

• Energy and Temperature: Animals that do not use energy to maintain their body temperature (cold-blooded) can tolerate some mixing of blood, but for homeotherms, the four chambers are an evolutionary necessity.
• Ventricle Evolution: The evolution from a single ventricle (in fish) to a partially divided ventricle (in reptiles) and finally to a completely divided ventricle (in mammals and birds) reflects the increasing demand for metabolic efficiency.
• Left Ventricle: In a four-chambered heart, the left ventricle usually has the thickest muscular wall because it must pump blood to the entire body.

The correct answer is Sieve tube.

Key Points

• Phloem is a complex permanent tissue in vascular plants that functions as the food-conducting complex tissue, transporting organic nutrients (mainly sucrose) from the leaves to other parts of the plant.
• Sieve tubes are the primary conducting elements of the phloem. They are long, tubular structures arranged longitudinally and are characterized by having perforated walls.
• The end walls of sieve tube cells are called sieve plates, which are perforated in a sieve-like manner to allow the continuous flow of food materials between adjacent cells. Hence, statement 1 is correct.
• A mature sieve tube element possesses a peripheral layer of cytoplasm and a large vacuole but lacks a nucleus. This structural adaptation allows for more efficient transport of nutrients.
• The functions of sieve tubes are controlled by the nucleus of companion cells, which are specialized parenchyma cells closely associated with them through pit fields.

Additional Information

• Components of Phloem:
• Companion Cells: These are specialized parenchymatous cells that help maintain the pressure gradient in the sieve tubes. They are living cells with a dense cytoplasm and a distinct nucleus.
• Phloem Parenchyma: This consists of elongated, tapering cylindrical cells that have dense cytoplasm and a nucleus. Their primary function is the storage of food and other substances like resins, latex, and mucilage.
• Phloem Fibers: Also known as bast fibers, these are made of sclerenchymatous cells. They are generally absent in primary phloem but present in secondary phloem. They provide mechanical strength and are dead at maturity.
• Vascular Bundle Context:
• In addition to phloem, plants possess xylem, which is responsible for the conduction of water and minerals from roots to leaves.
• Unlike phloem, which conducts material bidirectionally, xylem typically facilitates unidirectional transport.
• Sieve cells are found in gymnosperms, whereas sieve tubes are characteristic of angiosperms.

Important Points

• The sieve plate is the specific perforated end wall, but the sieve tube is the overall "part" or unit of the tissue that is described as tubular and having these perforated walls.
• Phloem transport is an active process that requires energy in the form of ATP to load sucrose into the sieve tube elements.
• The first formed primary phloem consists of narrow sieve tubes and is referred to as protophloem, whereas the later formed phloem has bigger sieve tubes and is called metaphloem.
• The sieve tube elements are unique because they are living cells that function without a nucleus, a condition known as being enucleated at maturity.

The correct answer is It increases genetic variation within the population.

Key Points

• Sexual reproduction involves the fusion of specialized cells called gametes (sperm and egg) from two different individuals, which combines their DNA to create a new organism.
• The most significant advantage of this process is the generation of genetic variation. Because the offspring receives a unique mix of alleles from both parents, they are genetically distinct from their parents and siblings.
• This variation is essential for the process of evolution. It provides the raw material upon which natural selection acts, allowing populations to adapt to changing environments, diseases, or predators. Hence, the statement "It increases genetic variation within the population" is correct.
• Combining DNA from two individuals allows for genetic recombination. During meiosis, homologous chromosomes exchange segments, further increasing the diversity of the genetic code passed to the next generation.
• While asexual reproduction is more efficient in stable environments, it produces clones. A lack of genetic diversity in asexual populations makes them highly susceptible to being wiped out by a single environmental change or pathogen.

Additional Information

• Reproductive Complexity: Contrary to reducing the need for specialized parts, sexual reproduction actually requires specialized reproductive organs (such as ovaries and testes) and complex hormonal systems to facilitate the production and fusion of gametes. Hence, statement 1 is incorrect.
• DNA Copying Errors: DNA replication is a biological process that naturally involves a small rate of error, leading to mutations. Sexual reproduction does not eliminate these errors; in fact, mutations are the ultimate source of all genetic variation, though sexual reproduction helps reshuffle these mutations. Hence, statement 2 is incorrect.
• Reproductive Speed: Sexual reproduction is generally much slower than asexual reproduction. It requires finding a mate, the energetic cost of producing gametes, and often a lengthy gestation or incubation period. Asexual reproduction (like binary fission in bacteria) is far more rapid. Hence, statement 4 is incorrect.
• The Red Queen Hypothesis: This evolutionary theory suggests that sexual reproduction is maintained because it allows hosts to evolve new genetic defenses rapidly enough to keep pace with the evolution of parasites and pathogens.

Important Points

• Meiosis: This is the key cellular process behind sexual reproduction. It involves two rounds of division to produce haploid cells (containing half the chromosomes of the parent).
• Independent Assortment: During meiosis, the random distribution of maternal and paternal chromosomes into gametes ensures that every gamete is unique, contributing to the genotypic diversity of the offspring.
• Adaptability: In a population with high genetic variation, there is a higher probability that some individuals will possess traits (such as drought resistance or immunity to a specific virus) that allow them to survive a catastrophe that might kill others.
• Muller's Ratchet: This concept explains that asexual populations tend to accumulate deleterious mutations over time because they cannot "purge" them through recombination. Sexual reproduction helps solve this problem by producing some offspring with fewer harmful mutations than their parents.

The correct answer is A proton is positively charged and its mass is taken as one unit.

Key Points

• A proton is a subatomic particle that carries a positive electrical charge of +1e (approximately 1.602 x 10^-19 Coulombs). Hence, statement 2 is correct.
• The mass of a proton is approximately 1.672 x 10^-27 kg. In the context of atomic chemistry, this mass is considered to be 1 unit (roughly 1 atomic mass unit or amu).
• Statement 1 suggests that a proton is neutral and has mass equal to an electron. This is incorrect because neutrons are the particles that are neutral, and a proton is about 1836 times heavier than an electron. Hence, statement 1 is incorrect.
• Statement 3 claims that a proton is negatively charged. This is the characteristic of an electron, not a proton. Hence, statement 3 is incorrect.
• Statement 4 mentions that a proton's mass is 1/2000 of a hydrogen atom. In reality, a hydrogen atom (specifically the protium isotope) consists of exactly one proton and one electron. Since the electron's mass is negligible, the proton's mass is nearly equal to that of a hydrogen atom, not a fraction of it. It is the electron whose mass is approximately 1/1840 (often rounded to 1/2000) of a proton. Hence, statement 4 is incorrect.

Additional Information

• Discovery of Protons:
• In 1886, E. Goldstein discovered the presence of new radiations in a gas discharge which he called canal rays. These rays were positively charged radiations.
• Later, Ernest Rutherford is credited with the discovery of the nucleus and identifying the proton as a fundamental particle in the year 1917 to 1919 through his gold foil experiment and subsequent nitrogen-alpha particle experiments.
• Location in the Atom:
• Protons are located inside the nucleus at the center of the atom.
• Along with neutrons, they are collectively referred to as nucleons.
• Atomic Number (Z):
• The Atomic Number of an element is defined by the total number of protons present in the nucleus of its atom.
• For example, Carbon has an atomic number of 6 because it contains 6 protons.
• Stability:
• The proton is a stable particle, meaning it does not decay into other particles under normal conditions. Its estimated half-life is greater than 10³⁴ years.

Important Points

• Comparison of Subatomic Particles:
• Proton: Positive charge (+1), Mass ~ 1 amu, Location: Inside Nucleus.
• Neutron: Neutral charge (0), Mass ~ 1 amu, Location: Inside Nucleus.
• Electron: Negative charge (-1), Mass ~ 1/1840 amu (negligible), Location: Outside Nucleus in orbits.
• Charge Neutrality: In a neutral atom, the number of protons is exactly equal to the number of electrons, making the net charge of the atom zero.

The correct answer is Square of the current flowing through the resistor.

Key Points

• The physical phenomenon where heat is generated by an electric current passing through a conductor is known as the Heating Effect of Electric Current.
• According to Joule’s Law of Heating, the heat produced in a resistor is directly proportional to the square of the current (I^2) for a given resistance. Hence, statement 1 is correct.
• The law also states that the heat produced is directly proportional to the resistance (R) for a given current. Hence, statement 4 is incorrect in this context because the question specifies a constant resistance, making length a factor already accounted for in the constant value.
• Furthermore, the magnitude of heat is directly proportional to the time (t) for which the electric current flows through the resistor.
• The mathematical representation of this law is H = I^2Rt, where H represents Heat in Joules, I represents Current in Amperes, R represents Resistance in Ohms, and t represents Time in Seconds.
• Heat is not linearly proportional to the current, but rather to its square; thus, if the current is doubled, the heat produced increases by 4 times. Hence, statement 3 is incorrect.

Additional Information

• James Prescott Joule: This law was first deduced by the English physicist James Prescott Joule in the year 1841. His work led to the First Law of Thermodynamics.
• Electric Power: The rate at which electrical energy is dissipated as heat is called Electric Power (P). It is expressed as P = I^2R or P = VI, where V is the potential difference.
• Applications: The heating effect is utilized in various household appliances such as electric heaters, electric irons, geysers, toasters, and electric kettles.
• Filament Lamps: In an incandescent light bulb, the Tungsten filament (having a high melting point of 3380 degrees Celsius) is heated to such a high temperature that it starts glowing and emitting light.
• Electric Fuse: A safety device used in electrical circuits that works on the heating effect of current. It is made of an alloy (like tin-lead) with a low melting point to protect the circuit from short circuits or overloading.

Important Points

• Resistance (R): It is the opposition offered by a material to the flow of electric current. The SI unit is Ohm (represented by Ω). While resistance depends on length and area, the question assumes it to be constant.
• Density of Material: The density of the resistor material does not directly appear in the Joule’s Law formula for heat dissipation. Hence, statement 2 is incorrect.
• Energy Conversion: In a purely resistive circuit, the entire electrical energy consumed is converted into thermal energy (heat).
• Nichrome: A commonly used alloy in heating elements because of its high resistivity and ability to withstand high temperatures without oxidation.
• Numerical Fact: 1 calorie is equal to approximately 4.184 Joules. Sometimes heat in these problems is requested in calories, requiring a division by 4.184.

The correct answer is They produce OH⁻ ions in water.

Key Points

• According to the Arrhenius theory of acids and bases, a base is a substance that dissociates in aqueous solution to form hydroxide ions (OH⁻). This increase in hydroxyl ion concentration is the defining characteristic of an Arrhenius base. Hence, statement 3 is correct.
• Bases have the property of turning red litmus paper blue. The statement that they turn blue litmus red is a characteristic of acids, not bases. Hence, statement 2 is incorrect.
• The physical property of sour taste is associated with acids (such as citric acid in lemons). Bases are known for having a bitter taste and a slippery or soapy feel to the touch. Hence, statement 4 is incorrect.
• While certain strong bases like Sodium Hydroxide (NaOH) can react with specific active metals like Zinc (Zn) or Aluminum (Al) to produce Hydrogen gas (H₂), this is not a general property of all bases. Generally, the reaction with metals to produce hydrogen is a classic identifying feature of acids. Hence, statement 1 is not the primary defining truth for bases in general.
• Common examples of bases in aqueous solutions include Sodium Hydroxide (NaOH), Calcium Hydroxide (Ca(OH)₂), and Potassium Hydroxide (KOH).

Additional Information

• The pH Scale: The strength of a base is measured on the pH scale. Bases have a pH value ranging from above 7 to 14. A pH of 7 is considered neutral (like pure water), while a pH of 14 indicates a very strong alkali.
• Strong vs. Weak Bases:
• Strong Bases: These are compounds that completely ionize or dissociate into their component ions in water. Examples include Group 1 hydroxides like Lithium Hydroxide (LiOH).
• Weak Bases: These do not dissociate completely in aqueous solution. Ammonia (NH₃) is a classic example of a weak base.
• Alkali: A base that is soluble in water is specifically called an alkali. It is important to note that while all alkalis are bases, not all bases are alkalis, as some do not dissolve in water.
• Neutralization Reaction: When a base reacts with an acid, they undergo a neutralization reaction to produce salt and water. For example, HCl + NaOH → NaCl + H₂O.
• Uses of Bases:
• Sodium Hydroxide (NaOH) is used in the manufacture of soap, paper, and synthetic fibers like rayon.
• Calcium Hydroxide (Ca(OH)₂), also known as slaked lime, is used in the manufacture of bleaching powder and for whitewashing.
• Magnesium Hydroxide (Mg(OH)₂) is commonly used as an antacid to neutralize excess stomach acid.

Important Points

• Bronsted-Lowry Definition: Apart from the Arrhenius definition, the Bronsted-Lowry theory defines a base as a proton (H⁺) acceptor.
• Lewis Definition: The Lewis theory provides a broader definition, stating that a base is an electron-pair donor.
• Indicators: Besides litmus, other indicators like Phenolphthalein turn pink in the presence of a base, while Methyl Orange turns yellow.
• Corrosive Nature: Strong bases are highly corrosive and can cause severe chemical burns, similar to strong acids. They are often referred to as caustic (e.g., caustic soda).

The correct answer is To remove impurities from the extracted metal.

Key Points

• Refining is the final stage in the metallurgical process used to obtain a metal of high purity.
• Metals obtained through reduction processes, such as smelting or electrowinning, are typically referred to as crude metals. These crude metals often contain impurities like unreacted oxides, other metals, non-metals (such as silicon or phosphorus), and residual slag. Hence, the statement regarding removing impurities is correct.
• The primary objective of refining is to remove these substances to ensure the metal possesses the desired physical and chemical properties, such as electrical conductivity and malleability.
• Converting a metal into an alloy (Option 2) is a separate process called alloying, which involves mixing the metal with other elements to enhance strength or corrosion resistance, rather than purifying it. Hence, statement 2 is incorrect.
• Reducing reactivity (Option 3) is generally achieved through passivation or coating, not the refining process itself. Hence, statement 3 is incorrect.
• Increasing the melting point (Option 4) is an inherent physical property or changed via alloying, but is not the main goal of purification. Hence, statement 4 is incorrect.

Additional Information

• Common Methods of Refining:
• Distillation: This method is used for volatile metals with low boiling points, such as Zinc (Zn) and Mercury (Hg). The impure metal is evaporated and the pure metal is obtained as condensate.
• Liquation: This is used for metals with a low melting point (like Tin (Sn) or Lead (Pb)) compared to their impurities. The metal is melted on a sloping hearth, allowing the liquid pure metal to flow away from the solid impurities.
• Electrolytic Refining: This is the most widely used method for metals like Copper (Cu), Gold (Au), and Silver (Ag). An anode of impure metal and a cathode of pure metal are placed in an electrolyte solution of the metal salt.
• Zone Refining: This process is based on the principle that impurities are more soluble in the melt than in the solid state of the metal. It is used to produce ultra-pure semiconductors like Germanium (Ge) and Silicon (Si).

Important Points

• Vapour Phase Refining: The metal is converted into a volatile compound and then decomposed to give pure metal.
• Mond Process: Used specifically for the refining of Nickel (Ni). The metal is reacted with Carbon Monoxide to form Nickel Tetracarbonyl, which is then decomposed at higher temperatures.
• Van Arkel Method: Used for refining Zirconium (Zr) and Titanium (Ti) by removing oxygen and nitrogen through the formation of volatile iodides.
• Purity Levels: Modern refining techniques can achieve purity levels exceeding 99.99%, which is critical for the electronics industry and aerospace engineering.
• Environmental Impact: Industrial refining often involves chemical reagents and high energy consumption; thus, modern environmental laws mandate strict waste management for refining by-products.

The correct answer is Newton.

Key Points

• The Newton (N) is the SI unit of weight. In the field of physics, weight is defined as the gravitational force exerted on an object by a large body, such as the Earth.
• Since weight is a type of force, it follows the same units as defined in the International System of Units (SI) for force. One Newton is the amount of force required to accelerate a mass of one kilogram at a rate of one meter per second squared (1 m/s²).
• The relationship between mass and weight is governed by the formula W = m × g, where W is the weight, m is the mass of the object, and g is the acceleration due to gravity.
• On the surface of the Earth, the value of g is approximately 9.8 m/s². Therefore, a mass of 1 kg has a weight of approximately 9.8 Newtons on Earth.
• Weight is a vector quantity because it has both magnitude and a specific direction, which is always directed toward the center of the planet or the source of gravity.

Additional Information

• Kilogram (kg):
• The kilogram is the SI base unit of mass. Unlike weight, mass is an intrinsic property of an object and represents the quantity of matter it contains.
• The mass of an object remains constant regardless of its location in the universe, whereas its weight changes depending on the local gravitational pull.
• Pascal (Pa):
• The Pascal is the SI derived unit of pressure. It is defined as one Newton per square meter (N/m²).
• It is used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. It is named after the French mathematician and physicist Blaise Pascal.
• Joule (J):
• The Joule is the SI unit of energy, work, or amount of heat.
• It is defined as the work done by a force of one Newton acting through a distance of one meter in the direction of the force.

Important Points

• Variability of Weight: Because weight depends on gravity, the weight of an object on the Moon is only about one-sixth (1/6th) of its weight on Earth, even though its mass remains exactly the same.
• Measurement Tools: A spring balance is typically used to measure weight (force), while a beam balance or physical balance is used to measure mass by comparison with known standards.
• Historical Context: The unit Newton was named in 1948 by the General Conference on Weights and Measures (CGPM) to honor Sir Isaac Newton for his contributions to classical mechanics and the Laws of Motion.
• Standard Gravity: For most calculations on Earth, the standard acceleration due to gravity is taken as 9.80665 m/s².

The correct answer is The New Cartesian Sign Convention must be used.

Key Points

• The Mirror Formula is a mathematical relationship between the object distance (u), image distance (v), and focal length (f) of a spherical mirror, expressed as 1/v + 1/u = 1/f.
• When substituting numerical values into this formula, it is mandatory to use the New Cartesian Sign Convention to distinguish between the directions and positions of the object and image. Hence, statement 2 is correct.
• All distances are measured from the Pole (P) of the mirror, which is treated as the origin of the coordinate system, rather than the centre of curvature. Hence, statement 1 is incorrect.
• The focal length (f) is not always positive; it is negative for a concave mirror and positive for a convex mirror according to the convention. Hence, statement 3 is incorrect.
• While it is necessary to maintain consistency in units (e.g., all values in cm or all in mm) to ensure the calculation is correct, it is not a strict prerequisite that they must only be in metres. Hence, statement 4 is incorrect.

Additional Information

• Rules of New Cartesian Sign Convention:
• The object is always placed to the left of the mirror, meaning light is incident from the left side.
• Distances measured in the direction of incident light (to the right of the pole) are taken as positive.
• Distances measured against the direction of incident light (to the left of the pole) are taken as negative.
• Heights measured upward and perpendicular to the principal axis are taken as positive, while those measured downward are negative.
• Concave Mirror:
• Commonly known as a converging mirror.
• The focal length (f) and radius of curvature (R) are always negative because the focus lies in front of the mirror.
• It can form both real and virtual images depending on the object distance (u).
• Convex Mirror:
• Commonly known as a diverging mirror.
• The focal length (f) and radius of curvature (R) are always positive because the focus lies behind the mirror.
• It always forms a virtual, erect, and diminished image.
• Linear Magnification (m):
• It is the ratio of the height of the image to the height of the object, also expressed as m = -v/u.
• A negative sign in magnification indicates a real image, while a positive sign indicates a virtual image.

Important Points

• The relationship between radius of curvature (R) and focal length (f) is given by R = 2f. This is applicable for spherical mirrors with small apertures.
• For a real object, the object distance (u) is always negative because it is placed to the left of the pole.
• The Mirror Formula is applicable to all types of spherical mirrors (concave and convex) for all positions of the object.

The correct answer is Ethanoic acid.

Key Points

• The reaction given is the oxidation of Ethanol (CH3-CH2OH). Ethanol is a primary alcohol characterized by the presence of one hydroxyl (-OH) group attached to a carbon atom that is bonded to only one other carbon.
• Alkaline Potassium Permanganate (KMnO4) and Acidified Potassium Dichromate (K2Cr2O7) are both classified as strong oxidizing agents. They are capable of donating oxygen or removing hydrogen from organic substrates.
• When Ethanol is treated with these reagents and subjected to Heat, it undergoes a two-step oxidation process. It first forms Ethanal (an aldehyde), which is then rapidly oxidized further because of the strength of the reagents.
• The final product of this complete oxidation is Ethanoic acid (CH3COOH). In this transformation, the carbon chain length remains constant; since the starting alcohol has two carbon atoms, the resulting carboxylic acid also has two carbon atoms. Hence, the product is Ethanoic acid.
• The reaction can be summarized as: CH3-CH2OH + 2[O] → CH3COOH + H2O. This chemical change involves the conversion of the primary alcoholic group into a carboxylic acid group.

Additional Information

• Ethanoic Acid (Acetic Acid):
• Ethanoic acid is the second member of the homologous series of carboxylic acids. It has the chemical formula CH3COOH.
• A 5% to 8% solution of Ethanoic acid in water is commonly known as Vinegar, which is widely used as a preservative in pickles and in various culinary applications.
• Pure Ethanoic acid is called Glacial Acetic Acid because its melting point is 290 K, and it often freezes during winter in cold climates, looking like ice.
• Role of Oxidizing Agents:
• KMnO4 is a purple colored compound that acts as a strong oxidant in alkaline medium. During the reaction, the purple color disappears, indicating that Ethanol has been oxidized.
• Acidified K2Cr2O7 is orange in color. When it oxidizes alcohol, the Chromium ions are reduced from Cr(+6) to Cr(+3), causing the solution to turn green.
• Differentiation of Products:
• Ethanal (CH3CHO) is the intermediate product. To obtain Ethanal as the final product, a mild oxidizing agent like PCC (Pyridinium Chlorochromate) would be required to prevent over-oxidation to the acid.
• Propanone (CH3COCH3) is a ketone with three carbons. It is typically formed by the oxidation of a secondary alcohol (like Propan-2-ol), not a primary alcohol like ethanol.
• Propanal (CH3CH2CHO) is an aldehyde with three carbons. It cannot be formed from ethanol because ethanol only contains two carbon atoms.

Important Points

• The reaction is categorized as a Combustion/Oxidation reaction in organic chemistry. Unlike combustion, which breaks the carbon chain to release CO2, controlled oxidation preserves the carbon skeleton.
• Alcohols can be identified through this reaction by observing the color change of the reagents. This is often used as a qualitative test in laboratories.
• Primary alcohols produce carboxylic acids, secondary alcohols produce ketones, and tertiary alcohols generally do not undergo oxidation under these standard conditions because they lack a hydrogen atom on the alpha-carbon.

The correct answer is Produce, transmit, and amplify sound.

Key Points

• Megaphones, horns, and stethoscopes are instruments designed to produce, transmit, and amplify sound signals using the physical principle of multiple reflection of sound.
• In these devices, a tube or a conical shape is used to reflect sound waves repeatedly in such a way that the waves are guided in a specific direction toward the audience or the listener's ear. Hence, the statement regarding their use to produce, transmit, and amplify sound is correct.
• A megaphone or speaking-trumpet consists of a conical opening that prevents the sound energy from spreading out in all directions, effectively amplifying the sound intensity in the forward direction.
• A stethoscope is a medical instrument used by doctors to listen to sounds produced within the body, primarily the heart and lungs. It works by transmitting the sound from the chest piece to the earpieces through multiple reflections inside the connecting rubber or plastic tubes.
• The primary function of these devices is not to block sound or change its pitch, but to ensure that the amplitude and clarity of the sound reaching the observer are significantly increased.

Additional Information

• Reflection of Sound: Just like light, sound reflects off surfaces. The laws of reflection apply here, where the angle of incidence is equal to the angle of reflection, and all three rays lie in the same plane.
• Echoes: An echo is the sensation of sound persisting in our brain for about 0.1 seconds. To hear a distinct echo, the minimum distance of the reflecting surface from the source of sound must be 17.2 meters at a temperature of 22 degrees Celsius.
• Reverberation: This refers to the persistence of sound due to repeated reflections in a large hall or auditorium. Excessive reverberation is usually reduced by using sound-absorbent materials like compressed fiberboards or rough plaster.
• Pitch vs. Loudness: Pitch depends on the frequency of the sound wave, while loudness or amplification depends on the amplitude. The devices mentioned in the question primarily affect the loudness and direction rather than the frequency (pitch).

Important Points

• Stethoscope Invention: The stethoscope was invented in 1816 by the French physician named René Laennec. Its modern version helps in diagnosing conditions of the respiratory and cardiovascular systems.
• Design of Concert Halls: The ceilings of concert halls, conference halls, and cinema halls are often curved so that sound, after reflection, reaches all corners of the hall uniformly.
• Sound Boards: Sometimes, a curved soundboard is placed behind the speaker in a large hall so that the sound, after reflecting from the soundboard, spreads evenly across the width of the hall.
• Applications of Multiple Reflection: Apart from medical and public speaking tools, the principle of multiple reflection is also utilized in ultrasonography and SONAR (Sound Navigation and Ranging) technologies used for underwater communication and detection.

The correct answer is Spirogyra.

Key Points

• Fragmentation is a form of asexual reproduction where a multicellular organism breaks into several pieces, and each piece grows into a new, complete individual.
• Spirogyra, which is a filamentous green alga found in freshwater, commonly reproduces through this method. Upon reaching maturity, the long filament simply breaks into smaller fragments. Hence, statement 1 is correct.
• This process does not involve the fusion of gametes, making it a simple and rapid way for the organism to colonize a habitat.
• In Spirogyra, the body design is relatively simple, allowing these broken fragments to undergo mitotic cell division to form a full-length filament again.
• Unlike regeneration, which is often a defense mechanism or a way to repair damaged parts, fragmentation in these organisms is a standard reproductive strategy.

Additional Information

• Bryophyllum:
• This plant is well-known for vegetative propagation through its leaves.
• Small buds (adventitious buds) develop along the notches of the leaf margins. When these leaves fall on moist soil, the buds develop into independent plantlets. Hence, statement 2 is incorrect.
• Jasmine:
• Jasmine plants are typically propagated through vegetative methods such as layering or stem cutting.
• In layering, a branch of the plant is bent to the ground and covered with soil while still attached to the parent, eventually growing its own roots. Hence, statement 3 is incorrect.
• Orange:
• Orange trees can reproduce sexually through seeds or via artificial vegetative propagation like grafting.
• Grafting is a technique where tissues of plants are joined so as to continue their growth together, commonly used in horticulture to ensure consistent fruit quality. Hence, statement 4 is incorrect.
• Other Modes of Asexual Reproduction:
• Budding: Observed in Yeast and Hydra, where an outgrowth or bud develops into a new individual.
• Fission: Common in unicellular organisms like Amoeba (binary fission) and Plasmodium (multiple fission).
• Spore Formation: Used by fungi like Rhizopus (bread mould) to survive unfavorable conditions and disperse.

Important Points

• Spirogyra belongs to the kingdom Plantae and the group Thallophyta.
• The name Spirogyra is derived from the spiral arrangement of its chloroplasts, which is a distinguishing cellular feature.
• It is often referred to as "water silk" or "pond scum" due to its slimy texture and appearance in stagnant water bodies.
• While fragmentation is the common asexual route, Spirogyra can also undergo sexual reproduction through a process called conjugation, where genetic material is exchanged between two filaments.

The correct answer is Manganese.

Key Points

• Essential elements for plant growth are divided into two categories based on their quantitative requirements: Macronutrients and Micronutrients.
• Micronutrients or trace elements are needed in very small amounts, generally less than 10 mmole kg⁻¹ of dry matter.
• Manganese (Mn) is a key micronutrient absorbed by plants in the form of manganous ions (Mn²⁺). It is essential for the photolysis of water to liberate oxygen during the process of photosynthesis. Hence, statement 1 is correct.
• It also plays a crucial role in activating several enzymes involved in respiration and nitrogen metabolism.
• While Manganese is vital, its toxicity can occur if absorbed in excess, often characterized by the appearance of brown spots surrounded by chlorotic veins.

Additional Information

• Nitrogen (N):
• It is a macronutrient required by plants in the greatest amount. It is absorbed mainly as nitrate (NO₃⁻), though some is taken up as nitrite (NO₂⁻) or ammonium (NH₄⁺).
• It is a major constituent of proteins, nucleic acids, vitamins, and hormones. Hence, statement 2 is incorrect.
• Phosphorous (P):
• It is a macronutrient absorbed by plants from the soil in the form of phosphate ions.
• It is a constituent of cell membranes, certain proteins, all nucleic acids, and nucleotides, and is required for all phosphorylation reactions. Hence, statement 3 is incorrect.
• Magnesium (Mg):
• It is a macronutrient absorbed as divalent Mg²⁺.
• It activates the enzymes of respiration and photosynthesis and is a central constituent of the ring structure of chlorophyll. It also helps in maintaining the ribosome structure. Hence, statement 4 is incorrect.

Important Points

• There are 17 essential elements required for plant growth and metabolism.
• Macronutrients (9): These include Carbon (C), Hydrogen (H), Oxygen (O), Nitrogen (N), Phosphorous (P), Potassium (K), Calcium (Ca), Magnesium (Mg), and Sulphur (S).
• Micronutrients (8): These include Iron (Fe), Manganese (Mn), Copper (Cu), Molybdenum (Mo), Zinc (Zn), Boron (B), Chlorine (Cl), and Nickel (Ni).
• Criteria for Essentiality: An element is considered essential if the plant cannot complete its life cycle without it, and the element is directly involved in the metabolism of the plant.
• Nutrients like Carbon, Hydrogen, and Oxygen are mainly obtained from CO₂ and H₂O, while others are absorbed from the soil as minerals.

The correct answer is the passengers have inertia and continue in their state of motion.

Key Points

• According to Newton’s First Law of Motion, an object will continue in its state of rest or uniform motion in a straight line unless compelled to change that state by an external force applied to it.
• When a car is in motion, the passengers sitting inside it are also moving at the same velocity as the car.
• When the brakes are applied suddenly, the car comes to a halt or slows down due to the frictional force between the tires and the road, but this force is not directly applied to the upper body of the passengers.
• Due to the property of Inertia of Motion, the passengers' bodies tend to maintain their original state of motion (moving forward), while the car (and the lower part of the passenger's body in contact with the seat) slows down.
• The air inside the car does not have enough mass or pressure change to push passengers forward significantly during braking. Hence, statement 1 is incorrect.
• The tendency to move forward is a direct result of the body's inertia resisting the change in velocity. Hence, statement 2 is correct.
• Gravity acts vertically downward and does not pull passengers forward horizontally when brakes are applied. Hence, statement 3 is incorrect.
• Brakes apply a backward force (deceleration) on the car, not a forward force on the passengers. Hence, statement 4 is incorrect.

Additional Information

• Types of Inertia:
• Inertia of Rest: The tendency of a body to remain at rest. For example, when a stationary bus starts suddenly, passengers fall backward because their lower body moves with the bus, but the upper body tries to stay at rest.
• Inertia of Motion: The tendency of a body to remain in uniform motion. The car braking scenario is a classic example of this.
• Inertia of Direction: The tendency of a body to resist a change in its direction of motion. For example, when a car takes a sharp curve, passengers are thrown toward the outside of the curve.
• Mass and Inertia:
• The inertia of an object is measured by its mass. An object with a larger mass (like a truck) has more inertia and is harder to stop or start than an object with a smaller mass (like a bicycle).
• The SI unit of mass is the kilogram (kg).
• Applications in Safety:
• Seatbelts are designed to provide the necessary external force to stop the passenger's body from moving forward due to inertia, preventing them from hitting the windshield or dashboard.
• Airbags provide a cushion to increase the time of impact, thereby reducing the force exerted on the passenger during a sudden stop.

Important Points

• The concept of Inertia was first proposed by Galileo Galilei through his experiments with inclined planes and was later incorporated by Sir Isaac Newton as the First Law of Motion.
• Newton published these laws in his famous book "Philosophiæ Naturalis Principia Mathematica" in the year 1687.
• It is important to note that Inertia is not a force but a property of matter.
• The Net External Force acting on an object is zero if the object is moving at a constant velocity or is at rest.

The correct answer is Add the atomic mass of sodium and the atomic mass of chlorine.

Key Points

• The formula unit mass of a substance is defined as the sum of the atomic masses of all atoms present in a formula unit of a compound.
• In the case of Sodium Chloride (NaCl), the crystal lattice consists of Sodium (Na+) and Chloride (Cl-) ions arranged in a 1:1 ratio.
• To calculate the formula unit mass, one must identify the atomic mass of each constituent element from the Periodic Table and add them together based on their proportions in the chemical formula. Hence, statement 1 is correct.
• Mathematical operations like division, subtraction, or multiplication of atomic masses do not represent the physical reality of combining masses in a chemical compound. Hence, statements 2, 3, and 4 are incorrect.
• For NaCl, the atomic mass of Sodium (Na) is approximately 23.0 u and the atomic mass of Chlorine (Cl) is approximately 35.5 u.
• The calculation is: 23.0 u + 35.5 u = 58.5 u. This total represents the mass of one formula unit of the ionic compound.

Additional Information

• Formula Unit Mass vs. Molecular Mass:
• The term molecular mass is used for substances that exist as discrete molecules (like H2O or CO2).
• The term formula unit mass is specifically used for ionic compounds (like NaCl, CaCl2, or KNO3) because they exist as continuous three-dimensional crystal lattices rather than individual molecules.
• Unified Atomic Mass Unit (u):
• Atomic masses are measured in u, where 1 u is exactly 1/12th of the mass of one atom of Carbon-12.
• This unit was previously known as amu (atomic mass unit).
• Structure of NaCl:
• Sodium Chloride, also known as Common Salt or Halite, crystallizes in a Face-Centered Cubic (FCC) structure.
• In this arrangement, each Na+ ion is surrounded by 6 Cl- ions, and vice versa, maintaining a coordination number of 6.
• Molar Mass:
• When the formula unit mass is expressed in grams per mole (g/mol), it is called the molar mass.
• The molar mass of NaCl is 58.5 g/mol, meaning 6.022 x 10^23 formula units of NaCl weigh 58.5 grams.

Important Points

• Calculating Multiple Atoms: If an ionic compound has a subscript in its formula (e.g., Calcium Chloride, CaCl2), the atomic mass of the element with the subscript must be multiplied by that number before adding.
• For CaCl2: (Mass of Ca) + 2 × (Mass of Cl) = 40.0 u + (2 × 35.5 u) = 111.0 u.
• Law of Conservation of Mass: The calculation of formula mass relies on the principle that the mass of a compound is the sum of the masses of its constituent atoms, as proposed by John Dalton in his Atomic Theory.
• Physical State: NaCl is a solid at room temperature with a high melting point of approximately 801°C (1,474°F) due to strong electrostatic forces between the oppositely charged ions.

The correct answer is N m2/kg2.

Key Points

• Newton's Law of Universal Gravitation states that the force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The mathematical expression for this law is F = G (m1 m2 / r2), where F represents the gravitational force, m1 and m2 are the masses of the two objects, r is the distance between them, and G is the Universal Gravitational Constant.
• To find the SI unit of G, we rearrange the formula: G = F r2 / (m1 m2).
• By substituting the standard units—Newton (N) for force, meter (m) for distance, and kilogram (kg) for mass—we derive the unit as N m2 / kg2. Hence, the correct answer is N m2/kg2.
• The accepted numerical value of the gravitational constant G is approximately 6.674 x 10^-11 N m2/kg2.

Additional Information

• Discovery and Measurement: The value of the Universal Gravitational Constant was first measured with high precision by the British scientist Henry Cavendish in the year 1798 using a torsion balance experiment.
• Dimensional Formula: The dimensions of G are represented as [M-1 L3 T-2], derived from the base units of force, length, and mass.
• Nature of G: It is a scalar quantity, meaning it only has magnitude and does not depend on direction. It is called a "universal" constant because its value is believed to be the same everywhere in the universe.
• Comparison with 'g': It is important to distinguish G from g (acceleration due to gravity). While G is a constant, the value of g (approximately 9.8 m/s2 on Earth) changes depending on the location, such as altitude or the celestial body one is standing on.
• Other Units in Options:
• kg/m3 is the SI unit of density, representing mass per unit volume.
• N/m2 is the SI unit of pressure, also known as the Pascal (Pa).

Important Points

• Independence of Properties: The value of G does not depend on the medium between the two bodies, the temperature, or the chemical composition of the objects.
• Role in Physics: The gravitational constant is a fundamental component in Einstein's Field Equations of General Relativity and is essential for calculating the orbital mechanics of planets, moons, and satellites.
• Weak Force: Gravity is the weakest of the four fundamental forces of nature (the others being electromagnetism, strong nuclear force, and weak nuclear force), which is why the value of G is so small (on the order of 10^-11).
• Calculation Applications: Using the value of G, scientists can determine the mass of the Earth (approx 5.97 x 10^24 kg) and other planets by observing the gravitational influence they exert on other objects.

The correct answer is zero.

Key Points

• Acceleration is defined as the rate of change of velocity with respect to time. In physics, it is a vector quantity that describes how quickly an object's velocity is changing.
• A body is said to be moving with uniform velocity when it covers equal displacements in equal intervals of time, meaning both its magnitude (speed) and direction remain constant over time.
• The mathematical formula for acceleration (a) is given by a = (v - u) / t, where v represents the final velocity, u represents the initial velocity, and t is the time interval.
• In the case of uniform velocity, the initial velocity is exactly equal to the final velocity (u = v). Therefore, the change in velocity (v - u) is 0.
• Because the change in velocity is 0, the result of the acceleration formula becomes 0 / t, which equals zero. Hence, for a body moving with uniform velocity, the acceleration is always zero.

Additional Information

• Velocity vs. Speed: While speed is a scalar quantity (only magnitude), velocity is a vector quantity. A change in either the speed or the direction of motion will result in a change in velocity, thereby producing acceleration.
• Uniform Circular Motion: This is a unique case where an object moves at a constant speed but its velocity is constantly changing because its direction is changing at every point. In this scenario, the acceleration is not zero; it is called centripetal acceleration.
• Graphical Representation:
• In a Velocity-Time (v-t) graph, uniform velocity is represented by a horizontal straight line parallel to the time axis. The slope of a v-t graph represents acceleration; since the slope of a horizontal line is 0, the acceleration is zero.
• In a Position-Time (s-t) graph, uniform velocity is represented by a straight line with a constant slope. The slope of this graph represents the velocity of the object.
• Types of Acceleration:
• Positive Acceleration: Occurs when the velocity of an object increases over time.
• Negative Acceleration (Retardation/Deceleration): Occurs when the velocity of an object decreases over time.
• Zero Acceleration: Occurs when the object is at rest or moving with uniform velocity.

Important Points

• Newton's First Law of Motion: Also known as the Law of Inertia, it states that an object will remain in its state of rest or uniform motion in a straight line unless compelled to change 그 state by an external force. This implies that if acceleration is zero, the net external force acting on the body is also zero.
• Newton's Second Law: Defined by the equation F = ma (Force = mass x acceleration). This relationship confirms that if acceleration (a) is 0, the resultant force (F) must be 0, regardless of the object's mass.
• Units and Dimensions: The S.I. unit of acceleration is m/s² (meters per second squared). Its dimensional formula is [M⁰ L¹ T⁻²].
• Instantaneous Acceleration: This refers to the acceleration of an object at a specific instant of time, calculated by taking the derivative of velocity with respect to time (dv/dt). For uniform velocity, this derivative is always 0.

The correct answer is Oxygen.

Key Points

• Ozone (O3) is a molecule composed of three oxygen atoms. It is formed naturally in the upper atmosphere through a photochemical process.
• The formation of Ozone begins when higher-energy ultraviolet (UV) radiation, specifically UV-C rays from the sun, strike an Oxygen molecule (O2).
• This high-energy radiation has enough energy to break the chemical bond holding the Oxygen molecule together, splitting it into two individual Oxygen atoms (O + O). This process is known as photolysis or photodissociation.
• These freed, highly reactive Oxygen atoms then quickly collide and combine with another intact Oxygen molecule (O2) to form Ozone (O3).
• The presence of a third body (usually Nitrogen or another Oxygen molecule) is often required in the collision to absorb excess energy and stabilize the newly formed Ozone molecule.
• Other options like Zinc, Iron, and Sodium are metallic elements that do not exist as gaseous diatomic molecules in the Earth's atmosphere to participate in this specific UV-driven ozone formation cycle.

Additional Information

• The Chapman Cycle: This is the set of four reactions, first proposed by Sydney Chapman in 1930, that describes how Ozone is continuously regenerated in the Earth's stratosphere.
• Location of Ozone: About 90% of the Earth's ozone is found in the Stratosphere (the atmospheric layer between 10 km and 50 km above the Earth's surface). This region is commonly called the Ozone Layer.
• Functions of the Ozone Layer: It acts as a protective shield by absorbing 97% to 99% of the sun's medium-frequency ultraviolet light (from about 200 nm to 315 nm wavelength), which otherwise would potentially damage exposed life forms on Earth.
• Types of UV Radiation:
• UV-A (315–400 nm): Not absorbed by Ozone; reaches the surface.
• UV-B (280–315 nm): Partially absorbed by Ozone; causes sunburn and skin cancer.
• UV-C (100–280 nm): Completely absorbed by Oxygen and Ozone in the upper atmosphere; it is the most energetic and dangerous type.

Important Points

• Measurement: The concentration of Ozone in the atmosphere is measured in Dobson Units (DU). A typical baseline value is approximately 300 DU.
• Ozone Hole: This refers to the thinning of the Ozone layer, primarily over Antarctica, caused by Ozone-Depleting Substances (ODS) like Chlorofluorocarbons (CFCs), halons, and carbon tetrachloride.
• Montreal Protocol: An international treaty signed in 1987 (effective from 1989) designed to protect the Ozone layer by phasing out the production of numerous substances that are responsible for ozone depletion.
• World Ozone Day: Observed every year on September 16th to commemorate the date of the signing of the Montreal Protocol.
• Ground-level Ozone: While stratospheric ozone is "good" for protection, tropospheric ozone (ground-level) is a "bad" pollutant and a key constituent of smog, formed by the reaction of NOx and VOCs in the presence of sunlight.

p data-end="57" data-start="0">The correct answer is A ray perpendicular to the principal axis that reflects through the focus.

 

Key Points

• To locate the image formed by a spherical mirror, certain specific rays are used because their paths after reflection are predictable based on the laws of reflection.
• Ray 1 (Parallel Ray): A ray that is parallel to the principal axis will, after reflection, pass through the principal focus in a concave mirror or appear to diverge from the focus in a convex mirror. Hence, statement 4 is correct.
• Ray 2 (Ray through Focus): A ray passing through the principal focus of a concave mirror (or directed towards the focus of a convex mirror) will emerge parallel to the principal axis after reflection. Hence, statement 1 is correct.
• Ray 3 (Ray through Center of Curvature): A ray passing through the center of curvature (C) of a spherical mirror reflects back along the same path. This happens because the ray strikes the mirror surface normally (at an angle of incidence of 0 degrees). Hence, statement 2 is correct.
• Incorrect Ray: A ray perpendicular to the principal axis is a vertical ray that does not follow the standard geometric path to reflect through the focus. For a ray to reflect through the focus, it must be parallel to the principal axis, not perpendicular. Hence, statement 3 is incorrect and is the ray NOT used.
• Hence, the correct answer is A ray perpendicular to the principal axis that reflects through the focus.

Additional Information

• Components of Spherical Mirrors:
• Pole (P): The center of the reflecting surface of a spherical mirror is called the pole. It lies on the surface of the mirror.
• Center of Curvature (C): The reflecting surface forms a part of a sphere. The center of this sphere is the center of curvature. For a concave mirror, it lies in front, while for a convex mirror, it lies behind the mirror.
• Principal Axis: An imaginary straight line passing through the pole and the center of curvature.
• Focal Length (f): The distance between the pole and the principal focus. It is exactly half of the radius of curvature (R), represented as f = R/2.
• Types of Spherical Mirrors:
• Concave Mirror: Also known as a converging mirror. It can form both real and virtual images depending on the object's position. Used in shaving mirrors, headlights, and by dentists.
• Convex Mirror: Also known as a diverging mirror. It always forms a virtual, erect, and diminished image. Used as rear-view mirrors in vehicles to provide a wider field of view.
• Sign Convention:
• According to the New Cartesian Sign Convention, the object is always placed to the left of the mirror. Distances measured in the direction of incident light are positive, and those opposite are negative.
• The focal length of a concave mirror is always negative, whereas the focal length of a convex mirror is always positive.

Important Points

• Laws of Reflection:
• The angle of incidence is always equal to the angle of reflection (i = r).
• The incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.
• Image Formation:
• To construct a ray diagram, at least two rays whose paths are known after reflection are required. The intersection of these reflected rays gives the position of the image of the point object.
• If the rays actually meet, the image is real and inverted. If they appear to meet when produced backwards, the image is virtual and erect.

The correct answer is Presence of appropriate environmental conditions.

Key Points

• Seed Germination is the biological process by which a plant grows from a seed after a period of dormancy. For this process to begin, the internal embryo must be activated by external triggers.
• The Presence of appropriate environmental conditions is the most critical requirement. These conditions primarily include moisture (water), oxygen, and a suitable temperature. Hence, the correct answer is option 2.
• Physical separation from the parent flower or plant is part of the seed dispersal mechanism, which helps reduce competition for resources. However, separation alone does not guarantee the biological start of growth if the soil or climate is unfavorable. Hence, statement 1 is incorrect.
• While Sunlight is essential for photosynthesis once the seedling develops leaves, many seeds actually germinate best in darkness (under the soil). Only specific "light-sensitive" seeds require exposure to light to break dormancy. Using the word "only" makes the statement scientifically inaccurate. Hence, statement 3 is incorrect.
• The decay of the fruit may help release the seed or provide some initial organic matter, but it is not a standalone essential condition for the physiological process of germination. Many seeds can germinate even if the fruit is manually removed or eaten by animals. Hence, statement 4 is incorrect.

Additional Information

• Essential Factors for Germination:
• Water: Germination begins with imbibition, where the seed rapidly absorbs water. This swells the seed, ruptures the seed coat, and activates hydrolytic enzymes that break down stored food.
• Oxygen: Early germination requires aerobic respiration. The seed needs energy to power cell division and growth, which it obtains by oxidizing stored nutrients using oxygen found in soil pores.
• Temperature: Every seed species has an optimum temperature range for germination, typically between 25°C and 30°C for many tropical plants. Extreme cold or heat can inhibit enzymatic activity.
• Seed Structure:
• The Radicle is the first part of the embryo to emerge, developing into the primary root to anchor the plant and absorb water.
• The Plumule is the embryonic shoot that eventually grows above the ground to form the stem and leaves.
• Cotyledons or the Endosperm serve as the primary food storage units that nourish the growing seedling until it can perform photosynthesis.

Important Points

• Seed Dormancy: This is a state where seeds do not germinate even when environmental conditions are favorable. It is a survival strategy to ensure seeds germinate only during the correct season.
• Scarification: This involves breaking or softening the seed coat (mechanically or via acid in animal guts) to allow water and oxygen to enter.
• Stratification: Some seeds require a period of cold treatment (simulating winter) before they are physiologically ready to germinate in the spring.
• Epigeal vs. Hypogeal Germination: In Epigeal germination, the cotyledons are pushed above the ground (e.g., bean seeds), whereas in Hypogeal germination, the cotyledons remain below the soil (e.g., pea seeds or maize).

The correct answer is Artery.

Key Points

• Arteries are the blood vessels responsible for carrying blood away from the heart to various tissues and organs throughout the body.
• Because the heart pumps blood into the arteries with significant force, the blood inside them flows under high pressure.
• To accommodate this high pressure, arteries have thick, muscular, and elastic walls. These walls consist of three distinct layers: the tunica intima, the tunica media (which is notably thick in arteries), and the tunica externa.
• The elasticity of the arterial walls allows them to expand during systole (heart contraction) and recoil during diastole (heart relaxation), which helps in maintaining a continuous flow of blood even when the heart is not beating.
• Unlike veins, most arteries do not have valves (except for the semilunar valves at the base of the aorta and the pulmonary artery) because the high pressure of the blood prevents it from flowing backward.

Additional Information

• Veins:
• Veins carry blood toward the heart from various parts of the body.
• Since the blood in veins is no longer under the direct pumping pressure of the heart, it flows at a lower pressure compared to arteries.
• Consequently, veins have thinner walls and a larger lumen (internal space). They are equipped with valves to ensure blood flows in only one direction, preventing backflow due to gravity.
• Capillaries:
• Capillaries are the smallest and thinnest blood vessels, connecting arterioles (small arteries) and venules (small veins).
• Their walls are extremely thin, usually only one cell thick (endothelium), which allows for the diffusion and exchange of oxygen, carbon dioxide, nutrients, and waste products between the blood and the surrounding interstitial fluid.
• Venules:
• Venules are very small blood vessels in the microcirculation that allow blood to return from capillary beds to drain into the larger veins.
• Pulmonary Exceptions:
• While most arteries carry oxygenated blood, the Pulmonary Artery is the exception, carrying deoxygenated blood from the right ventricle to the lungs.
• Similarly, the Pulmonary Vein carries oxygenated blood from the lungs back to the left atrium of the heart.

Important Points

• The Aorta is the largest artery in the human body, originating from the left ventricle and distributing blood to the entire systemic circulation.
• The elasticity of arteries decreases with age or due to certain health conditions like atherosclerosis, which can lead to increased blood pressure (hypertension).
• In a healthy adult, the typical systolic blood pressure measured in the arteries is about 120 mmHg, while the diastolic pressure is about 80 mmHg.
• The smooth muscle in the tunica media of arteries allows them to perform vasoconstriction and vasodilation, which are essential for regulating blood flow and body temperature.

The correct answer is R/3.

Key Points

• The resistance of a conductor is the property by virtue of which it opposes the flow of electric current through it. It is mathematically defined by the formula R = P × (L / A), where R is the resistance, P is the resistivity of the material, L is the length of the wire, and A is the area of cross-section.
• According to the formula, resistance (R) is directly proportional to the length (L) of the wire and inversely proportional to the area of cross-section (A). This means if the area increases, the resistance decreases, provided the length and material (resistivity) remain unchanged.
• In this specific case, the initial resistance is R = P(L / A). The problem states that the area of cross-section is tripled, meaning the new area A′ = 3A, while the length (L) and resistivity (P) are kept the same.
• Substituting the new area into the formula, the new resistance R′ = P × (L / 3A). This can be rewritten as R′ = (1/3) × [P × (L / A)]. Since the term in the brackets is the original resistance R, the new resistance becomes R′ = R/3.
• Therefore, when the area of cross-section of a wire is increased by three times, its resistance is reduced to one-third of its original value.

Additional Information

• Ohm’s Law: Discovered by Georg Simon Ohm in 1827, it states that the current passing through a conductor is directly proportional to the potential difference (V) across its ends, provided temperature remains constant (V = IR). The SI unit of resistance is the Ohm (Ω).
• Resistivity (P): Also known as specific resistance, it is a characteristic property of the material of the wire. Unlike resistance, it does not change with the length or area of the wire but depends on the nature of the material and its temperature. Its SI unit is Ohm-meter (Ω·m).
• Effect of Length: If the length (L) of the wire were doubled while keeping the area constant, the resistance would also double (2R) because of the direct proportionality.
• Conductivity: This is the reciprocal of resistivity and measures a material's ability to conduct electricity. Materials like Silver and Copper have very low resistivity and high conductivity, making them excellent choices for electrical wiring.

Important Points

• Temperature Factor: For most metallic conductors, the resistance increases with an increase in temperature. This is due to the increased vibration of atoms, which causes more frequent collisions with the flowing electrons.
• Volume Conservation: It is important to note that the question specifies "keep all other parameters the same." If a wire is stretched to triple its area (which is physically impossible without changing length), the volume (V = A × L) would remain constant. However, in this theoretical problem, only area is modified while length is held constant, leading directly to the R/3 result.
• Practical Application: Thicker wires (greater area A) are used in high-power appliances because they offer lower resistance, which minimizes heat loss (calculated by H = I²Rt) and allows higher current (I) to flow safely.

The correct answer is 500 J.

Key Points

• The work done by an object when it is lifted against the force of gravity is calculated using the formula for gravitational potential energy, which is W = m * g * h.
• In this formula, m represents the mass of the object, g represents the acceleration due to gravity, and h represents the height or displacement.
• According to the question, the mass (m) of the luggage is 25 kg and the height (h) to which it is lifted is 2.0 m.
• The value for acceleration due to gravity (g) is provided as 10 m/s².
• By substituting the values into the formula: Work Done = 25 kg * 10 m/s² * 2.0 m.
• The calculation results in 500 Joules (J). Hence, the correct answer is 500 J.

Additional Information

• Definition of Work:
• In physics, work is defined as the product of the force applied to an object and the displacement caused by that force in the direction of the force.
• The mathematical expression is W = F * d * cos(θ), where θ is the angle between the force and the displacement vector.
• When lifting an object vertically, the angle θ is 0 degrees, and since cos(0) = 1, the work is simply Force * Displacement.
• Energy Transformation:
• The work done by the porter is not "lost" but is transferred to the luggage in the form of Potential Energy.
• Potential Energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
• The specific type of energy here is Gravitational Potential Energy, which increases as the height of the object increases from the ground.
• Units and Measurements:
• The SI unit of work and energy is the Joule (J).
• 1 Joule is equal to the work done by a force of 1 Newton acting through a distance of 1 meter.
• In base units, 1 J = 1 kg·m²/s².

Important Points

• Scalar Nature: Work is a scalar quantity, meaning it has only magnitude and no specific direction, unlike force or velocity.
• Zero Work Condition: If the porter were to carry the luggage and walk on a horizontal platform, the work done by the force of gravity would be zero because the force (downward) and displacement (horizontal) are perpendicular (90 degrees).
• Acceleration due to Gravity: While the standard value of g on Earth is approximately 9.8 m/s², it is common in competitive exams to use 10 m/s² for ease of calculation.
• Mass vs Weight: It is important to distinguish between mass (25 kg) and weight. Weight is a force calculated as mass * gravity, which in this case is 250 Newtons. Work is then Weight * Height.

Shortcut Trick

Income is always 100%.

Savings % = 100% − Spending % = 100% − 50% = 50%.

Given Savings = ₹42,000 (unit value).

50% → ₹42,000

100% → ₹42,000 × 2 = ₹84,000

∴ The correct answer is ₹84,000.

Alternate Method

Given:

Spending = 50% of Income

Savings = ₹42,000

Formula Used:

Income = Expenditure + Savings

Income = Savings / (1 − Spending Fraction)

Let Total Income be x.

⇒ Expenditure = 50% of x = 0.5x

⇒ Savings = x − 0.5x = 0.5x

⇒ 0.5x = 42,000

⇒ x = 42,000 / 0.5

⇒ x = 42,000 × 2

⇒ x = 84,000

∴ The correct answer is ₹84,000.

Additional Information

Income-Expenditure Relationship

Income is always the sum of Expenditure and Savings. Ratio of Income : Expenditure : Savings is used for complex calculations.

Percentage Change in Income

If Income increases by a% and Expenditure increases by b%, the change in Savings depends on the initial ratio of E:S.

Fractional Conversions

50% = 1/2 | 25% = 1/4 | 33.33% = 1/3 | 20% = 1/5. Using fractions simplifies division steps.

Shortcut Trick

Arrange data: 6, 8, 9, 11, 16, 22

Total terms (n) = 6 (Even number)

Median = Average of the two middle terms

Middle terms are 9 and 11

Median = (9 + 11) ÷ 2 = 10

∴ The correct answer is 10.

Alternate Method

Given: Data set = 9, 6, 8, 11, 16, 22

Formula Used:

If n is even, Median = [ (n/2)^th term + (n/2 + 1)^th term ] ÷ 2

⇒ Step 1: Arrange data in ascending order: 6, 8, 9, 11, 16, 22

⇒ Step 2: Count the number of observations (n) = 6

⇒ Step 3: Identify the positions of the middle terms

⇒ Position 1 = n/2 = 6/2 = 3^rd term

⇒ Position 2 = (n/2) + 1 = 3 + 1 = 4^th term

⇒ Step 4: Value of 3^rd term = 9 and 4^th term = 11

⇒ Median = (9 + 11) ÷ 2

⇒ Median = 20 ÷ 2 = 10

∴ The correct answer is 10.

Additional Information

Median for Odd Number of Terms

If the total number of terms (n) is odd, the median is the middle-most term located at the ((n + 1)/2)^th position after sorting.

Arithmetic Mean

Mean is the sum of all values divided by the number of values. Formula: Σx ÷ n.

Mode of Data

Mode is the value that appears most frequently in the data set. A set can be unimodal, bimodal, or multimodal.

Relationship between Mean, Median, and Mode

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Shortcut Trick

Divisibility rule of 4: The last two digits of the number must be divisible by 4.

The last two digits of 322A4828 are 28.

Since 28 ÷ 4 = 7 (Remainder = 0), the number is already divisible by 4.

The value of A (at the ten-thousands place) does not affect divisibility by 4.

The minimum possible digit that can be assigned to A is 0.

∴ The correct answer is 0.

Alternate Method

Given number: 322A4828

Divisibility Rule of 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. This is because any number N can be written as (100 × k) + (last two digits), and 100 is always divisible by 4.

⇒ Number formed by last two digits = 28 ⇒ 28 ÷ 4 = 7 ⇒ Since the remainder is 0, the number 322A4828 is divisible by 4 for any digit value of A. ⇒ Possible values for A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ⇒ The question asks for the minimum value of A. ⇒ Minimum value = 0

∴ The correct answer is 0.

Additional Information

Divisibility by 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Divisibility by 3 and 9

A number is divisible by 3 (or 9) if the sum of all its digits is divisible by 3 (or 9).

Divisibility by 11

The difference between the sum of digits at odd places and the sum of digits at even places must be 0 or a multiple of 11.

Shortcut Trick

Speed of first lap (S₁) = 450 / 150 = 3 m/s.

Speed of second lap (S₂) = 450 / 75 = 6 m/s.

When distances are equal, Average Speed = (2 × S₁ × S₂) / (S₁ + S₂).

Average Speed = (2 × 3 × 6) / (3 + 6) = 36 / 9 = 4 m/s.

∴ The correct answer is 4.

Alternate Method

Given:

Distance of Lap 1 (D₁) = 450 m

Time for Lap 1 (T₁) = 150 sec

Distance of Lap 2 (D₂) = 450 m

Time for Lap 2 (T₂) = 75 sec

Formula Used:

Average Speed = Total Distance ÷ Total Time

⇒ Total Distance = D₁ + D₂ = 450 + 450 = 900 m

⇒ Total Time = T₁ + T₂ = 150 + 75 = 225 sec

⇒ Average Speed = 900 ÷ 225

⇒ Average Speed = 4 m/sec

∴ The correct answer is 4.

Additional Information

Average Speed (Equal Distances)

If a body travels two equal distances with speeds x and y, the average speed is given by (2xy) / (x + y).

Average Speed (Equal Time Intervals)

If a body travels for two equal time intervals with speeds x and y, the average speed is (x + y) / 2.

Unit Conversion

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

Shortcut Trick

Use the complementary angle identity: cos² θ + cos² (90° − θ) = 1.

Here, 30° + 60° = 90°, so cos² 30° + cos² 60° = 1.

The value of cos 90° is 0, so cos² 90° = 0.

Total sum = 1 + 0 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

Expression = cos² 30° + cos² 60° + cos² 90°

Formula Used:

cos θ = Adjacent / Hypotenuse

Calculations:

⇒ cos 30° = √3 / 2

⇒ cos 60° = 1 / 2

⇒ cos 90° = 0

Substitute these values into the expression:

⇒ (√3 / 2)² + (1 / 2)² + (0)²

⇒ (3 / 4) + (1 / 4) + 0

⇒ 4 / 4

⇒ 1

∴ The correct answer is 1.

Additional Information

Trigonometric Identity

The fundamental identity states that sin²θ + cos²θ = 1 for any angle θ.

Complementary Angle Rule

For complementary angles A and B (where A + B = 90°), sin A = cos B and cos A = sin B.

Standard Cosine Values

Standard values include cos 0° = 1, cos 45° = 1/√2, and cos 90° = 0.

Shortcut Trick

Total Surface Area (TSA) = Curved Surface Area (CSA) + 2 × Base Area

Base Area = πr² = (22/7) × 14 × 14 = 616 m²

2 × Base Area = 2 × 616 = 1232 m²

CSA = TSA − 2 × Base Area = 3080 − 1232 = 1848 m²

∴ The correct answer is 1848 m².

Alternate Method

Given:

Total Surface Area (TSA) = 3080 m²

Radius (r) = 14 m

Formula Used:

TSA of cylinder = 2πr(h + r)

Curved Surface Area (CSA) of cylinder = 2πrh

Calculations:

⇒ 3080 = 2 × (22/7) × 14 × (h + 14)

⇒ 3080 = 2 × 22 × 2 × (h + 14)

⇒ 3080 = 88 × (h + 14)

⇒ h + 14 = 3080 ÷ 88

⇒ h + 14 = 35

⇒ h = 35 − 14 = 21 m

⇒ CSA = 2 × (22/7) × 14 × 21

⇒ CSA = 2 × 22 × 2 × 21

⇒ CSA = 88 × 21

⇒ CSA = 1848 m²

∴ The correct answer is 1848 m².

Additional Information

Volume of Cylinder

The space occupied by a cylinder is given by Volume = πr²h.

Hollow Cylinder

For a hollow cylinder with outer radius R and inner radius r, TSA = 2πh(R + r) + 2π(R² − r²).

Base Perimeter

The perimeter or circumference of the base of a cylinder is 2πr, which when multiplied by height gives the CSA.

Shortcut Trick

SP = (100 − Discount%)% of MP

SP = (100 − 12)% of 775

SP = 88% of 775 = 0.88 × 775

SP = 682

∴ The correct answer is ₹682.

Alternate Method

Given: Marked Price (MP) = ₹775 | Discount % = 12%

Formula: Selling Price (SP) = Marked Price − Discount

⇒ Discount amount = (Discount% / 100) × MP

⇒ Discount amount = (12 / 100) × 775

⇒ Discount amount = 0.12 × 775 = 93

⇒ SP = MP − Discount amount

⇒ SP = 775 − 93 = 682

∴ The correct answer is ₹682.

Additional Information

Discount Percentage

Discount % is always calculated on the Marked Price (MP) of an article: Discount % = (Discount / MP) × 100.

Successive Discounts

If two successive discounts of a% and b% are given, the net discount is [a + b − (a × b)/100]%.

Relationship between MP and CP

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

Shortcut Trick

Rate 15% = 3/20 ⇒ Ratio of P : A for 1 year = 20 : 23

For 2 years, Ratio = 20² : 23² = 400 : 529

Given Principal 400 units = ₹6,400

1 unit = 6,400 ÷ 400 = 16

Amount = 529 units × 16 = ₹8,464

∴ The correct answer is ₹8,464.

Alternate Method

Given: Principal (P) = ₹6,400, Rate (R) = 15%, Time (n) = 2 years

Formula Used: Amount (A) = P(1 + R/100)ⁿ

⇒ A = 6400 × (1 + 15/100)²

⇒ A = 6400 × (115/100) × (115/100)

⇒ A = 6400 × (23/20) × (23/20)

⇒ A = 6400 × (529/400)

⇒ A = 16 × 529

⇒ A = 8464

∴ The correct answer is ₹8,464.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate of interest can be calculated as x + y + (x × y)/100. Here, 15 + 15 + (15 × 15)/100 = 32.25%.

Difference between CI and SI

The difference between CI and SI for 2 years is P(R/100)². In this case, 6400 × (15/100)² = ₹144.

Compound Interest Formula

The interest alone can be calculated using CI = P[(1 + R/100)ⁿ − 1].

Shortcut Trick

Ratio of Mother : Daughter (Present) = 2 : 1 (Difference = 1 unit)

Ratio of Mother : Daughter (10 yrs ago) = 3 : 1 (Difference = 2 units)

Make differences equal: Present Ratio = (2 : 1) × 2 = 4 : 2

Unit difference (4 → 3 or 2 → 1) is 1 unit = 10 years.

Mother's present age = 4 units = 4 × 10 = 40 years.

∴ The correct answer is 40 years.

Alternate Method

Given:

Present age Mother = 2 × Daughter's age

10 years ago Mother's age = 3 × (Daughter's age − 10)

Formula Used:

Linear Equation based on Age conditions.

Let the present age of the Daughter be x years.

⇒ Present age of the Mother = 2x years.

Age 10 years ago:

⇒ Daughter's age = (x − 10)

⇒ Mother's age = (2x − 10)

According to the condition:

⇒ (2x − 10) = 3 × (x − 10)

⇒ 2x − 10 = 3x − 30

⇒ 3x − 2x = 30 − 10

⇒ x = 20

Present age of the Mother:

⇒ Mother = 2x = 2 × 20

⇒ Mother = 40 years.

∴ The correct answer is 40 years.

Additional Information

Constant Age Difference

The difference between the ages of two persons remains constant throughout their lives. Always use this to balance ratios.

Ratio Balancing Method

If the age difference in units is not the same, multiply the ratios by the opposite difference values to make the gap equal.

Averages of Ages

If the average age of n people is A, after t years, the new average age will be (A + t).

Shortcut Trick

Remaining distance fraction = 1 − 2/3 = 1/3 of 228 km = 76 km

Remaining time fraction = 1 − 5/8 = 3/8 of 8 hours = 3 hours

Required speed = Remaining distance ÷ Remaining time

Speed = 76 ÷ 3 = 25.33... ≈ 25.3 km/hr

∴ The correct answer is 25.3.

Alternate Method

Given: Total Distance = 228 km, Total Time = 8 hours

Formula: Speed = Distance ÷ Time

⇒ Total Distance = 228 km

⇒ Distance covered = (2/3) × 228 = 152 km

⇒ Remaining Distance = 228 − 152 = 76 km

⇒ Total Time = 8 hours

⇒ Time spent = (5/8) × 8 = 5 hours

⇒ Remaining Time = 8 − 5 = 3 hours

⇒ Required Speed = 76 ÷ 3

⇒ Required Speed = 25.333...

∴ The correct answer is 25.3.

Additional Information

Average Speed

Average speed is defined as the Total Distance covered divided by the Total Time taken. Average Speed = Total Distance ÷ Total Time.

Unit Conversion

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

Constant Distance Relation

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

Shortcut Trick

Let the total enrolled voters be x.

Successful candidates votes = x × (Cast%) × (Valid%) × (Candidate's share%)

⇒ 13572 = x × 0.78 × 0.96 × 0.58

⇒ x = 13572 ÷ (0.78 × 0.96 × 0.58) = 31,250

∴ The correct answer is 31,250.

Alternate Method

Given:

Voters cast = 78%

Invalid votes = 4% (So, Valid votes = 96% of cast votes)

Candidate's share = 58% of valid votes = 13,572

Formula:

Total Voters × (Cast/100) × (Valid/100) × (Share/100) = Candidate Votes

⇒ Let total enrolled voters = x

⇒ x × (78/100) × (96/100) × (58/100) = 13572

⇒ x × (78 × 96 × 58) / 1000000 = 13572

⇒ x = (13572 × 1000000) / (78 × 96 × 58)

⇒ Dividing 13572 by 58: 13572 ÷ 58 = 234

⇒ x = (234 × 1000000) / (78 × 96)

⇒ Dividing 234 by 78: 234 ÷ 78 = 3

⇒ x = (3 × 1000000) / 96

⇒ x = 1000000 / 32

⇒ x = 31250

∴ The correct answer is 31,250.

Additional Information

Election-based Percentages

Valid Votes = Total Cast Votes − Invalid Votes. The winner and loser share is always calculated from the valid votes unless stated otherwise.

Successive Percentage Application

When multiple reductions or shares are applied in sequence, use the chain rule: Final Value = Initial Value × a% × b% × c%.

Majority Margin

Margin of victory = (Votes of Winner − Votes of Loser). Usually expressed as a percentage of the total valid votes.

Shortcut Trick

Cost Price for the collector = ₹1,440

Profit = 25% of 1,440 = 1/4 × 1,440 = ₹360

Auction Selling Price = 1,440 + 360 = ₹1,800

Alternatively: SP = 1,440 × 125% = 1,440 × 5/4 = ₹1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given:

Price at which dealer sells to collector (CP for collector) = ₹1,440

Profit percentage at auction = 25%

Formula used: Selling Price (SP) = CP × (1 + Profit%/100)

⇒ Profit = 25% of ₹1,440

⇒ Profit = (25/100) × 1440

⇒ Profit = 1/4 × 1440 = 360

⇒ Final Price = Cost Price + Profit

⇒ Final Price = 1440 + 360

⇒ Final Price = 1800

∴ The correct answer is ₹1,800.

Additional Information

Profit and Loss Basics

Profit = Selling Price − Cost Price. Profit percentage is always calculated on the CP: (Profit/CP) × 100.

Successive Transactions

When an item passes through multiple owners, the SP for one becomes the CP for the next. The final price is CP × (1 + p₁/100) × (1 + p₂/100)...

Fractional Equivalents

Common profit percentages: 20% = 1/5, 25% = 1/4, 33.33% = 1/3, 50% = 1/2. Use (1 + fraction) as a multiplier for SP.

Shortcut Trick

Sum of exterior angles is always 360°.

Given ratio (Exterior : Interior) = 2 : 43

2 units = 360° ⇒ 1 unit = 180°

Sum of interior angles = 43 × 180°

Since Sum = (n − 2) × 180°, we have n − 2 = 43 ⇒ n = 45

Measure of interior angle = (43 × 180°) / 45 = 43 × 4 = 172°

∴ The correct answer is 172°.

Alternate Method

Given:

Ratio of sum of exterior angles to sum of interior angles = 2 : 43

Formula Used:

Sum of exterior angles of a polygon = 360°

Sum of interior angles of a polygon = (n − 2) × 180°

Each interior angle of a regular polygon = [(n − 2) × 180°] / n

According to the question:

⇒ 360 / [(n − 2) × 180] = 2 / 43

⇒ 2 / (n − 2) = 2 / 43

⇒ n − 2 = 43

⇒ n = 45

Now, calculate the measure of each interior angle:

⇒ Interior Angle = [(45 − 2) × 180°] / 45

⇒ Interior Angle = (43 × 180°) / 45

⇒ Interior Angle = 43 × 4°

⇒ Interior Angle = 172°

∴ The correct answer is 172°.

Additional Information

Sum of Angles Property

For any convex polygon, the sum of all exterior angles (one at each vertex) is always 360°, regardless of the number of sides.

Interior and Exterior Relation

The sum of an interior angle and its adjacent exterior angle at any vertex of a polygon is always 180° (Linear Pair).

Number of Diagonals

The number of diagonals in a polygon with n sides is given by the formula: [n × (n − 3)] / 2.

Shortcut Trick

Net effective increase = x + y + (xy/100)

⇒ Net % = 15 + 30 + (15 × 30)/100 = 45 + 4.5 = 49.5%

Population after 2 years = 100% + 49.5% = 149.5% of 1,60,000

⇒ 1.495 × 1,60,000 = 2,39,200

∴ The correct answer is 2,39,200.

Alternate Method

Given: Initial Population (P) = 1,60,000

Increase in Year 1 (R₁) = 15%

Increase in Year 2 (R₂) = 30%

Formula: Population after n years = P × (1 + R₁/100) × (1 + R₂/100)

⇒ Population after 1st year = 1,60,000 × (115/100)

⇒ Population after 1st year = 1,600 × 115 = 1,84,000

⇒ Population after 2nd year = 1,84,000 × (130/100)

⇒ Population after 2nd year = 1,840 × 130 = 2,39,200

∴ The correct answer is 2,39,200.

Additional Information

Successive Percentage Change

When a value changes by x% and then y% successively, the net percentage change is given by [x + y + (xy/100)]%.

Population Decrease (Depreciation)

If the population decreases, the value of R becomes negative in the formula: Final = P × (1 − R/100)ⁿ.

Constant Growth Rate

If the growth rate remains constant for n years, the formula simplifies to: Final = P × (1 + R/100)ⁿ, which is identical to the Compound Interest formula.

Shortcut Trick

Sum of Length (L) and Breadth (B) = Perimeter ÷ 2 = 40 ÷ 2 = 20 cm.

Difference between Length and Breadth (L − B) = 5 cm.

Length (L) = (Sum + Difference) ÷ 2 = (20 + 5) ÷ 2 = 12.5 cm.

Breadth (B) = (Sum − Difference) ÷ 2 = (20 − 5) ÷ 2 = 7.5 cm.

∴ The correct answer is 12.5 cm, 7.5 cm.

Alternate Method

Given:

Perimeter (P) = 40 cm

Length (L) = Breadth (B) + 5 cm

Formula Used:

Perimeter of a rectangle = 2 × (Length + Breadth)

Calculations:

⇒ 2 × (L + B) = 40

⇒ L + B = 20

⇒ (B + 5) + B = 20

⇒ 2B + 5 = 20

⇒ 2B = 20 − 5

⇒ 2B = 15

⇒ B = 7.5 cm

⇒ L = B + 5 = 7.5 + 5 = 12.5 cm

∴ The correct answer is 12.5 cm, 7.5 cm.

Additional Information

Area of Rectangle

The total space occupied by the rectangle is given by Area = Length × Breadth.

Diagonal of Rectangle

The distance between opposite corners is d = √(L² + B²).

Square Property

A rectangle becomes a square when Length = Breadth, and its perimeter is 4 × Side.

Shortcut Trick

Maximum Selling Price is obtained when the total effective discount is minimum.

Check the sum of discounts for each scheme: (1) 38%, (2) 58%, (3) 61%, (4) 20%.

Since Scheme 4 has the lowest sum and lowest individual values, its effective discount will be the minimum.

Minimum Discount ⇒ Maximum Selling Price.

∴ The correct answer is Two successive discounts of 10%.

Alternate Method

Successive Discounts: d₁% and d₂%

Effective Discount % = d₁ + d₂ − (d₁ × d₂)/100

Calculating effective discount for each scheme:

⇒ Scheme 1: 23 + 15 − (23 × 15)/100 = 38 − 3.45 = 34.55%

⇒ Scheme 2: 12 + 46 − (12 × 46)/100 = 58 − 5.52 = 52.48%

⇒ Scheme 3: 25 + 36 − (25 × 36)/100 = 61 − 9 = 52.00%

⇒ Scheme 4: 10 + 10 − (10 × 10)/100 = 20 − 1 = 19.00%

⇒ To maximize Selling Price, we need the Minimum Effective Discount.

⇒ Minimum discount is 19% from Scheme (4).

∴ The correct answer is Two successive discounts of 10%.

Additional Information

Effective Discount Formula

For three successive discounts a%, b%, and c%, the effective discount is calculated by applying the two-discount formula twice or using the multiplier method: 1 − [(1−a/100)(1−b/100)(1−c/100)].

Relationship between MP and CP

MP/CP = (100 + Profit%)/(100 − Discount%). This ratio is useful for finding the marked price when profit and discount are known.

Maximum vs Minimum Discount

For a fixed sum of two discounts (a+b = constant), the effective discount is maximum when the difference between a and b is maximum, and minimum when the difference is minimum.

Shortcut Trick

Total Work (LCM of 8, 12, 16) = 48 units.

Efficiencies: A = 6, B = 4, C = 3. Total efficiency (A+B+C) = 13 units/hr.

Work done in 2 hours = 13 × 2 = 26 units.

Remaining work = 48 − 26 = 22 units.

Time for A and B to finish = 22 ÷ (6 + 4) = 2.2 hours.

Total time = 2 + 2.2 = 4.2 hours.

∴ The correct answer is 4.2 hours.

Alternate Method

Given:

Time taken by A = 8 hrs, B = 12 hrs, C = 16 hrs.

C leaves after 2 hours.

Calculations:

⇒ Total units of work = LCM(8, 12, 16) = 48 units

⇒ Efficiency of A = 48 ÷ 8 = 6 units/hr

⇒ Efficiency of B = 48 ÷ 12 = 4 units/hr

⇒ Efficiency of C = 48 ÷ 16 = 3 units/hr

⇒ Combined efficiency of A + B + C = 6 + 4 + 3 = 13 units/hr

⇒ Work done in the first 2 hours = 13 × 2 = 26 units

⇒ Remaining work = 48 − 26 = 22 units

⇒ Workers remaining = A + B

⇒ Efficiency of A + B = 6 + 4 = 10 units/hr

⇒ Time taken to finish remaining work = 22 ÷ 10 = 2.2 hours

⇒ Total time = 2 + 2.2 = 4.2 hours

∴ The correct answer is 4.2 hours.

Additional Information

Work and Efficiency Relation

Total Work = Efficiency × Time. Efficiency is inversely proportional to the time taken to complete a task.

Pipes and Cisterns

The same logic applies where filling pipes have positive efficiency and outlet pipes (leaks) have negative efficiency.

Work and Wages

Wages are distributed based on the ratio of work done by each individual, which depends on their efficiency and time worked.

Shortcut Trick

Profit Ratio of Suman : Tariq : Neha = 6 : 11 : 12

Total units = 6 + 11 + 12 = 29 units

Difference between Neha and Suman = 12 − 6 = 6 units

Value of 6 units = (79721 ÷ 29) × 6 = 2749 × 6

Value of 6 units = ₹16,494

∴ The correct answer is ₹16,494.

Alternate Method

Given:

Investment ratio of Suman, Tariq and Neha = 6 : 11 : 12

Total annual profit = ₹79,721

Formula Used:

Profit Share = (Individual Ratio Part ÷ Sum of Ratio Parts) × Total Profit

Sum of ratio parts = 6 + 11 + 12 = 29

⇒ 29 units = ₹79,721

⇒ 1 unit = 79721 ÷ 29

⇒ 1 unit = ₹2,749

⇒ Suman share = 6 × 2749 = ₹16,494

⇒ Neha share = 12 × 2749 = ₹32,988

⇒ Difference = 32988 − 16494

⇒ Difference = ₹16,494

∴ The correct answer is ₹16,494.

Additional Information

Partnership Profit Rule

Profit is distributed based on the product of Capital (C) and Time (T). Profit Ratio P₁ : P₂ = C₁ × T₁ : C₂ × T₂.

Total Ratio Unitary Method

To find the value of any share or difference, divide the total amount by the sum of all ratio parts to find the value of 1 unit.

Working vs Sleeping Partner

A working partner manages the business and may receive a salary before the remaining profit is divided according to the investment ratio.

Shortcut Trick

Recognize x = 3 + 2√2 as a perfect square: (√2 + 1)².

∴ √x = √2 + 1.

Reciprocal 1/√x = 1/(√2 + 1) = √2 − 1.

Substitute in expression: (√2 + 1) − (√2 − 1).

Value = 2.

∴ The correct answer is 2.

Alternate Method

Given: x = 3 + √8

Formula: (√x − 1/√x)² = x + 1/x − 2

⇒ 1/x = 1 / (3 + √8) × (3 − √8) / (3 − √8) ⇒ 1/x = (3 − √8) / (9 − 8) = 3 − √8 ⇒ x + 1/x = (3 + √8) + (3 − √8) = 6 ⇒ (√x − 1/√x)² = 6 − 2 ⇒ (√x − 1/√x)² = 4 ⇒ √x − 1/√x = √4 = 2

∴ The correct answer is 2.

Additional Information

Conjugate Surds

For any surd of the form a + √b, its conjugate is a − √b. Their product results in a rational number a² − b.

Square Root of a Surd

If a surd is in the form (a + b) + 2√(ab), its square root is simply √a + √b.

Algebraic Identity

The expansion of (a − b)² = a² + b² − 2ab is frequently used to find values of √x − 1/√x by setting a = √x and b = 1/√x.

Shortcut Trick

Smallest number leaving same remainder k = LCM(divisors) + k

LCM(18, 35, 140):

LCM(35, 140) = 140

LCM(140, 18) = 1260

Required number = 1260 + 7 = 1267

∴ The correct answer is 1267.

Alternate Method

Given values: Divisors = 18, 35, 140 and Remainder = 7.

Formula: Smallest number = LCM(x, y, z) + r

⇒ Prime factors of 18 = 2 × 3²

⇒ Prime factors of 35 = 5 × 7

⇒ Prime factors of 140 = 2² × 5 × 7

⇒ LCM = 2² × 3² × 5 × 7

⇒ LCM = 4 × 9 × 35 = 1260

⇒ Number = 1260 + 7 = 1267

∴ The correct answer is 1267.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is perfectly divisible by each of the given numbers.

Case of Different Remainders

If a number divided by x, y, z leaves remainders a, b, c such that (x−a) = (y−b) = (z−c) = k, the number is LCM(x, y, z) − k.

Relationship between HCF and LCM

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

Shortcut Trick

Ratio of articles P : Q : R = 4 : 2 : 5

Total cost per ratio unit = (4 × 280) + (2 × 185) + (5 × 105)

Total cost per ratio unit = 1120 + 370 + 525 = 2015

Common multiplier = Total budget ÷ Unit cost = 8060 ÷ 2015 = 4

Number of articles of type Q = 2 × 4 = 8

∴ The correct answer is 8.

Alternate Method

Given prices: P = ₹280, Q = ₹185, R = ₹105

Given ratio of quantity P : Q : R = 4 : 2 : 5

Let the quantities of P, Q, and R be 4x, 2x, and 5x respectively.

Total Expenditure = (Price_P × Quantity_P) + (Price_Q × Quantity_Q) + (Price_R × Quantity_R)

⇒ 8060 = (280 × 4x) + (185 × 2x) + (105 × 5x)

⇒ 8060 = 1120x + 370x + 525x

⇒ 8060 = 2015x

⇒ x = 8060 ÷ 2015

⇒ x = 4

⇒ Number of articles of type Q = 2x

⇒ Number of articles of type Q = 2 × 4 = 8

∴ The correct answer is 8.

Additional Information

Ratio of Values

The ratio of the total values of different items is the product of the ratio of their individual prices and the ratio of their quantities (Value Ratio = Price Ratio × Quantity Ratio).

Unitary Method in Ratios

When the total sum and the ratio are known, the value of one ratio unit is calculated as: Total Sum ÷ Sum of ratio parts (weighted by their respective values).

Compound Ratio

For two ratios a:b and c:d, the compound ratio is (a×c):(b×d), which is often used in problems involving price and quantity to find the total expenditure ratio.

Shortcut Trick

Efficiency Ratio (Amit : Gopal) = 100 : 130 = 10 : 13

Time Ratio is inverse of Efficiency Ratio ⇒ Amit : Gopal = 13 : 10

Given Amit's time = 13 days (13 units = 13 days)

⇒ Gopal's time = 10 units = 10 days

∴ The correct answer is 10.

Alternate Method

Given:

Time taken by Amit = 13 days

Efficiency of Gopal = Efficiency of Amit + 30% of Efficiency of Amit

Formula Used:

Total Work = Efficiency × Time

⇒ Let Amit's efficiency = 10 units/day

⇒ Gopal's efficiency = 10 + 30% of 10 = 13 units/day

⇒ Total Work = Amit's time × Amit's efficiency

⇒ Total Work = 13 × 10 = 130 units

⇒ Time taken by Gopal = Total Work ÷ Gopal's Efficiency

⇒ Time taken by Gopal = 130 ÷ 13 = 10 days

∴ The correct answer is 10.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a task. If efficiency increases by x%, the time decreases proportionally.

Work Equation

Total Work = (Efficiency₁ × Time₁) = (Efficiency₂ × Time₂). This constant work principle allows solving for any missing variable.

Ratio Conversion

If the ratio of efficiency of A and B is a : b, then the ratio of time taken by them to complete the same work will be b : a.

Shortcut Trick

The average of the first n odd natural numbers is always equal to n.

Here, n = 12.

∴ Average = 12.

∴ The correct answer is 12.

Alternate Method

Given:

First 12 odd natural numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23.

Formula Used:

Sum of the first n odd natural numbers = n²

Average = Total Sum ÷ Number of terms

Calculations:

⇒ Total terms (n) = 12

⇒ Sum of 12 terms = 12² = 144

⇒ Average = 144 ÷ 12

⇒ Average = 12

Alternatively, for an Arithmetic Progression (AP):

⇒ Average = (First term + Last term) ÷ 2

⇒ Average = (1 + 23) ÷ 2 = 24 ÷ 2 = 12

∴ The correct answer is 12.

Additional Information

Average of first n even numbers

The average of the first n even natural numbers (2, 4, 6... 2n) is always (n + 1).

Average of first n natural numbers

The average of the first n natural numbers (1, 2, 3... n) is given by (n + 1) ÷ 2.

Sum of first n natural numbers

The sum of the first n natural numbers is calculated using the formula [n × (n + 1)] ÷ 2.

Shortcut Trick

Surface Area (SA) = 4πr² = 616 ⇒ r² = 49 ⇒ r = 7 cm.

Radius decreases by 25% ⇒ Ratio of radii = 1 : 0.75 = 4 : 3.

Ratio of volumes = (4)³ : (3)³ = 64 : 27.

Original Volume = (4/3) × (22/7) × 7³ = 4312/3 cm³.

New Volume = (4312/3) × (27/64) = 135.375 × 4.48 ≈ 606.375 cm³.

∴ The correct answer is 606.375 cm³.

Alternate Method

Given:

Surface area of the original sphere = 616 cm²

Reduction in radius = 25%

Formula Used:

Surface area of sphere = 4πr²

Volume of sphere = (4/3)πr³

Step 1: Finding the radius of the original sphere (r).

⇒ 4 × (22/7) × r² = 616

⇒ r² = (616 × 7) / (4 × 22)

⇒ r² = (616 × 7) / 88

⇒ r² = 7 × 7 = 49

⇒ r = 7 cm

Step 2: Finding the radius of the new sphere (r_new).

⇒ r_new = r − 25% of r

⇒ r_new = 7 − (25/100 × 7)

⇒ r_new = 7 − 1.75 = 5.25 cm

Step 3: Finding the volume of the new sphere.

⇒ Volume = (4/3) × π × (r_new)³

⇒ Volume = (4/3) × (22/7) × (5.25) × (5.25) × (5.25)

⇒ Volume = (4/3) × (22/7) × (21/4) × (21/4) × (21/4)

⇒ Volume = (22 × 21 × 21) / (4 × 4)

⇒ Volume = (11 × 441) / 8 = 4851 / 8

⇒ Volume = 606.375 cm³

∴ The correct answer is 606.375 cm³.

Additional Information

Properties of a Sphere

For a sphere of radius r, the Surface Area is 4πr² and the Volume is (4/3)πr³.

Ratio Relationship

If the radius of a sphere is changed by a factor of k, the surface area changes by k² and the volume changes by k³.

Hemisphere Calculations

For a hemisphere, the Curved Surface Area is 2πr² and the Total Surface Area is 3πr².

Shortcut Trick

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

In this expression, a = 5pq and b = 16q.

Required value = 4 × 5pq × 16q = 320pq².

∴ The correct answer is 320pq².

Alternate Method

Given: (5pq + 16q)² − (5pq − 16q)²

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

⇒ [(5pq)² + 2 × 5pq × 16q + (16q)²] − [(5pq)² − 2 × 5pq × 16q + (16q)²]

⇒ [25p²q² + 160pq² + 256q²] − [25p²q² − 160pq² + 256q²]

⇒ 25p²q² + 160pq² + 256q² − 25p²q² + 160pq² − 256q²

⇒ 160pq² + 160pq²

⇒ 320pq²

∴ The correct answer is 320pq².

Additional Information

Algebraic Identity for Difference of Squares

x² − y² = (x + y)(x − y). This can also be used to solve the problem by letting x = (5pq + 16q) and y = (5pq − 16q).

Algebraic Identity for Sum of Squares

(a + b)² + (a − b)² = 2(a² + b²). This identity is used when the middle sign is positive.

Simplification Rule

While simplifying algebraic expressions, always check for common factors or standard identities to reduce calculation time.

The logic followed here is based on the alphabetical position pattern:

Option 1: LF - PJ

Option 2: JR - NV

Option 3: YA - CF

Option 4: TI - XM

Thus, YA - CF does not follow the uniform shift of +4 for both letters.

Hence, the odd one out is Option 3.

The logic followed here is:

So, 'YKF 50' will replace the question mark in the given series.

Hence, the correct answer is "Option 3".

Given:

READ is coded as 4628

AIMS is coded as 5716

So, the code for A = 6

Hence, the correct answer is Option 1.

The logic followed here is:

Given:

Town A is to the West of Town B.

Town D is to the South of Town A.

Town C is to the East of Town D.

Town E is to the North of Town C.

So, with respect to Town D: Town B lies towards the North-East direction.

Hence, the correct answer is Option 2.

Given Statements:

All kettles are ovens.

All ovens are toasters.

The least possible Venn diagram for the given statements is:

Conclusions:

I. Some toasters are kettles → Follows (Since all kettles are toasters, there is definitely an overlap between toasters and kettles.)

II. No toaster is a kettle → Does not follow (This contradicts the given relationship that all kettles are inside toasters.)

So, only conclusion I follows.

Hence, the correct answer is "Option 2".

The logic followed here is:

thus, "118" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

The logic followed here is:

For, XTBH → BXFL

For, MFKP → QJOT

Now, applying the same pattern to BXLR:

So, BXLR → FBPV

Hence, the correct answer is Option 4.

The logic followed here is:

Logic: Third Number = First Number × Second Number.

For, 12 - 4 - 48

⇒ 12 × 4 = 48

For, 15 - 3 - 45

⇒ 15 × 3 = 45

Now, checking the options using the same pattern:

Option 1) 3 - 4 - 14

⇒ 3 × 4 = 12 ≠ 14

Option 2) 21 - 3 - 61

⇒ 21 × 3 = 63 ≠ 61

Option 3) 19 - 2 - 38

⇒ 19 × 2 = 38

Option 4) 16 - 2 - 34

⇒ 16 × 2 = 32 ≠ 34

Hence, the correct answer is Option 3.

Given Statements:

All cars are buses.

All buses are ships.

No car is a train.

The least possible Venn diagram for the given statements is:

Conclusions:

I. No bus is a train → Does not follow (Only cars are excluded from trains; buses may still overlap with trains except the car portion)

II. No ship is a train → Does not follow (Since ships contain buses and cars, and only cars are restricted from trains, ships may still overlap with trains)

So, neither conclusion I nor II follows.

Hence, the correct answer is "Option 1".

Given: Seven people S, Q, U, A, R, E and D are sitting in a row, facing north.

Only two people sit to the right of A.

Only two people sit between A and R.

Only two people sit between E and D.

D sits to the immediate left of A.

S sits to the immediate right of U.

Now, the third position from the left end is: Q

Hence, the correct answer is "Option 2".

The logic followed here is:

Given numbers: 700, 496, 723, 316, 639

Logic: Identify the highest and lowest numbers, extract the required digits, and perform subtraction.

Arranging the numbers:

Highest number = 723

Lowest number = 316

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Result 6 − 2 = 4

Hence, the correct answer is Option 1.

The logic followed here is:

Given numbers: 853, 139

Logic: Identify the highest and lowest numbers, then extract the required digits and compute their sum.

Arranging the numbers:

Highest number = 853

Lowest number = 139

<br>

Result: 3 + 3 = 6

Hence, the correct answer is Option 2.

The logic followed here is:

Given:

Gyalmu drives 15 km East

He turns right → 16 km South

He turns right → 17 km West

He turns right → 18 km North

He turns right → 2 km East

So, To reach Point A, he must go 2 km South.

Hence, the correct answer is Option 1.

Given:

Total number of people = 144

Mr.Lkj is 65th from the left end

and Mr. Ihg is 8th from the right end.

Number of people between them = 144 − (65 + 8) = 144 − 73 = 71.

So, there are 71 people between Mr. Lkj and Mr. Ihg.

Hence, the correct answer is "Option 2".

Given number: 28369475

Logic:

If digit is even → add 1

If digit is odd → subtract 2

Original number: 2 8 3 6 9 4 7 5

New number formed: 39177553

Digits appearing more than once: 3, 7, 5

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "77" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

Given:

Total students = 50

Meha is 15th from the right end,

and Priya is 8 places to the left of Meha.

Priya’s position from the right end = 15 + 8 = 23rd.

Position from the left end = (50 − 23) + 1 = 28.

So, Priya’s position from the left end is 28th.

Hence, the correct answer is "Option 4".

The logic followed here is:

For, GUSB → LZXV

For, NHIN → SMNH

Now, applying the same pattern to MLKO:

So, MLKO → RQPI

Hence, the correct answer is Option 3.

BODMAS Table: 

Decoding the given information:

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Now, replacing the sign from left to right using BODMAS Rule, we get:

⇒ 405 ÷ 5 + 32 − 16 × 6 = ?

⇒ 81 + 32 − 96 = ?

⇒ 81 + 32 = 113

⇒ 113 − 96 = ?

⇒ 17

Hence, the correct option is "Option 3".

Given: (Left) C H T M L K N B P G Q D V W R A X Z F Y E S (Right)

Now, we identify the distinct vowels (A, E, I, O, U) present in the series. Each vowel is counted only once, even if repeated.

(Left) C H T M L K N B P G Q D V W R A X Z F Y E S (Right)

Distinct vowels found: A, E

So, total unique vowels = 2

Hence, the correct answer is "Option 3".

The logic followed here is:

For, VANE → DIVM

For, BGTK → JOBS

Now, applying the same pattern to ZERI:

So, ZERI → HMZQ

Hence, the correct answer is Option 2.

Given: Seven people A, B, C, D, E, F and G are sitting in a row, facing north.

B sits sixth from the left end of the row.

Only two people sit to the left of D.

F sits to the right of A but left of D.

E sits third to the left of G.

So, only 1 person sits between C and G.

Hence, the correct answer is "Option 2".

Given: Seven people A, B, C, D, E, F and G are sitting in a row, facing north.

Only six people sit to the left of F.

D sits fourth from the right end of the row.

Only two people sit between A and D.

G is the immediate neighbour of C and F.

B does not sit to the left of E.

Now, people between E and G: B, D, C → 3 people

Hence, the correct answer is "Option 4".

The logic followed here is:

thus, "168" will replace the question mark in the given series.

Hence, the correct option is "Option 4".

Given:

“bees buzz close” → “zd st cq”

 “hear bees hum” → “cq ge vm”

So, the code for bees = cq

Hence, the correct answer is Option 1.

The logic followed here is:

So, 'KG9' will replace the question mark in the given series.

Hence, the correct answer is "Option 1".

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

Now,

We need to find an even digit which is immediately preceded by an odd digit and immediately followed by an even digit.

So, Needed pattern is: Odd - Even - Even.

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

Total valid even digits satisfying the condition = 3.

Hence, the correct answer is 'Option 4'.

Given number: 5346188

Logic:

Odd digits → +3

Even digits → −4

Original number: 5 3 4 6 1 8 8

New number formed: 8 6 0 2 4 4 4 → 8602444

<br>

Now, sum: 0 + 2 = 2

Hence, the correct answer is Option 1.

Family Chart:

Decoding the Symbols:

Given: L @ O # V + E × R

It means: L is the father of O, O is the brother of V, V is the sister of E, and E is the wife of R.

Thus,  L is R’s wife’s father.

Hence, the correct answer is "Option 1".