The correct answer is When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it..

Key Points

• Archimedes' Principle states that when a body is fully or partially submerged in a fluid (liquid or gas), it experiences an upward buoyant force. This force is equal to the weight of the fluid displaced by the body.
• The upward force, commonly referred to as the buoyant force, acts in the opposite direction of gravity and is responsible for the floating or sinking of objects in fluids.
• This principle explains why some objects float while others sink. If the weight of the displaced fluid is greater than or equal to the weight of the object, the object will float. Otherwise, it will sink.
• The principle is widely used in designing ships, submarines, hot air balloons, and other engineering applications where buoyancy is a critical factor.
• The formula used in Archimedes' Principle is: Buoyant Force (F_b) = Weight of Displaced Fluid = ρ × V × g, where:
• ρ = Density of the fluid
• V = Volume of the fluid displaced
• g = Acceleration due to gravity

Additional Information

• Explanation of Options:
• Option 1: Incorrect. This statement is wrong because Archimedes' Principle describes an upward force, not a downward force.
• Option 2: Correct. This option accurately explains Archimedes' Principle that the body experiences an upward force equal to the weight of the fluid displaced.
• Option 3: Incorrect. The downward force (weight of the body) is not less than the weight of the displaced fluid; instead, the upward buoyant force balances the weight of the object in certain cases (floating).
• Option 4: Incorrect. The upward force is not less than the weight of the displaced fluid; it is equal to the weight of the displaced fluid as per Archimedes' Principle.
• Applications of Archimedes' Principle:
• Shipbuilding: Ships float because the weight of the water displaced by the ship is equal to the ship's weight.
• Submarines: Submarines use Archimedes' Principle to dive and resurface by controlling the amount of water in their ballast tanks.
• Hot Air Balloons: Hot air balloons rise because the air displaced by the balloon weighs more than the balloon itself.
• Hydrometers: Used to measure the density of liquids based on the principle of buoyancy.
• Important Terms:
• Buoyant Force: The upward force exerted by a fluid on a submerged or partially submerged object.
• Displaced Fluid: The volume of fluid pushed aside by a submerged object.
• Density: The mass per unit volume of a substance, which determines whether an object will float or sink.
• Historical Background: Archimedes' Principle was discovered by the ancient Greek mathematician and inventor Archimedes in the 3rd century BCE.

Important Points

• Key Applications in Modern Science:
• Archimedes' Principle is used extensively in fluid mechanics and engineering design to ensure stability and functionality of devices like ships, submarines, and balloons.
• It is also used in laboratories to calculate the density and specific gravity of objects by measuring the weight of displaced fluid.

The correct answer is Sum of individual resistances.

Key Points

• When resistors are connected in series, the total or equivalent resistance is the sum of all individual resistances in the circuit.
• The formula to calculate the total resistance in a series circuit is: R_total = R₁ + R₂ + R₃ + ... + R_n, where R₁, R₂, R₃, etc., are the resistances of individual resistors.
• This is because, in a series connection, the current remains the same across all resistors, but the voltage drop adds up across each resistor.
• The concept of adding resistances in series ensures that the total resistance is always greater than any individual resistance.
• The series connection is commonly used in circuits where a higher resistance is required or to divide the voltage across multiple components.

Additional Information

• Characteristics of Resistors in Series:
• In a series circuit, the current is the same through all the resistors.
• The voltage drop across each resistor depends on its resistance and can be calculated using Ohm's Law: V = I × R, where I is the current, and R is the resistance.
• Adding more resistors in series increases the total resistance of the circuit.
• Applications of Series Circuits:
• Used in devices like string lights, where the same current flows through all bulbs.
• Series resistors are often used to reduce current or limit voltage in circuits.
• Comparison with Parallel Circuits:
• In a parallel circuit, the total resistance is calculated as the reciprocal of the sum of reciprocals of individual resistances: 1/R_total = 1/R₁ + 1/R₂ + ...
• Parallel circuits have a lower total resistance compared to series circuits for the same resistors.
• Practical Example:
• If three resistors of 5 Ω, 10 Ω, and 15 Ω are connected in series, the total resistance is: R_total = 5 + 10 + 15 = 30 Ω.

Important Points

• Option 1 (Reciprocal of sum of reciprocals): This is the formula for calculating total resistance in a parallel circuit, not a series circuit. Hence, this option is incorrect.
• Option 2 (Product of individual resistances): The product of resistances is not used to calculate total resistance in any standard circuit configuration. Hence, this option is incorrect.
• Option 3 (Sum of individual resistances): This is the correct formula for calculating total resistance in a series circuit. Hence, this option is correct.
• Option 4 (Always less than any single resistor): In a series circuit, the total resistance is always greater than the highest individual resistance. Hence, this option is incorrect.

The correct answer is Option 3: farther distance in Case-I and moves forever in Case-II.

Key Points

• Case-I: Inclined Plane with Small Angle: In Case-I, the ball is released from a plane inclined on the left and rolls onto a plane inclined on the right, but with a smaller angle. The smaller angle on the right reduces the slope, thereby extending the rolling distance before the ball comes to a stop. Hence, the ball travels farther compared to an equally inclined case.
• Case-II: Flat Surface (Zero Inclination): When the ball transitions from the inclined plane on the left to a flat surface on the right, there is no opposing force to stop its motion. Since the plane is frictionless, the ball continues to move indefinitely due to the absence of friction or any other resisting force. Hence, the ball moves forever.
• Comparison with Equally Inclined Case: In an equally inclined scenario, the ball's motion would be symmetrical, and the ball would stop at an equal distance on both sides due to gravity's equal opposing forces. Case-I modifies this condition, causing the ball to travel farther on the smaller inclined plane. Case-II, with no inclination, eliminates the stopping mechanism altogether. Thus, option 3 correctly identifies the outcomes.
• Physics of Motion on Frictionless Planes: The ball's behavior is determined solely by gravitational forces and the plane's inclination. A smaller inclination reduces the gravitational component acting against the motion, and a flat surface eliminates it entirely, allowing perpetual motion. This matches the behavior described in the question.
• Hence, the correct answer is Option 3.

Additional Information

• Frictionless Plane: A frictionless plane is an idealized surface where no resistance opposes the motion of objects. In reality, such surfaces are rare, but they serve as a useful approximation for studying concepts of motion and forces in physics.
• Gravitational Component: On an inclined plane, the force acting on an object due to gravity can be divided into two components: one parallel to the plane (causing motion) and one perpendicular to the plane (pressing the object against the surface). The parallel component decreases with the angle of inclination.
• Perpetual Motion: On a frictionless flat surface, an object in motion will continue moving indefinitely according to Newton's First Law of Motion, which states that an object will remain in its state of motion unless acted upon by an external force.
• Inclined Plane Dynamics: Inclined planes are simple machines that modify the effect of gravity on objects. The steeper the incline, the greater the gravitational force acting to accelerate the object down the slope.
• Practical Applications: Concepts involving frictionless planes and inclined surfaces are used in designing systems like roller coasters, ramps, and experimental setups in physics labs to study motion and forces.

Important Points

• Newton's Laws of Motion: These laws govern the behavior of objects in motion. In this scenario, Newton's First Law explains why the ball moves forever on a frictionless flat surface, and Newton's Second Law helps calculate the acceleration caused by gravity on the inclined plane.
• Energy Conservation: The ball's motion on the inclined plane follows the principle of conservation of energy, where potential energy converts to kinetic energy as it rolls down. On a flat surface, the kinetic energy remains constant without losses due to friction.
• Ideal Conditions: The question assumes ideal conditions like a perfectly smooth ball and frictionless planes, which are theoretical but simplify understanding of motion concepts.
• Real-World Examples: While frictionless planes are theoretical, low-friction surfaces like ice or air tracks can approximate such conditions and are used in physics experiments.

The correct answer is Most of the alpha particles passed straight through the gold foil.

Key Points

• Rutherford's alpha-particle scattering experiment demonstrated that most of the alpha particles passed straight through the gold foil. This indicated that most of the atom's volume is empty space, as the particles encountered no obstruction.
• The alpha particles, which are positively charged helium nuclei, moved through the gold foil without significant deflection because the atomic structure is largely empty.
• A few alpha particles were deflected at small angles, which suggested the presence of a positive charge in the atom but did not affect the conclusion that most of the atom is empty space.
• Very few alpha particles were deflected by 180°, indicating a highly concentrated and positively charged nucleus at the center of the atom. However, the rarity of this event further supported the notion that the atom's interior is mostly empty.
• The experiment disproved the earlier "plum pudding model" of the atom, which suggested that positive and negative charges were distributed evenly throughout the atom.

Additional Information

• Rutherford's Experiment Details:
• Conducted by Ernest Rutherford in 1911, the experiment involved bombarding a thin gold foil with alpha particles emitted from a radioactive source.
• A fluorescent screen was used to detect the scattering of alpha particles.
• The experiment provided evidence for the nuclear model of the atom, replacing J.J. Thomson's plum pudding model.
• Structure of the Atom:
• The atom consists of a dense, positively charged nucleus at its center, containing protons and neutrons.
• Electrons, which are negatively charged, revolve around the nucleus in defined orbits.
• Most of the atom's volume is empty space, as evidenced by the large fraction of alpha particles passing through the foil without deflection.
• Key Observations from the Experiment:
• Most alpha particles passed straight through the foil, indicating empty space in the atom.
• A small fraction of alpha particles were deflected at small angles, suggesting the presence of a dense positive charge.
• A very few particles rebounded back, confirming the existence of a concentrated nucleus.
• Impact of the Experiment:
• It led to the development of the nuclear model of the atom, which forms the basis of modern atomic theory.
• The discovery of the nucleus paved the way for further research into subatomic particles, such as protons, neutrons, and later electrons.

Important Points

• Alpha Particles:
• Alpha particles are helium nuclei, consisting of 2 protons and 2 neutrons, with a positive charge.
• Their high energy and positive charge made them suitable for the scattering experiment.
• Gold Foil:
• The gold foil used in the experiment was extremely thin, about 0.00004 cm thick.
• Gold was chosen because it is malleable and can be beaten into very thin sheets.
• Nucleus Characteristics:
• The nucleus occupies a very small volume of the atom (approximately 1/10,000th of the atom's diameter).
• It contains nearly all the atom's mass and is positively charged.
• Significance of the Experiment:
• It confirmed the existence of a nucleus and laid the foundation for the Bohr model of the atom.
• The findings eventually contributed to the development of quantum mechanics and nuclear physics.

The correct answer is Force is a vector quantity measured in Newton..

Key Points

• Force is a vector quantity: A force has both magnitude and direction, which makes it a vector quantity. This distinguishes it from scalar quantities, which only have magnitude.
• SI Unit of Force: The SI unit of force is the Newton (N). It is defined as the force required to accelerate a 1 kg mass by 1 m/s².
• Force and Energy are Different: Force is not measured in Joules. Joule is the unit of energy, while Newton is the unit of force. Hence, energy and force are fundamentally different physical quantities.
• Directionality of Force: Since force is a vector quantity, it requires both direction and magnitude for its complete specification. The direction determines the way the force acts.
• Examples: Pushing a door, pulling a rope, and the gravitational force acting on an object are examples that demonstrate the vector nature of force.

Additional Information

• Newton’s Second Law of Motion: Force is mathematically expressed using Newton's second law of motion as F = ma, where F is force, m is mass, and a is acceleration.
• Types of Forces: Forces can be broadly categorized into two types: contact forces (e.g., friction, tension) and non-contact forces (e.g., gravitational force, magnetic force).
• Representation of Force: Forces are represented using arrows in diagrams. The length of the arrow corresponds to the magnitude of the force, and the arrow's direction indicates the direction of the force.
• Force and Work: Work is done when a force is applied to an object and the object moves in the direction of the force. The formula for work is W = F × d × cos(θ), where θ is the angle between the force and the displacement direction.
• Gravitational Force: A common example of force is the gravitational force, which acts between two masses. Its magnitude is given by Newton's law of gravitation: F = G(m₁m₂)/r², where G is the gravitational constant, m₁ and m₂ are the masses, and r is the distance between them.

Important Points

• Scalar Quantity: Scalar quantities have only magnitude and no direction. Examples include speed, mass, time, and energy.
• Joule as a Unit: The Joule (J) is the SI unit of energy and work. One Joule is defined as the energy transferred when a force of one Newton displaces an object by one meter in the direction of the force.
• Misconception About Force: Force is often misunderstood as a scalar quantity, but its vectorial nature is critical in understanding phenomena like motion, equilibrium, and dynamics.
• Practical Applications of Force: Forces are integral to various fields such as engineering (e.g., structural analysis), physics (e.g., mechanics), and daily life (e.g., using tools, driving vehicles).
• Historical Context: The concept of force was formalized by Sir Isaac Newton in the 17th century through his laws of motion, which form the foundation of classical mechanics.

The correct answer is Ketone.

Key Points

• Propanone, commonly known as acetone, belongs to the ketone group. Ketones are organic compounds characterized by the presence of a carbonyl group (C=O) bonded to two carbon atoms.
• The chemical formula of propanone is CH3COCH3, where the carbonyl group is bonded to two methyl groups.
• Ketones are a part of a larger class of organic compounds known as carbonyl compounds, which include aldehydes and carboxylic acids.
• Propanone is the simplest ketone and is widely used as a solvent in industries for applications such as nail polish remover and cleaning agents.
• The distinguishing feature of ketones is that the carbonyl group is not bonded to a hydrogen atom, which differentiates them from aldehydes where the carbonyl group is bonded to at least one hydrogen atom.

Additional Information

• Alcohol:
• Alcohols are organic compounds containing one or more hydroxyl groups (-OH) attached to a saturated carbon atom.
• Examples include ethanol (C2H5OH) and methanol (CH3OH).
• Alcohols are classified into primary, secondary, and tertiary alcohols based on the number of carbon atoms attached to the carbon holding the hydroxyl group.
• Aldehyde:
• Aldehydes are characterized by the presence of a carbonyl group (C=O) bonded to at least one hydrogen atom.
• An example is formaldehyde (HCHO), which is used in disinfectants and preservatives.
• Aldehydes have a distinct odor and are commonly found in flavoring agents and perfumes.
• Ester:
• Esters are organic compounds formed by the reaction of an alcohol and an acid, with the elimination of water.
• They contain the functional group -COOR, where R is an alkyl or aryl group.
• Esters are widely used in perfumes, flavorings, and as solvents in industries.
• Ketones:
• Ketones are used as solvents in industries for applications such as cleaning agents, paint thinners, and nail polish removers.
• They are also precursors in the synthesis of various compounds, including pharmaceuticals.
• Propanone (acetone) is a well-known example and is widely used due to its effectiveness as a solvent.
• Ketones are produced naturally in the human body during fat metabolism in the form of ketone bodies.

Important Points

• Propanone (Acetone):
• Propanone is highly volatile and flammable, making it suitable for use in laboratory and industrial processes.
• It is miscible with water and many organic solvents, making it a versatile solvent.
• The chemical structure of propanone is planar due to the sp2 hybridization of the carbonyl carbon.
• Propanone plays a role in the pharmaceutical industry as a precursor in drug synthesis.
• Applications of Ketones:
• Ketones are used in the production of plastics and synthetic fibers.
• They are intermediates in the synthesis of perfumes, dyes, and explosives.
• Ketones serve as solvents for paints, varnishes, and adhesives.

The correct answer is Muscular tissue.

Key Points

• Muscular tissue contains specialized contractile proteins such as actin and myosin, which are essential for contraction and relaxation, facilitating movement in the body.
• These proteins interact in a highly coordinated manner, generating the force required for activities such as walking, running, breathing, and digestion.
• Muscular tissue is classified into three types: skeletal, smooth, and cardiac muscles, each of which plays a distinct role in movement and function.
• Skeletal muscles are involved in voluntary movements such as limb movement, while smooth muscles control involuntary processes like digestion and blood vessel contraction. Cardiac muscles are responsible for the rhythmic contraction of the heart.
• The presence of contractile proteins makes muscular tissue uniquely capable of generating force and motion, distinguishing it from other tissue types. Hence, the correct answer is muscular tissue.

Additional Information

• Cuboidal epithelium tissue:
• This type of tissue consists of cube-shaped cells and is primarily involved in secretion and absorption.
• It is found in regions like the kidney tubules and glands.
• Unlike muscular tissue, cuboidal epithelium lacks contractile proteins and does not participate in movement.
• Nervous tissue:
• Composed of neurons and glial cells, nervous tissue is specialized for communication and signal transmission.
• It facilitates the transfer of electrical signals between different parts of the body, enabling coordination and response to stimuli.
• Although nervous tissue is essential for muscle activation, it does not directly cause movement since it lacks contractile proteins.
• Squamous epithelium tissue:
• This tissue consists of flat, thin cells and is adapted for diffusion and filtration.
• It is found lining structures such as alveoli in the lungs and blood vessels, where rapid exchange of gases and nutrients occurs.
• Squamous epithelium does not have contractile properties and is not involved in movement.
• Important Points:
• The key difference between muscular tissue and other tissue types lies in the presence of contractile proteins that enable movement.
• Muscular tissue also consumes a significant amount of energy (ATP) during contraction, highlighting its distinct metabolic demands compared to other tissues.
• Each type of muscle tissue (skeletal, smooth, cardiac) has specific roles and structural adaptations suited to its function.

The correct answer is The line passing through the centre of curvature and pole..

Key Points

• The principal axis of a spherical mirror is defined as the straight line that passes through the centre of curvature (C) and the pole (P) of the mirror.
• This line acts as the reference line for all measurements and is perpendicular to the surface of the spherical mirror at the pole.
• The principal axis is significant because it determines the symmetry of the mirror and is the axis along which the mirror's properties, such as focus and curvature, are defined.
• In both concave and convex mirrors, the principal axis is the central line that helps in locating key points like the focus (F) and the centre of curvature (C).
• Hence, the correct definition of the principal axis is the line passing through the centre of curvature and pole, making Option 2 the right answer.

Additional Information

• Pole (P):
• The pole is the geometric center of the mirror's reflecting surface.
• It is the point where the principal axis intersects the mirror's surface.
• Centre of Curvature (C):
• The centre of curvature is the center of the sphere of which the spherical mirror is a part.
• It lies outside the mirror for both concave and convex mirrors and is located on the principal axis.
• Focus (F):
• The focus is the point on the principal axis where parallel rays of light, after reflection, converge (in concave mirrors) or appear to diverge from (in convex mirrors).
• The distance from the pole to the focus is called the focal length (f).
• Radius of Curvature (R):
• The radius of curvature is the distance between the pole and the centre of curvature.
• It is equal to twice the focal length, i.e., R = 2f.
• Convex vs. Concave Mirrors:
• Concave mirrors are converging mirrors that reflect light inward, while convex mirrors are diverging mirrors that spread light outward.
• The principal axis remains the same in both types, helping locate key points like the focus and the centre of curvature.

Important Points

• Option 1: "The line passing through the focus and object" is incorrect because the principal axis does not depend on the object's position.
• Option 3: "The tangent at the pole" is incorrect as a tangent is a line that touches the surface at one point and does not represent the principal axis.
• Option 4: "The perpendicular line drawn from the pole" is vague and does not define the axis passing through the centre of curvature.
• Always remember, the principal axis is the line that passes through the centre of curvature and the pole of the mirror.

The correct answer is One version from each parent.

Key Points

• Sexual reproduction involves the fusion of male and female gametes (sperm and egg), each of which carries one set of chromosomes. The child inherits one version of each trait from each parent.
• Each parent contributes one allele for every trait. An allele is a version of a gene that determines specific characteristics, such as eye color or blood type.
• The combination of alleles inherited from both parents determines the genotype (genetic makeup) of the offspring. This may lead to the expression of dominant or recessive traits.
• For example, in humans, each individual has 23 pairs of chromosomes, for a total of 46. One chromosome in each pair is inherited from the mother, and the other is inherited from the father.
• Traits such as height, hair color, and even predisposition to certain diseases are determined by the interaction of the alleles contributed by both parents, following principles of Mendelian inheritance.

Additional Information

• Chromosomes and Genes:
• Humans have 23 pairs of chromosomes in each cell. Chromosomes carry genes, which are the basic units of heredity.
• Each chromosome contains numerous genes. These genes exist in pairs, with one copy coming from the father and the other from the mother.
• Alleles and Genetic Variation:
• An allele is a variant form of a gene. For example, the gene for eye color may have alleles for blue, green, or brown eyes.
• The presence of different alleles contributes to genetic diversity, which is essential for survival and adaptation in changing environments.
• Mendel's Laws of Inheritance:
• Law of Segregation: Each parent contributes only one allele for a trait to their offspring. This occurs because alleles separate during the formation of gametes.
• Law of Independent Assortment: Alleles for different traits are inherited independently of each other, provided the genes are located on different chromosomes.
• Dominant and Recessive Traits:
• A dominant allele masks the effect of a recessive allele when both are present. For example, in pea plants, the allele for tallness is dominant over the allele for shortness.
• Recessive traits are expressed only when an individual inherits two copies of the recessive allele.
• Examples of Inherited Traits:
• Eye Color: Controlled by multiple genes, with dominant and recessive alleles determining the final color.
• Blood Type: Determined by three alleles (A, B, and O), with combinations like AB, A, B, and O resulting from inheritance from both parents.
• Genetic Disorders: Some conditions, like sickle cell anemia and cystic fibrosis, are inherited when both parents contribute a recessive allele for the disorder.

Important Points

• Gregor Mendel: Known as the "Father of Genetics," Mendel's experiments with pea plants laid the foundation for understanding inheritance patterns.
• Genotype vs. Phenotype: Genotype refers to the genetic makeup of an organism, while phenotype refers to the observable traits resulting from the genotype.
• Heredity and Environment: While genes determine potential traits, environmental factors can influence their expression, such as height being affected by nutrition.
• Mutation: A mutation is a change in the DNA sequence. While some mutations are harmful, others can introduce beneficial traits that aid evolution.
• Sex-linked Traits: Traits like color blindness and hemophilia are linked to genes found on sex chromosomes (X and Y) and are often inherited differently in males and females.

The correct answer is 142 u.

Key Points

• The formula unit mass of a compound is calculated by summing up the atomic masses of all the atoms present in its chemical formula.
• The chemical formula of sodium sulfate is Na₂SO₄, which consists of:
• 2 sodium (Na) atoms, each with an atomic mass of 23 u.
• 1 sulfur (S) atom, with an atomic mass of 32 u.
• 4 oxygen (O) atoms, each with an atomic mass of 16 u.
• Calculation of formula unit mass:
• Mass of sodium: 2 × 23 = 46 u
• Mass of sulfur: 1 × 32 = 32 u
• Mass of oxygen: 4 × 16 = 64 u
• Total formula unit mass: 46 + 32 + 64 = 142 u
• Thus, the calculated formula unit mass of Na₂SO₄ is 142 u.

Additional Information

• Atomic Mass:
• The atomic mass of an element is the average mass of its atoms, typically measured in atomic mass units (u).
• For example:
• Sodium (Na): 23 u
• Sulfur (S): 32 u
• Oxygen (O): 16 u
• Formula Unit Mass:
• The formula unit mass is the sum of the atomic masses of all atoms in a given formula unit of a compound.
• It is expressed in atomic mass units (u).
• Sodium Sulfate (Na₂SO₄):
• Sodium sulfate is an ionic compound composed of sodium cations (Na⁺) and sulfate anions (SO₄^2-).
• It is used in various industries, such as in detergents, paper production, and as a drying agent.
• Applications of Formula Unit Mass:
• It helps in determining the proportion of elements in a compound.
• It is essential for stoichiometric calculations in chemical reactions.

Important Points

• Difference Between Atomic Mass and Molecular Mass:
• Atomic mass: The mass of a single atom of an element.
• Molecular mass: The sum of atomic masses of all atoms in a molecule.
• Units of Measurement:
• The atomic mass unit (u) is based on the carbon-12 isotope, where 1 u = 1/12th the mass of a carbon-12 atom.
• Relevance in Chemistry:
• Understanding formula unit mass is crucial for analyzing chemical formulas and performing stoichiometric calculations.
• It is widely used in academic and industrial chemistry applications.

The correct answer is Few reptiles.

Key Points

• Sex determination in certain reptiles is influenced by environmental temperature during a critical period of embryo development, known as the thermosensitive period (TSP).
• This phenomenon is termed temperature-dependent sex determination (TSD) and is observed in species such as turtles, crocodiles, and some lizards.
• In TSD, the sex of the offspring is not determined by genetic factors but by the temperature at which the eggs are incubated. For example, in many turtle species, warmer temperatures produce females, while cooler temperatures produce males.
• Unlike mammals and birds, which have chromosomal sex determination, these reptiles lack sex chromosomes and instead rely on environmental cues for sex determination.
• This adaptation is thought to be linked to the evolutionary success of these species, as it allows populations to adjust the sex ratio in response to environmental conditions.

Additional Information

• Human beings:
• The sex of humans is determined by chromosomal sex determination, specifically the XX/XY system.
• Males have one X chromosome and one Y chromosome (XY), while females have two X chromosomes (XX).
• This system is not influenced by environmental factors.
• All fishes:
• Some fish species exhibit environmental sex determination (ESD), but it is not universally observed in all fishes.
• For example, in clownfish, sex can change based on social hierarchy rather than temperature.
• However, sex determination in most fishes is genetically determined.
• All birds:
• Birds follow a ZW sex determination system, where males are ZZ and females are ZW.
• This system is genetically determined and is not influenced by environmental temperature.
• Temperature-dependent sex determination (TSD):
• TSD is primarily observed in reptiles such as turtles, crocodiles, and some lizards.
• The critical temperature range that determines sex varies across species. For example, in the American alligator, males are produced at intermediate temperatures, while females are produced at both high and low temperatures.
• This mechanism allows reptiles to adapt their reproductive output to environmental changes.

Important Points

• Significance of TSD:
• TSD plays an important role in the ecology and evolution of reptiles by allowing populations to adjust their sex ratio based on environmental conditions.
• However, rising global temperatures due to climate change could disrupt this balance, leading to skewed sex ratios and threatening the survival of certain species.
• Examples of TSD species:
• Red-eared slider turtle: Females are produced at higher temperatures, while males are produced at lower temperatures.
• American alligator: Males are produced at intermediate temperatures, while females are produced at both high and low temperatures.
• Research and conservation:
• Understanding TSD is crucial for conservation efforts, especially in the context of climate change.
• Artificial incubation and temperature control are used in conservation programs to ensure balanced sex ratios in endangered reptile populations.

The correct answer is A charged atom or molecule.

Key Points

• An ion is a particle, atom, or molecule that carries a net electrical charge due to the loss or gain of electrons. This charge can be either positive (cation) or negative (anion).
• Cations are positively charged ions formed when an atom or molecule loses electrons. Example: Na⁺ (Sodium ion).
• Anions are negatively charged ions formed when an atom or molecule gains electrons. Example: Cl⁻ (Chloride ion).
• Ions play a crucial role in chemical reactions, particularly in ionic bonding and electrochemical processes.
• Electrolytes, substances that produce ions in solution, are essential for biological processes like nerve transmission and muscle contraction.

Additional Information

• Formation of Ions:
• Atoms achieve stability by following the Octet Rule, which states that atoms prefer to have eight electrons in their outer shell.
• To achieve this, atoms may lose or gain electrons, resulting in the formation of ions.
• Example: Sodium (Na) loses one electron to form Na⁺, and Chlorine (Cl) gains one electron to form Cl⁻.
• Types of Ions:
• Monoatomic ions: Ions consisting of only one atom (e.g., K⁺, Mg²⁺).
• Polyatomic ions: Ions consisting of two or more atoms bonded together (e.g., SO₄²⁻, NH₄⁺).
• Applications of Ions:
• In industrial processes, ions are crucial for electroplating, electrolysis, and battery operation.
• In biology, ions like Na⁺, K⁺, and Ca²⁺ are essential for cell signaling and maintaining osmotic balance.
• In water chemistry, ions determine water hardness and pH levels (e.g., carbonate ions in water hardness).
• Role in Ionic Bonding:
• Ionic bonds form when electrons are transferred from one atom to another, creating oppositely charged ions.
• Example: Sodium chloride (NaCl) forms via the transfer of an electron from Na to Cl, resulting in Na⁺ and Cl⁻ ions.

Important Points

• Characteristics of Neutral Molecules:
• Neutral molecules have no net electrical charge as the number of protons equals the number of electrons.
• They can participate in covalent bonding but do not form ions unless chemical or physical conditions cause electron transfer.
• Misconceptions about Ions:
• Ions are not necessarily radioactive or unstable; many ions, such as Na⁺ or Cl⁻, are stable and play important roles in everyday chemistry.
• Atoms with only neutrons do not exist as independent entities; neutrons are subatomic particles found within the nucleus of atoms.
• Examples of Ionic Compounds:
• Common table salt (NaCl) consists of Na⁺ and Cl⁻ ions.
• Magnesium oxide (MgO) forms from Mg²⁺ and O²⁻ ions.
• Calcium carbonate (CaCO₃) contains Ca²⁺ and CO₃²⁻ ions.

The correct answer is 0.5 A.

Key Points

• The given resistors of 4 Ω, 8 Ω, and 12 Ω are connected in series. In a series connection, the total or equivalent resistance is the sum of the individual resistances.
• The formula to calculate the total resistance in a series circuit is:
  

R_total = R₁ + R₂ + R₃.

Substituting the given values:

R_total = 4 Ω + 8 Ω + 12 Ω = 24 Ω.

• The circuit is powered by a 12 V battery. Using Ohm's Law, which is V = IR, we can calculate the current in the circuit. Rearranging the formula:
  

I = V / R.

• Substitute the values for voltage and total resistance:
  

I = 12 V / 24 Ω = 0.5 A.

• Thus, the current in the circuit is 0.5 A, which matches the given correct option.

Additional Information

• Ohm's Law:
• Ohm's Law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor.
• The formula is expressed as: V = IR, where V is voltage, I is current, and R is resistance.
• This law is fundamental in understanding the behavior of electrical circuits.
• Series Circuit:
• In a series circuit, the current remains the same across all components, but the voltage is divided among them.
• The total resistance in a series circuit is the sum of all individual resistances: R_total = R₁ + R₂ + ... + R_n.
• Resistors in Series vs. Parallel:
• In a series connection, resistors add up to increase the total resistance, while in a parallel connection, the total resistance decreases.
• The formula for total resistance in a parallel circuit is:
  

1/R_total = 1/R₁ + 1/R₂ + ... + 1/R_n.

• Electric Current:
• Electric current is the rate of flow of electric charge through a conductor.
• The unit of electric current is the ampere (A), named after French physicist André-Marie Ampère.
• 1 ampere is equivalent to 1 coulomb of charge passing through a point in 1 second.

Important Points

• Power in an Electrical Circuit:
• The power (P) dissipated in an electrical circuit is calculated using the formula: P = VI, where V is voltage and I is current.
• For resistors, power can also be expressed as: P = I²R or P = V²/R.
• Applications of Series Circuits:
• Series circuits are commonly used in applications where the same current is required to flow through multiple components, such as decorative lights.
• However, if one component in the series circuit fails, the entire circuit is interrupted.
• Units of Measurement:
• Voltage: Measured in volts (V).
• Resistance: Measured in ohms (Ω).
• Current: Measured in amperes (A).

The correct answer is Pesticide.

Key Points

• Pesticides are chemical substances used to eliminate or control pests in agricultural practices. These include herbicides, insecticides, fungicides, and rodenticides.
• Pesticides are non-biodegradable and persist in the environment, leading to their accumulation in the food chain. This process is termed biological magnification, where concentrations increase as one moves up the trophic levels.
• The phenomenon of biological magnification is particularly harmful as it affects organisms at the top of the food chain, such as humans, by causing severe health risks like cancer, neurological disorders, and reproductive issues.
• Other organic substances like rose petals, neem bark, and orange peel do not contribute to biological magnification as they are biodegradable and decompose naturally in the environment. Hence, these are incorrect options.
• Examples of pesticides that cause biological magnification include DDT, Aldrin, Heptachlor, and Mercury-based compounds.

Additional Information

• Biological Magnification:
• Definition: Biological magnification refers to the increase in concentration of toxic substances, such as pesticides or heavy metals, as they move up the food chain.
• Impact: Organisms at higher trophic levels, such as predatory birds, fish, and humans, accumulate higher concentrations of toxins, leading to severe ecological and health impacts.
• Examples: Persistent Organic Pollutants (POPs) like DDT and mercury compounds are common substances involved in magnification.
• Global Concerns: The Stockholm Convention on Persistent Organic Pollutants aims to eliminate or restrict the production and use of POPs to prevent biological magnification.
• Pesticides and Environmental Impact:
• Types: Pesticides include chemical categories like organophosphates, carbamates, and pyrethroids.
• Health Risks: Long-term exposure to pesticides can cause cancer, developmental disorders, and endocrine disruption.
• Regulation: Countries have established strict guidelines for pesticide use. For instance, in India, the Insecticides Act of 1968 regulates pesticide production and use.
• Biodegradable Substances:
• Substances like rose petals, neem bark, and orange peel are biodegradable and decompose naturally through microbial action.
• They do not persist in the environment or accumulate in the food chain, making them environmentally safe.
• Neem bark, for example, is widely used as a natural pesticide due to its antibacterial and antifungal properties, but it does not contribute to biological magnification.
• Examples of Biological Magnification:
• The pesticide DDT was widely used until its ecological impact was recognized. It caused eggshell thinning in birds like eagles and falcons, leading to population declines.
• Mercury poisoning in aquatic ecosystems has led to fish contamination, impacting human health when consumed.
• Other examples include toxic accumulation of PCBs (Polychlorinated Biphenyls) in marine food chains.

Important Points

• Stockholm Convention: An international treaty signed in 2001 to eliminate or restrict the use of Persistent Organic Pollutants (POPs).
• Silent Spring: A book by Rachel Carson, published in 1962, which raised awareness about the environmental impact of pesticides, especially DDT.
• DDT Ban: Many countries, including India, banned the use of DDT in agriculture due to its ecological and health hazards.
• Alternative Solutions: The use of biopesticides, integrated pest management (IPM), and organic farming are promoted to reduce reliance on harmful chemical pesticides.

The correct answer is Sodium bicarbonate (baking soda).

Key Points

• Washing soda (sodium carbonate, Na₂CO₃·10H₂O) is prepared by heating sodium bicarbonate (NaHCO₃).
• The process involves thermal decomposition of sodium bicarbonate, which produces sodium carbonate, carbon dioxide (CO₂), and water.
• The chemical equation for this process is as follows:
  

2NaHCO₃ → Na₂CO₃ + CO₂ + H₂O.

• The sodium carbonate formed is then recrystallized to obtain washing soda, which contains 10 molecules of water of crystallization (Na₂CO₃·10H₂O).
• Washing soda is commonly used in cleaning and laundry applications due to its ability to soften water and remove grease and stains.

Additional Information

• Sodium Bicarbonate (Baking Soda):
• It is a white crystalline compound with the formula NaHCO₃.
• It decomposes when heated to release carbon dioxide (CO₂) and water, leaving behind sodium carbonate.
• Commonly used in baking as a leavening agent, in fire extinguishers, and for neutralizing acids.
• Washing Soda (Sodium Carbonate):
• It is a hydrated salt of sodium carbonate with 10 molecules of water (Na₂CO₃·10H₂O).
• Has applications in laundry, cleaning agents, and water softening.
• It is alkaline in nature and is used in various industrial processes, including glass manufacturing.
• Other Options:
• Sodium Chloride: Common salt, primarily used as a seasoning and preservative. Not involved in producing washing soda.
• Calcium Carbonate: Found in limestone and chalk; decomposes to form calcium oxide and carbon dioxide but is unrelated to washing soda preparation.
• Calcium Oxide: Also known as quicklime, used in construction and chemical processes. It does not form washing soda.

Important Points

• Applications of Washing Soda:
• Used as a water softener to remove calcium and magnesium ions in hard water.
• Essential in the production of glass, paper, and detergents.
• Helps in cleaning tough stains and grease in household cleaning tasks.
• Difference Between Washing Soda and Baking Soda:
• Baking soda: Sodium bicarbonate (NaHCO₃), mild alkaline, used in food and as an antacid.
• Washing soda: Sodium carbonate (Na₂CO₃), strong alkaline, used for cleaning and industrial purposes.

The correct answer is Plastic bags.

Key Points

• Biodegradable substances are materials that can decompose naturally through the action of microorganisms like bacteria and fungi.
• Plastic bags are made from synthetic polymers like polyethylene, which are derived from petroleum. These materials are non-biodegradable as they cannot be broken down by natural processes. Hence, they persist in the environment for hundreds of years.
• Spoilt food, waste paper, and vegetable peels are examples of biodegradable substances. They decompose naturally and contribute to the formation of compost or organic matter. Therefore, these options are incorrect.
• The accumulation of plastic waste poses significant environmental hazards, including soil and water pollution, harm to wildlife, and a contribution to microplastic pollution in ecosystems.
• The use of biodegradable alternatives to plastic bags, such as cloth or jute bags, is encouraged to reduce environmental impact.

Additional Information

• Plastic Pollution:
• Plastic pollution occurs when plastic products accumulate in the environment, causing harm to ecosystems and human health.
• Microplastics, which are tiny plastic particles, have been found in oceans, rivers, and even in the food chain, posing a risk to marine life and humans.
• Globally, only a small percentage of plastic waste is recycled, while the rest ends up in landfills or as litter in the environment.
• Biodegradable Substances:
• Examples include food waste, paper, wood, and natural fibers like cotton and wool.
• These materials decompose naturally through biological processes, producing nutrient-rich compost beneficial for soil health.
• The decomposition process reduces environmental waste and supports sustainable waste management practices.
• Non-Biodegradable Substances:
• These are materials that do not decompose naturally, including plastics, metals, and synthetic chemicals.
• They persist in the environment and often require specialized recycling or disposal methods to mitigate their impact.
• The slow breakdown of these materials contributes to long-term environmental challenges, including pollution and habitat destruction.
• Efforts to Reduce Plastic Use:
• Many countries have implemented bans or restrictions on single-use plastics to combat plastic pollution.
• Innovative solutions, such as biodegradable plastics and recycling technologies, are being developed to address the issue.
• Public awareness campaigns encourage individuals to adopt sustainable practices, such as using reusable bags and reducing plastic consumption.
• Government Initiatives in India:
• The Plastic Waste Management Rules, 2016, mandate proper disposal and recycling of plastic waste.
• India pledged to eliminate single-use plastics by 2022 as part of its commitment to environmental sustainability.
• Campaigns like Swachh Bharat Abhiyan promote waste segregation and the use of biodegradable materials.

The correct answer is Nitrogen prevents rancidity..

Key Points

• Nitrogen prevents rancidity: Chips manufacturers flush bags with nitrogen gas to prevent rancidity caused by the oxidation of fats and oils present in the chips. This helps maintain their freshness and taste.
• Oxygen removal: By replacing oxygen with nitrogen, the chances of oxidative reactions that cause spoilage are minimized, as oxygen is a key contributor to oxidation.
• Preservation of quality: Nitrogen acts as an inert gas, which creates a protective atmosphere inside the bag and preserves the flavor, texture, and aroma of the chips.
• Packaging protection: Nitrogen filling also prevents the bag from collapsing or crushing the chips, as it creates a cushioning effect.
• Non-reactive nature: Unlike oxygen, nitrogen is chemically inert, meaning it does not react with the chips or their packaging material, ensuring extended shelf life.

Additional Information

• Rancidity:
• Rancidity is a chemical process where fats and oils are oxidized, leading to an unpleasant smell, taste, and sometimes harmful by-products.
• The oxidation of unsaturated fatty acids is the primary cause of rancidity, and exposure to oxygen accelerates this process.
• Role of nitrogen in food packaging:
• Nitrogen is widely used in the food industry to preserve the freshness and quality of perishable items such as chips, coffee, and nuts.
• It is also used in Modified Atmosphere Packaging (MAP), where the food is stored in an atmosphere of gases like nitrogen and carbon dioxide to extend shelf life.
• Alternative preservation methods:
• Other preservation techniques include vacuum sealing, freezing, and the use of preservatives such as antioxidants (e.g., vitamin E and BHT).
• While these methods are effective, nitrogen flushing is particularly favored for dry snacks due to its simplicity and efficiency.
• Other benefits of nitrogen:
• Nitrogen is non-toxic, odorless, and tasteless, making it a safe choice for food preservation.
• It also prevents the growth of aerobic microorganisms, further ensuring food safety and quality.
• Incorrect options explained:
• Nitrogen acts as a preservative: While nitrogen helps preserve chips, it does not directly act as a chemical preservative like additives; its role is to prevent oxidation.
• Nitrogen prevents reduction of oil: This is incorrect as nitrogen does not prevent the reduction of oil content but rather prevents its oxidation.
• Nitrogen acts as a taste enhancer: Nitrogen does not enhance taste; it simply maintains the original flavor by preventing spoilage.

Important Points

• Industrial use of nitrogen: Nitrogen is used in multiple industries, including electronics, pharmaceuticals, and food packaging, because of its inert nature.
• Environmental impact: Nitrogen flushing is environmentally friendly as it does not release harmful by-products.
• Shelf life improvement: Nitrogen-filled packaging can significantly enhance the shelf life of products by creating a stable, oxygen-free environment.

The correct answer is More work done in less time means more power..

Key Points

• Power is defined as the rate at which work is done or energy is transferred over time. The formula for power is: Power = Work Done / Time.
• When more work is done in less time, the numerator (work done) increases while the denominator (time) decreases, resulting in a higher value of power. Hence, this statement is correct.
• Statement 1: Power and work are not independent of each other, as power is directly dependent on the amount of work done and the time taken. Hence, this statement is incorrect.
• Statement 2: Power does not necessarily decrease when work increases. If the time remains constant or decreases, power will increase. Hence, this statement is incorrect.
• Statement 4: Work done directly affects power as they are mathematically related. Power increases when work increases, provided time remains constant or decreases. Hence, this statement is incorrect.

Additional Information

• Power:
• Power is measured in Watts (W), where 1 Watt equals 1 Joule of work done per second.
• Power can also be expressed in terms of energy transfer: Power = Energy Transferred / Time.
• Work:
• Work is defined as the product of force and displacement in the direction of the force: Work = Force × Displacement × cos(θ), where θ is the angle between force and displacement.
• Work is measured in Joules (J).
• Time:
• Time is a crucial parameter in determining power. A shorter time for the same amount of work results in higher power output.
• Examples of Power:
• A 60-Watt bulb consumes 60 Joules of energy per second.
• A car engine with a power rating of 100 horsepower (~74600 Watts) does more work in a shorter time compared to a less powerful engine.

Important Points

• Relation Between Power, Work, and Time:
• If work increases and time remains constant, power increases proportionally.
• If work remains constant but time decreases, power also increases.
• Units of Power:
• The SI unit of power is the Watt (W).
• Other units include horsepower (1 HP = 746 W) and kilowatts (1 kW = 1000 W).
• Practical Applications:
• In mechanical systems, power ratings determine the efficiency of machines, such as engines and motors.
• In electrical systems, power ratings indicate energy consumption, as seen in household appliances like air conditioners and refrigerators.

The correct answer is It allows propagation of seedless plants such as banana.

Key Points

• Vegetative propagation is a method of plant reproduction that occurs through vegetative parts of the plant such as roots, stems, or leaves, without involving seeds.
• This technique is particularly beneficial for plants like bananas, sugarcane, and potatoes, which are seedless or have seeds that are not viable for propagation.
• Since vegetative propagation produces genetically identical plants to the parent, it helps maintain desirable traits such as taste, size, and disease resistance.
• It is a faster method of reproduction compared to growing plants from seeds, as it skips the germination phase and directly establishes a mature plant.
• Vegetative propagation is widely used in horticulture and agriculture for commercial cultivation of high-yield or disease-resistant varieties.

Additional Information

• Methods of Vegetative Propagation:
• Natural Vegetative Propagation: Occurs through plant structures like runners (e.g., grass), tubers (e.g., potato), rhizomes (e.g., ginger), and bulbs (e.g., onion).
• Artificial Vegetative Propagation: Techniques include grafting, cutting, layering, and tissue culture, which are often used in commercial agriculture.
• Advantages of Vegetative Propagation:
• Ensures uniformity in crop quality, as all offspring are clones of the parent plant.
• Allows cultivation of plants that are sterile or have non-viable seeds, such as bananas, seedless grapes, and some citrus varieties.
• Speeds up the growth and production process as plants bypass the germination and seedling stages.
• Disadvantages of Vegetative Propagation:
• Since the plants are genetically identical, they are vulnerable to the same diseases or pests.
• Does not introduce genetic variation, which can limit adaptability to changing environmental conditions.
• Examples of Plants Propagated Vegetatively:
• Bananas: Propagated using suckers or tissue culture due to their sterility.
• Potatoes: Propagated through tubers that have "eyes."
• Sugarcane: Propagated using cuttings of stems with nodes.
• Ornamental Plants: Roses and hibiscus are often propagated through cuttings.

Important Points

• Explanation of Incorrect Options:
• Option 1: "Plants grown this way take longer to bear fruits." – Incorrect, as vegetative propagation often accelerates fruiting compared to seed-based growth.
• Option 2: "All new plants are genetically different from the parent." – Incorrect, as vegetative propagation produces genetically identical offspring.
• Option 4: "It requires cross-pollination to occur." – Incorrect, as vegetative propagation does not involve pollination or sexual reproduction.
• Applications in Agriculture and Horticulture: Vegetative propagation is a crucial technique for rapidly multiplying desirable plant varieties, ensuring consistent quality and yield in crops.

The correct answer is Hypertonic solution.

Key Points

• A hypertonic solution is one where the concentration of solutes outside the cell is higher than inside the cell. This difference in solute concentration creates an osmotic gradient.
• When a cell is placed in a hypertonic solution, water moves out of the cell through the plasma membrane via osmosis to balance the solute concentration. This results in the cell shrinking, a process known as crenation in animal cells.
• The movement of water occurs because of the principle of osmosis, where water flows from a region of lower solute concentration to a region of higher solute concentration.
• The shrinkage of the cell due to water loss can disrupt its normal functions and, in extreme cases, lead to cell death.
• Hypertonic solutions are commonly used in medical treatments, such as reducing edema or managing certain infections, but they must be administered carefully to avoid cellular damage.

Additional Information

• Hypotonic Solution:
• A hypotonic solution has a lower solute concentration outside the cell compared to the inside.
• In such a solution, water moves into the cell, causing it to swell and potentially burst in a process called lysis.
• Hypotonic solutions are often used in treatments where hydration of cells is required, such as in patients suffering from severe dehydration.
• Isotonic Solution:
• An isotonic solution has an equal concentration of solutes inside and outside the cell, resulting in no net movement of water.
• Isotonic conditions are ideal for maintaining cell shape and function, and such solutions are commonly used in intravenous fluids like saline (0.9% NaCl).
• Osmosis:
• Osmosis is the movement of water molecules across a semipermeable membrane from a region of low solute concentration to a region of high solute concentration.
• This process is critical for maintaining cellular homeostasis and plays a key role in various physiological processes.
• Applications of Hypertonic Solutions:
• Hypertonic saline solutions are used in medical treatments to reduce intracranial pressure or treat pulmonary edema.
• In food preservation, hypertonic solutions like concentrated sugar or salt are used to prevent microbial growth by drawing water out of cells.

Important Points

• Cellular Effects of Different Solutions:
• In a hypertonic solution, cells shrink (crenation in animal cells, plasmolysis in plant cells).
• In a hypotonic solution, cells swell and may burst (lysis in animal cells, turgidity in plant cells).
• In an isotonic solution, cells maintain their normal size and shape.
• Osmoregulation:
• Organisms regulate water and solute concentrations to prevent damage caused by hypertonic, hypotonic, or isotonic conditions.
• For example, freshwater organisms actively expel excess water, while marine organisms regulate salt intake to maintain balance.
• Medical Implications:
• Understanding the effects of hypertonic, hypotonic, and isotonic solutions is critical in administering intravenous (IV) fluids to patients.
• Incorrect use of these solutions can lead to severe conditions like cellular dehydration or swelling.

The correct answer is Gangue.

Key Points

• Gangue refers to the unwanted impurities such as soil, sand, and rocks that are present along with the desired mineral in an ore.
• These impurities do not hold any economic value and must be separated from the ore to extract the valuable minerals.
• The process of removing gangue from ore is done through various techniques such as crushing, grinding, magnetic separation, froth flotation, or gravity separation.
• The presence of gangue increases the cost of extraction and processing of the ore, as additional steps are required to isolate the valuable minerals.
• Examples of gangue materials include silica (SiO₂), alumina (Al₂O₃), and other non-metallic impurities that are found alongside metallic ores like iron, copper, and gold.

Additional Information

• Slag:
• Slag is the byproduct formed during the smelting process of extracting metal from its ore. It consists of flux and impurities that are separated from the molten metal.
• Unlike gangue, slag is formed after the processing of ore and can sometimes be used in construction or cement industries.
• For example, in iron extraction, slag contains calcium silicate formed by the reaction of limestone (flux) with silica impurities.
• Flux:
• Flux is a substance added during smelting to remove impurities or gangue by forming a liquid slag.
• Common flux materials include limestone (CaCO₃) and silica (SiO₂), which react with gangue to form slag.
• Example: In the extraction of iron from hematite, limestone acts as a flux to remove silica impurities.
• Alloy:
• An alloy is a homogeneous mixture of two or more metals or a metal and a non-metal.
• Alloys have enhanced properties like strength, corrosion resistance, and hardness compared to pure metals.
• Examples include steel (iron and carbon), bronze (copper and tin), and brass (copper and zinc).
• Ore:
• An ore is a naturally occurring solid material from which a metal or valuable mineral can be profitably extracted.
• Examples of ores include hematite (iron), bauxite (aluminum), and chalcopyrite (copper).
• Ores consist of two components: valuable minerals and gangue (impurities).

Important Points

• Difference between Gangue and Slag:
• Gangue refers to impurities present in the ore before processing, while slag is a byproduct formed after processing during smelting.
• Gangue is separated through physical or chemical processes, while slag is separated during the metallurgical process.
• Economic Impact of Gangue:
• The presence of gangue in ore increases the cost of extraction due to additional beneficiation processes.
• Industries aim to minimize gangue by selecting high-grade ores to reduce production costs.
• Examples of Processes to Remove Gangue:
• Magnetic Separation: Used to separate magnetic materials (e.g., iron ores) from non-magnetic gangue.
• Froth Flotation: Used for sulfide ores like zinc sulfide, where gangue is separated using frothing agents.
• Gravity Separation: Relies on differences in density between ore and gangue particles.

The correct answer is Option 2.

Key Points

• Statement A: Salts derived from strong acids and strong bases are neutral with a pH of 7.
• A strong acid completely dissociates to release hydrogen ions (H⁺), while a strong base dissociates to release hydroxide ions (OH⁻).
• When combined, their ions neutralize each other, forming water and a neutral salt with no acidic or basic character.
• For example, sodium chloride (NaCl) is formed from hydrochloric acid (HCl) and sodium hydroxide (NaOH), both strong compounds. This salt has a pH of 7 in an aqueous solution.
• Statement A is correct.
• Statement B: Sodium chloride is a neutral salt.
• Sodium chloride (NaCl) is formed by the neutralization reaction between hydrochloric acid (HCl) and sodium hydroxide (NaOH).
• Neither ion of NaCl (Na⁺ and Cl⁻) undergoes hydrolysis in water, keeping the solution neutral.
• Therefore, it is an example of a neutral salt.
• Statement B is correct.
• Both Statements A and B are correct, making Option 2 the correct answer.

Additional Information

• Types of Salts:
• Neutral salts: Formed from strong acids and strong bases; examples include NaCl and KNO₃.
• Acidic salts: Formed from a strong acid and a weak base; examples include NH₄Cl and AlCl₃.
• Basic salts: Formed from a weak acid and a strong base; examples include Na₂CO₃ and CH₃COONa.
• Neutralization Reaction: - This chemical reaction occurs when an acid and a base react to form water and a salt. - Example: HCl + NaOH → NaCl + H₂O
• pH Scale: - The pH scale ranges from 0 to 14, where 7 is neutral. Values below 7 indicate acidity, and values above 7 indicate basicity. - Neutral salts have a pH of 7 because they do not affect the concentration of H⁺ or OH⁻ ions in solution.
• Hydrolysis of Salts: - Salts derived from weak acids or weak bases may undergo hydrolysis, altering the pH of the solution. - For example, NH₄Cl (ammonium chloride) hydrolyzes to form a slightly acidic solution.
• Examples of Neutral Salts: - Sodium chloride (NaCl) - Potassium nitrate (KNO₃) - Calcium chloride (CaCl₂)

Important Points

• Strong Acids: HCl, H₂SO₄, HNO₃, etc.
• Strong Bases: NaOH, KOH, Ca(OH)₂, etc.
• Neutralization Reaction Applications: Used in water treatment, antacid production, and chemical manufacturing.
• pH Monitoring: Essential in industries like agriculture, cosmetics, and pharmaceuticals to ensure product efficacy and safety.

The correct answer is Xylem and phloem are complex tissues that help in conduction and form vascular bundles..

Key Points

• Complex permanent tissues are composed of more than one type of cell, working together to perform a common function. This makes them distinct from simple permanent tissues like parenchyma, collenchyma, and sclerenchyma.
• Xylem is a type of complex tissue responsible for the transportation of water and minerals from roots to other parts of the plant. It consists of different elements like tracheids, vessels, xylem fibers, and xylem parenchyma.
• Phloem is the tissue responsible for transporting food (sugar and organic nutrients) from the leaves to other parts of the plant. Its components include sieve tubes, companion cells, phloem fibers, and phloem parenchyma.
• The combination of xylem and phloem forms vascular bundles, which are essential for the conduction of water, minerals, and food in plants. These bundles also provide mechanical support to the plant body.
• The other options are incorrect as they misrepresent the structure and function of complex permanent tissues. For instance, complex tissues are not made of identical cells, nor are they exclusive to animals.

Additional Information

• Xylem:
• Tracheids: Long, tube-like cells that aid in the conduction of water and provide mechanical support.
• Vessels: Shorter, wider cells aligned end-to-end to form continuous tubes for water conduction.
• Xylem parenchyma: Stores food and helps in lateral conduction of water.
• Xylem fibers: Provide mechanical strength to the plant.
• Phloem:
• Sieve tubes: Cylindrical cells responsible for transporting food.
• Companion cells: Assist sieve tubes in the transport of nutrients and maintain their functionality.
• Phloem parenchyma: Stores food and other substances like resin and latex.
• Phloem fibers: Provide mechanical support to the plant.
• Vascular bundles:
• They are classified into three types based on arrangement: collateral, bicollateral, and radial vascular bundles.
• These bundles are vital for the growth and survival of plants by ensuring the efficient transport of essential substances.
• Incorrect options explained:
• Option 1: Complex tissues are not made of identical cells; they consist of different cell types working together.
• Option 2: Complex tissues are not exclusive to animals; they are found in plants.
• Option 4: Complex tissues like xylem and phloem are tightly packed, not loosely packed like parenchyma cells.

Important Points

• Other plant tissues:
• Simple permanent tissue: Includes parenchyma, collenchyma, and sclerenchyma, each with distinct functions like storage, support, and strength.
• Meristematic tissue: Responsible for plant growth as it contains actively dividing cells.
• Transport in plants:
• Water and minerals are transported through xylem via processes like root pressure, cohesion-tension, and capillary action.
• Food is transported in phloem through a process called translocation, driven by osmotic pressure.
• Applications: Understanding plant tissues is crucial for fields like agriculture, horticulture, and forestry, where plant growth and health are of utmost importance.

The correct answer is Angle of deviation.

Key Points

• The angle of deviation is defined as the angle between the direction of the incident ray and the direction of the emergent ray as it exits a prism.
• When a light ray passes through a prism, it undergoes refraction twice: once when it enters the prism and once when it exits the prism. This bending of light results in the angle of deviation.
• The magnitude of the angle of deviation depends on factors like the angle of incidence, the refractive index of the material of the prism, and the apex angle of the prism.
• For a given prism, there is a specific angle of incidence at which the angle of deviation is minimum. This is called the minimum deviation angle, and it is often used to calculate the refractive index of the prism.
• It is an important concept in optics and is widely studied in the context of prism-based dispersion and light refraction.

Additional Information

• Angle of Prism:
• The angle of prism refers to the angle between the two refracting surfaces of the prism. It is also known as the apex angle.
• The angle of prism plays a significant role in determining the extent of deviation of light passing through the prism.
• Angle of Incidence:
• The angle of incidence is the angle between the incident ray and the normal to the surface at the point of incidence.
• This angle affects the refraction of light and consequently influences the angle of deviation in a prism.
• Critical Angle:
• The critical angle is the angle of incidence above which total internal reflection occurs for light traveling from a denser to a rarer medium.
• It is not directly related to the angle of deviation but is a key concept in understanding refraction and reflection phenomena.
• Applications of Angle of Deviation:
• It is used in the design of optical instruments like spectrometers to measure the refractive index of materials.
• The concept is also applied in the study of dispersion, where light is separated into its constituent colors using a prism.

Important Points

• Light Dispersion:
• The angle of deviation is responsible for the splitting of white light into its constituent colors when it passes through a prism. This phenomenon is called dispersion.
• The deviation angle is different for different wavelengths of light, which leads to the formation of a spectrum.
• Minimum Deviation:
• At minimum deviation, the refracted ray inside the prism is symmetric with respect to the prism's base, and the path of light is at its most efficient.
• This condition is often used in experiments to determine the refractive index of the prism material.

The correct answer is 0.004 s.

Key Points

• The time period of a wave is the reciprocal of its frequency. It is calculated using the formula:
  

Time Period (T) = 1 / Frequency (f)

• The given frequency of the sound wave is 250 Hz.
  

Substituting this into the formula:

T = 1 / 250 = 0.004 seconds.

• Therefore, the time period of the sound wave is 0.004 seconds.
• The unit of frequency is Hertz (Hz), which represents the number of cycles per second. The unit of time period is seconds (s).
• This calculation confirms that the correct option is 0.004 s, which corresponds to Option 1.

Additional Information

• Frequency and Time Period Relationship:
• Frequency (f) and time period (T) are inversely proportional. This means as frequency increases, the time period decreases, and vice versa.
• The mathematical expression for their relationship is:
  

T = 1 / f or f = 1 / T.

• Definition of Frequency:
• Frequency is the number of complete oscillations or cycles a wave completes in one second.
• It is measured in Hertz (Hz). For example, 1 Hz = 1 cycle per second.
• Definition of Time Period:
• The time period is the time taken to complete one oscillation or cycle of a wave.
• It is measured in seconds (s).
• Applications of Frequency and Time Period:
• These concepts are widely used in fields like acoustics, electronics, and signal processing.
• In sound waves, frequency determines the pitch of the sound, while the time period provides insights into the wave's periodicity.
• Real-life Examples:
• A guitar string vibrating at 440 Hz has a time period of approximately 0.00227 seconds.
• The human ear can typically hear frequencies ranging from 20 Hz to 20,000 Hz.

Important Points

• Speed of Sound:
• The speed of sound in air is approximately 343 m/s at room temperature (20°C).
• The speed depends on factors like medium, temperature, and pressure.
• Wavelength and Frequency:
• Wavelength (λ) is inversely proportional to frequency when the speed of the wave is constant:
  

λ = v / f, where v is the speed of the wave.

• Higher frequency waves have shorter wavelengths.
• Practical Use:
• Frequency and time period calculations are vital in designing musical instruments, speakers, and communication devices.
• They are also crucial in medical imaging techniques like ultrasound.

If x² + 1/x² = k, then x + 1/x = √(k + 2).

Given: k = 79

x + 1/x = √(79 + 2) = √81

x + 1/x = 9

∴ The correct answer is 9.

Given: x² + 1/x² = 79

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

⇒ (x + 1/x)² = x² + (1/x)² + 2 × x × (1/x)

⇒ (x + 1/x)² = x² + 1/x² + 2

⇒ (x + 1/x)² = 79 + 2

⇒ (x + 1/x)² = 81

⇒ x + 1/x = ±√81

⇒ x + 1/x = 9 or −9

From the given options, the possible value is 9.

∴ The correct answer is 9.

Reciprocal Subtraction Identity

(x − 1/x)² = x² + 1/x² − 2. Therefore, x − 1/x = √((x² + 1/x²) − 2).

Higher Power Relation

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k. This is derived from the (a + b)³ expansion.

Standard Algebraic Identity

The base for these formulas is (a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab.

Shortcut Trick

Direct formula for Speed = Distance ÷ Time

Speed = 1200 ÷ 3

Speed = 400 km/hr

∴ The correct option is C.

Alternate Method

Given:

Distance traveled by flight = 1200 km

Time taken = 3 hours

Formula Used:

Speed = Distance ÷ Time

Start End  FLIGHT 1200 km 3 hours Speed = 1200 ÷ 3

Calculations:

⇒ Speed = 1200 ÷ 3

⇒ Speed = 400 km/hr

∴ The correct option is 3.

Additional Information

Unit Conversion

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

Average Speed

Average Speed = Total Distance ÷ Total Time. If a journey is divided into two equal distances with speeds x and y, Average Speed = 2xy ÷ (x + y).

Relative Speed

When two objects move in the same direction, Relative Speed = S₁ − S₂. When they move in opposite directions, Relative Speed = S₁ + S₂.

Shortcut Trick

Price Ratio (Lemon : Strawberry) = 22 : 6

Quantity Ratio (Lemon : Strawberry) = 14 : 23

Expenditure Ratio (S : L) = (6 × 23) : (22 × 14)

Ratio = 138 : 308 = 69 : 154

∴ The correct option is C.

Alternate Method

Given:

Price ratio of Lemon (P_L) to Strawberry (P_S) = 22 : 6

Quantity ratio of Lemon (Q_L) to Strawberry (Q_S) = 14 : 23

Formula Used:

Expenditure = Price per unit × Quantity consumed

Lemon (L) Price: 22 units Qty: 14 units Strawberry (S) Price: 6 units Qty: 23 units Exp (L) = 308 Ratio S:L Exp (S) = 138 Final Ratio = 69 : 154

Calculations:

⇒ Expenditure on Lemon = Price_L × Quantity_L

⇒ Expenditure on Lemon = 22 × 14 = 308 units

⇒ Expenditure on Strawberry = Price_S × Quantity_S

⇒ Expenditure on Strawberry = 6 × 23 = 138 units

⇒ Required Ratio = Expenditure (Strawberry) : Expenditure (Lemon)

⇒ Required Ratio = 138 : 308

⇒ Dividing both sides by 2:

⇒ Ratio = 69 : 154

∴ The correct option is C.

Additional Information

Compounding of Ratios

When two ratios a:b and c:d are multiplied, the resulting ratio is (a × c) : (b × d), which is called the compounded ratio.

Expenditure Relationship

If the price changes by a ratio x:y and consumption changes by a ratio m:n, the total expenditure changes by the ratio (x × m) : (y × n).

Ratio Simplification

A ratio a : b remains unchanged if both terms are multiplied or divided by the same non-zero constant k. (a/k : b/k = a : b).

Shortcut Trick

Convert composite bases to prime factors: 6 = 2 × 3 and 10 = 2 × 5.

Denominator = (2 × 3)⁻¹ × (2 × 5)² = 2⁻¹ × 3⁻¹ × 2² × 5² = 2¹ × 3⁻¹ × 5².

Subtract denominator exponents from numerator exponents for like bases: 2³⁻¹, 3^{−2−(−1)}, 5⁴⁻².

Result = 2² × 3⁻¹ × 5².

∴ The correct option is B.

Alternate Method

Given: (2³ × 3⁻² × 5⁴) / (6⁻¹ × 10²)

Formula Used: (a × b)n = an × bn and am ÷ an = am−n

⇒ Expression = (2³ × 3⁻² × 5⁴) / ((2 × 3)⁻¹ × (2 × 5)²)

⇒ Expression = (2³ × 3⁻² × 5⁴) / (2⁻¹ × 3⁻¹ × 2² × 5²)

⇒ Expression = (2³ × 3⁻² × 5⁴) / (2⁻¹⁺² × 3⁻¹ × 5²)

⇒ Expression = (2³ × 3⁻² × 5⁴) / (2¹ × 3⁻¹ × 5²)

⇒ Expression = 2⁽³⁻¹⁾ × 3^{(−2−(−1))} × 5⁽⁴⁻²⁾

⇒ Expression = 2² × 3⁻²⁺¹ × 5²

⇒ Expression = 2² × 3⁻¹ × 5²

∴ The correct option is 2.

Additional Information

Laws of Exponents (Product and Quotient)

For any non-zero base, a^m × aⁿ = a^m+n and a^m ÷ aⁿ = a^m−n.

Negative Exponents

A term with a negative exponent can be moved to the denominator (or vice versa) to make the exponent positive: a⁻ⁿ = 1/aⁿ.

Power of a Product Rule

When a product is raised to a power, each factor is raised to that power: (ab)ⁿ = aⁿ × bⁿ.

Shortcut Trick

When profit is equal to the Selling Price (SP) of x items for n items sold:

Profit % = [x ÷ (n − x)] × 100

Profit % = [20 ÷ (100 − 20)] × 100

Profit % = (20 ÷ 80) × 100 = 25%

In decimal form: 25 ÷ 100 = 0.25

∴ The correct answer is 0.25.

Alternate Method

Given: Total cupcakes sold = 100, Profit = SP of 20 cupcakes.

Formula: Profit = Selling Price (SP) − Cost Price (CP)

Total Selling Price (100 units) Cost Price (80 units) Profit (20 units) Base for % Calculation (CP)

Let the Selling Price (SP) of 1 cupcake = 1 unit.

⇒ Total SP of 100 cupcakes = 100 units.

⇒ Profit = SP of 20 cupcakes = 20 units.

⇒ Cost Price (CP) = SP − Profit

⇒ CP = 100 − 20 = 80 units.

⇒ Profit % = (Profit ÷ CP) × 100

⇒ Profit % = (20 ÷ 80) × 100 = 25%.

⇒ Profit in decimal = 0.25.

∴ The correct answer is 0.25.

Additional Information

Profit based on Cost Price

If profit is equal to the CP of x items while selling n items, then Profit % = (x ÷ n) × 100.

Loss based on Selling Price

If loss is equal to the SP of x items while selling n items, then Loss % = [x ÷ (n + x)] × 100.

Marked Price (MP)

Discount is always calculated on the Marked Price. Selling Price = MP × [1 − (Discount % ÷ 100)].

Shortcut Trick

Apply the Reverse Operation method (backtracking).

Start from the final result 29.

Subtract the added value: 29 − 8 = 21.

Divide by the multiplier: 21 ÷ 3 = 7.

∴ The correct answer is 7.

Alternate Method

Given:

Result = 29

Multiplier = 3

Addition constant = 8

Formula: (Number × 3) + 8 = Result

x × 3 3x + 8 29 Reverse: (29 − 8) ÷ 3

Let the required number be x.

⇒ 3 × x + 8 = 29

⇒ 3x = 29 − 8

⇒ 3x = 21

⇒ x = 21 ÷ 3

⇒ x = 7

∴ The correct answer is 7.

Additional Information

Linear Equations

An equation of the form ax + b = c is a linear equation in one variable where x is the unknown value to be determined.

Order of Operations (BODMAS)

When solving or forming expressions, follow Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Inverse Operations

To isolate the variable, perform the opposite operation: use subtraction for addition and division for multiplication.

Shortcut Trick

Scheme A: Discount = 2 ÷ 5 × 100 = 40%

Scheme B: Discount = 65%

Scheme C: Discount = 31 + 40 − (31 × 40 ÷ 100) = 58.6%

Comparing discounts: 40% (A) < 58.6% (C) < 65% (B)

Ascending order: A, C, B

∴ The correct option is B.

Alternate Method

Given:

Scheme A = Buy 3 get 2 free

Scheme B = Single discount of 65%

Scheme C = Successive discounts of 31% and 40%

Formula Used:

1. Discount % (Buy x get y free) = [y ÷ (x + y)] × 100
2. Equivalent Single Discount for x% and y% = [x + y − (xy ÷ 100)]%

Calculations:

⇒ Discount % for Scheme A = [2 ÷ (3 + 2)] × 100

⇒ Discount % for Scheme A = (2 ÷ 5) × 100 = 40%

⇒ Discount % for Scheme B = 65%

⇒ Discount % for Scheme C = [31 + 40 − (31 × 40 ÷ 100)]%

⇒ Discount % for Scheme C = [71 − 1240 ÷ 100]% = 71 − 12.4 = 58.6%

⇒ Arranging in ascending order: 40% < 58.6% < 65%

⇒ Order: Scheme A < Scheme C < Scheme B

∴ The correct option is B.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the equivalent single discount is given by (a + b − ab/100)%.

Buy X Get Y Free Concept

In this scheme, the customer gets (X + Y) items but pays only for X items. The discount percentage is calculated as [Free Items ÷ Total Items] × 100.

Successive Multiplier Method

For multiple successive discounts, the final Selling Price = Marked Price × (1 − d₁/100) × (1 − d₂/100) × ... × (1 − d_n/100).

Shortcut Trick

Let Cost Price (CP) = 100 units.

Marked Price (MP) = 100 + 36% of 100 = 136 units.

Selling Price (SP) = 100 + 20% of 100 = 120 units.

Discount = MP − SP = 136 − 120 = 16 units.

Discount % = (16 ÷ 136) × 100 = (2 ÷ 17) × 100 ≈ 11.76%.

Decimal value = 11.76%.

∴ The correct option is D.

Alternate Method

Given:

Marked Price (MP) = ₹744

Mark-up Percentage = 36%

Profit Percentage = 20%

Formula Used:

MP = CP × (1 + Markup%/100)

SP = CP × (1 + Profit%/100)

Discount % = [(MP − SP) ÷ MP] × 100

 CP SP MP (744)  Mark-up: +36%  Profit: +20%  Discount: ?

⇒ CP = 744 ÷ (1 + 36/100) = 744 ÷ 1.36

⇒ CP = 547.0588

⇒ SP = CP × (1 + 20/100) = 547.0588 × 1.20

⇒ SP = 656.4705

⇒ Discount = MP − SP = 744 − 656.4705 = 87.5295

⇒ Discount % = (87.5295 ÷ 744) × 100

⇒ Discount % ≈ 11.7647%

⇒ In decimal form = 11.76%

∴ The correct option is D.

Additional Information

Relation between CP and MP

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

Marked Price Formula

MP = SP + Discount = CP + Mark-up amount. Mark-up is always calculated on the Cost Price.

Successive Discounts

If two successive discounts d₁ and d₂ are given, the single equivalent discount is (d₁ + d₂ − (d₁ × d₂)/100)%.

Shortcut Trick

Using the formula: (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

(30 × 4) / 4140 = (55 × 10) / W₂

120 / 4140 = 550 / W₂

W₂ = (550 × 4140) / 120 = 550 × 34.5

W₂ = 18975

∴ The correct answer is 18975.

Alternate Method

Given:

Initial workers (M₁) = 30

Initial days (D₁) = 4

Initial earnings (W₁) = ₹4140

Final workers (M₂) = 55

Final days (D₂) = 10

Formula Used:

The total earning is directly proportional to the number of workers and the number of days: W ∝ M × D

⇒ (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂

Case 1 30 Workers 4 Days ₹4140 Case 2 55 Workers 10 Days ₹18975 Earning ∝ (Workers × Days)

Calculations:

⇒ (30 × 4) / 4140 = (55 × 10) / W₂

⇒ 120 / 4140 = 550 / W₂

⇒ 1 / 34.5 = 550 / W₂

⇒ W₂ = 550 × 34.5

⇒ W₂ = 18975

∴ The correct answer is 18975.

Additional Information

Chain Rule Formula

The general formula is (M₁×D₁×H₁)/W₁ = (M₂×D₂×H₂)/W₂, where M = men, D = days, H = hours, and W = work or wages.

Direct Proportion

Wages are directly proportional to the total work done. If number of workers or days increase, the total wage also increases linearly.

Efficiency Aspect

If efficiencies are different, include it in the numerator: (M₁×D₁×E₁)/W₁ = (M₂×D₂×E₂)/W₂.

Shortcut Trick

Teacher’s age = New Average + (Initial Number × Increase in Average)

New Average = 28, Initial Students = 25, Increase = 28 − 27 = 1

Teacher’s age = 28 + (25 × 1)

Teacher’s age = 28 + 25 = 53 years

∴ The correct answer is 53.

Alternate Method

Given:

Number of students = 25

Average age of students = 27 years

Average age including teacher = 28 years

Formula Used:

Average = Sum of values ÷ Total number of values

Sum of values = Average × Total number of values

Initial Group Students: 25 Avg Age: 27 + Teacher New Group Total People: 26 Avg Age: 28 Teacher Age = (26 × 28) − (25 × 27)

⇒ Initial sum of ages of 25 students = 25 × 27 = 675 years

⇒ Total number of people including teacher = 25 + 1 = 26

⇒ New sum of ages including teacher = 26 × 28

⇒ New sum = 728 years

⇒ Teacher’s age = New sum − Initial sum

⇒ Teacher’s age = 728 − 675

⇒ Teacher’s age = 53 years

∴ The correct answer is 53.

Additional Information

Deviation Method

If the average increases by x when a new member joins, the new member's value = Old Average + New Count × x.

Weighted Average

The combined average of two groups is given by (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n is count and A is average.

Effect of Constant Change

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

Shortcut Trick

Let Income = 100 units, Savings = 22.5 units, Expenditure = 77.5 units.

Income increase = 14% of 100 = 14 units.

Expenditure increase = 40% of 77.5 = 31 units.

Savings change = 14 − 31 = −17 units (Decrease).

Given 100 units = ₹68,500 ⇒ 17 units = 17 × 685 = ₹11,645.

∴ The correct option is C.

Alternate Method

Given:

Income = ₹68,500, Savings Percentage = 22.5%

Formula Used:

Income = Expenditure + Savings

Income: ₹68,500 (100%) Exp: 77.5% Sav: 22.5% Income ↑ 14% (₹9,590) | Exp ↑ 40% (₹21,235) Savings Decrease = ₹11,645

Calculations:

⇒ Original Savings = 22.5% of 68,500 = (22.5 ÷ 100) × 68,500 = ₹15,412.5

⇒ Original Expenditure = 68,500 − 15,412.5 = ₹53,087.5

⇒ Increase in Income = 14% of 68,500 = (14 ÷ 100) × 68,500 = ₹9,590

⇒ Increase in Expenditure = 40% of 53,087.5 = (40 ÷ 100) × 53,087.5 = ₹21,235

⇒ Change in Savings = Increase in Income − Increase in Expenditure

⇒ Change in Savings = 9,590 − 21,235 = −₹11,645

⇒ The negative value denotes a decrease in savings.

∴ The correct option is C.

Additional Information

Income-Expenditure Identity

The fundamental relation is Income = Expenditure + Savings. Any change in two components directly determines the change in the third.

Net Change in Savings

Change in Savings = (Income × Income increase %) − (Expenditure × Expenditure increase %). If the result is negative, savings have decreased.

Percentage Change Formula

Percentage change in savings = (Change in Savings ÷ Original Savings) × 100. This helps in identifying the growth or reduction rate.

Shortcut Trick

Efficiency: A = 1/2, B = 1/5, C = 1/20

Total efficiency = 1/2 + 1/5 + 1/20 = 10/20 + 4/20 + 1/20 = 15/20 = 3/4

Time for 1 work = 1 ÷ (3/4) = 4/3 days

Time for 3 works = 3 × (4/3) = 4 days

∴ The correct answer is 4 days.

Alternate Method

⇒ Total efficiency = 10 + 4 + 1 = 15 units/day

⇒ Time for 1 work = 20 ÷ 15 = 4/3 days

⇒ Time for 3 works = 3 × (4/3) = 4 days

∴ The correct answer is 4 days.

Additional Information

LCM Method

Take LCM of days as total work. Efficiency = Total Work ÷ Days.

Combined Work Formula

If efficiencies are a, b, c then total = a + b + c and Time = Total Work ÷ Combined Efficiency.

Multiple Work Concept

If 1 work takes T days, then n works take n × T days.

Efficiency Relation

Efficiency is inversely proportional to time: Efficiency ∝ 1/Time.

Shortcut Trick

Net % change = a + b + ab/100 where a = +18, b = −22

= 18 + (−22) + (18 × −22)/100

= 18 − 22 − 396/100

= −4 − 3.96 = −7.96%

Negative sign means decrease.

∴ The correct answer is 7.96% decrease.

Alternate Method

Given:

First change = +18% (increase)

Second change = −22% (decrease)

Formula used:

Net % change = a + b + (a × b) ÷ 100

Where a = first % change | b = second % change

Use + for increase and − for decrease.

Original ₹100 +18% After +18% ₹118 −22% After −22% ₹92.04 Net change = 92.04 − 100 = −7.96% a + b + ab/100 = 18 − 22 − 3.96 = −7.96% Decrease of 7.96%

Calculations:

Let original price = ₹100

⇒ After 18% increase = 100 × 118/100 = ₹118

⇒ After 22% decrease on ₹118 = 118 × 78/100 = ₹92.04

⇒ Net change = 92.04 − 100 = −7.96

⇒ % change = −7.96%

Using direct formula: a + b + ab/100 = 18 + (−22) + (18 × −22)/100

⇒ = 18 − 22 − 396/100

⇒ = −4 − 3.96

⇒ = −7.96%

Since value is negative ⇒ it is a decrease.

∴ The correct answer is 7.96% decrease.

Additional Information

Successive Percentage Change — Direct Formula

For two successive % changes a and b: Net % = a + b + ab/100. Use + for increase and − for decrease. Example: +20% then −20% ⇒ net = 20 − 20 + (20 × −20)/100 = −4% — always results in a net decrease when same % is applied both ways.

Multiplying Factor Method

x% increase → multiply by (100+x)/100 | x% decrease → multiply by (100−x)/100. For this question: 100 × 118/100 × 78/100 = 92.04. Net factor = 0.9204 ⇒ decrease of 7.96%. This method works for any number of successive changes.

Three Successive % Changes

For three changes a, b, c: apply the formula twice. First find net of a and b ⇒ then apply formula again with c. Or use multiplying factor method directly: Original × (1±a/100) × (1±b/100) × (1±c/100). This is the fastest method for three or more changes.

% Increase Needed to Recover After Decrease

If price drops by x%, the % increase needed to recover = x/(100−x) × 100. For this question final price is 92.04% of original — to recover: 7.96/92.04 × 100 ≈ 8.65% increase is needed. Recovery % is always greater than the decrease %.

Shortcut Trick

Profit Ratio of Amit : Total = 2,900 : 8,700 = 1 : 3

Since time is constant, Investment Ratio = Profit Ratio.

Amit’s Investment = (1/3) × Total Investment

Amit’s Investment = 46,800 ÷ 3 = ₹15,600

∴ The correct option is C.

Alternate Method

Given:

Total Investment = ₹46,800

Total Profit = ₹8,700

Amit’s Profit Share = ₹2,900

Formula Used:

Profit Ratio = (Investment₁ × Time₁) : (Investment₂ × Time₂)

Amit: ₹2,900 Deeksha: ₹5,800 Profit Ratio = 2900 : 5800 = 1 : 2 Total Investment = ₹46,800 Amit’s Share = 1/3 of 46,800

Calculations:

⇒ Deeksha’s Profit = Total Profit − Amit’s Profit

⇒ Deeksha’s Profit = 8,700 − 2,900 = ₹5,800

⇒ Ratio of Profit (Amit : Deeksha) = 2,900 : 5,800

⇒ Ratio of Profit (Amit : Deeksha) = 1 : 2

⇒ Since time period is same, Investment Ratio = Profit Ratio = 1 : 2

⇒ Sum of ratio parts = 1 + 2 = 3

⇒ 3 units = ₹46,800

⇒ 1 unit = 46,800 ÷ 3 = ₹15,600

⇒ Amit’s Investment = 1 unit = ₹15,600

∴ The correct option is C.

Additional Information

Simple Partnership

When partners invest their capital for the same duration, the profit or loss is divided in the ratio of their capital (P₁ : P₂ = C₁ : C₂).

Compound Partnership

When partners invest capital for different time periods, profit is divided in the ratio of (Capital × Time), i.e., P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Ratio and Proportion in Partnership

Total Investment = Individual Investment × (Total Profit ÷ Individual Profit Share), provided time is constant for all partners.

Shortcut Trick

Ratio 5 yrs ago = 7 : 2 (diff 5) | Ratio 5 yrs hence = 3 : 1 (diff 2)

Equalize difference: (7:2) × 2 = 14 : 4 and (3:1) × 5 = 15 : 5

Increase of 1 unit (15−14 or 5−4) = 10 years (5 years ago to 5 years hence)

Newer building age 5 years ago = 4 units = 4 × 10 = 40 years

Present age = 40 + 5 = 45 years

∴ The correct answer is 45 years.

Alternate Method

Given:

Ratio 5 years ago = 7 : 2

Ratio 5 years hence = 3 : 1

Formula Used:

Age = Ratio value × Constant factor (x)

 5 Yrs Ago Present 5 Yrs Hence  7 : 2  3 : 1  Total Gap = 10 Yrs

Let the ages of the two buildings 5 years ago be 7x and 2x.

Present ages = (7x + 5) and (2x + 5)

Ages 5 years hence = (7x + 5 + 5) and (2x + 5 + 5) = (7x + 10) and (2x + 10)

According to the question:

⇒ (7x + 10) / (2x + 10) = 3 / 1

⇒ 7x + 10 = 3 × (2x + 10)

⇒ 7x + 10 = 6x + 30

⇒ 7x − 6x = 30 − 10

⇒ x = 20

Age of newer building 5 years ago = 2x

⇒ 2 × 20 = 40 years

Present age of newer building = 40 + 5 = 45 years

∴ The correct answer is 45 years.

Additional Information

Constant Age Difference

The difference between the ages of two individuals or objects always remains constant, regardless of how many years pass.

Ratio Balancing Method

To solve age problems quickly, multiply the ratios so that the numerical difference between the ratio parts (e.g., 7−2=5 and 3−1=2) becomes equal.

Time Gap Calculation

Total time gap between 'n' years ago and 'm' years hence is always (n + m) years.

Shortcut Trick

Equate volumes: Volume of Hollow Sphere = Volume of Cone

⇒ (4/3) × π × (R³ − r³) = (1/3) × π × r_cone² × h

⇒ 4 × (10³ − 6³) = 8² × h

⇒ 4 × (1000 − 216) = 64 × h

⇒ 4 × 784 = 64 × h ⇒ 784 = 16 × h

⇒ h = 49 cm

∴ The correct answer is 49 cm.

Alternate Method

Given:

External radius of sphere (R) = 10 cm

Internal radius of sphere (r) = 6 cm

Base radius of cone (r_c) = 8 cm

Value of π = 22/7

R=10  r=6 Hollow Sphere   Recast    r=8  h=? Solid Cone

Formula Used:

Volume of hollow sphere = (4/3) × π × (R³ − r³)

Volume of cone = (1/3) × π × r² × h

Calculations:

⇒ Volume of hollow sphere = (4/3) × (22/7) × (10³ − 6³)

⇒ Volume of hollow sphere = (4/3) × (22/7) × (1000 − 216)

⇒ Volume of hollow sphere = (4/3) × (22/7) × 784

⇒ Volume of cone = (1/3) × (22/7) × 8² × h

⇒ (4/3) × (22/7) × 784 = (1/3) × (22/7) × 64 × h

⇒ 4 × 784 = 64 × h

⇒ 3136 = 64 × h

⇒ h = 3136 / 64

⇒ h = 49 cm

∴ The correct answer is 49 cm.

Additional Information

Volume of Solid Sphere

The volume of a solid sphere with radius r is given by V = (4/3) × π × r³.

Curved Surface Area of Cone

The curved surface area (CSA) of a cone is π × r × l, where l = √(r² + h²) is the slant height.

Total Surface Area of Hollow Sphere

The total surface area of a hollow sphere is the sum of inner and outer surface areas: 4 × π × (R² + r²).

Volume of Cylinder

A solid cylinder has a volume equal to π × r² × h, which is three times the volume of a cone with the same base and height.

Shortcut Trick

Volume = 100 × 80 × 120 = 9,60,000 cm³

Outlet Rate = 9,60,000 ÷ 200 min = 4800 cm³/min

Net Emptying Rate = 4800 − 2000 = 2800 cm³/min

Time = 9,60,000 ÷ 2800 = 40/7 hours = 5 ⁵/₇ hours

∴ The correct option is A.

Alternate Method

Given:

Dimensions = 100 cm × 80 cm × 120 cm

Inlet rate = 2000 cm³/min

Outlet time = 3 hours 20 minutes

Formula Used:

Volume of Cistern = Length × Breadth × Height

Emptying Rate = Total Volume ÷ Time to empty

Net Rate (Emptying) = Outlet Rate − Inlet Rate

Inlet: 2000 cm3/min Outlet: 4800 cm3/min Vol: 9,60,000 cm3

Calculations:

⇒ Volume of the cistern = 100 × 80 × 120 = 9,60,000 cm³

⇒ Time taken by outlet pipe to empty = 3 × 60 + 20 = 200 min

⇒ Rate of outlet pipe = 9,60,000 ÷ 200 = 4800 cm³/min

⇒ Rate of inlet pipe = 2000 cm³/min

⇒ Net emptying rate = 4800 − 2000 = 2800 cm³/min

⇒ Time required to empty the full cistern = 9,60,000 ÷ 2800 = 2400/7 min

⇒ Time in hours = (2400 ÷ 7) × (1 ÷ 60) = 40/7 hours

⇒ Time = 5 ⁵/₇ hours

∴ The correct option is A.

Additional Information

Net Rate Logic

When both pipes are open and the tank empties, the net rate of emptying is calculated as (Outlet Rate − Inlet Rate). If the result is positive, the tank will eventually empty.

Volume to Rate Relationship

Rate is defined as the volume of liquid flowing per unit time. Rate = Total Volume ÷ Total Time. In this case, 1 cm³ is equivalent to 1 milliliter (ml).

Mixed Fraction Conversion

To convert an improper fraction like 40/7 to a mixed fraction, divide 40 by 7 to get the quotient (5) and the remainder (5), written as 5 ⁵/₇.

Shortcut Trick

Inverse Proportion: More pupils → Fewer days.

Product constant: M₁ × D₁ = M₂ × D₂

42 × 12 = 63 × Days

Days = (42 × 12) ÷ 63 = 504 ÷ 63 = 8 days

∴ The correct option is A.

Alternate Method

Given: Initial Pupils (M₁) = 42, Initial Days (D₁) = 12, New Pupils (M₂) = 63

Formula: Total Provisions = Number of Pupils × Number of Days

42 Pupils → 12 Days 63 Pupils → x Days Total Provisions = 42 × 12 = 504 pupil-days ⇒ M₁ × D₁ = M₂ × D₂ ⇒ 42 × 12 = 63 × D₂ ⇒ 504 = 63 × D₂ ⇒ D₂ = 504 ÷ 63 ⇒ D₂ = 8

∴ The correct option is A.

Additional Information

Inverse Proportion

When one value increases while the other decreases such that their product remains constant. Formula: x₁ × y₁ = x₂ × y₂.

Work & Time Relationship

The number of workers and the time taken to complete a fixed amount of work are inversely proportional.

Chain Rule

General formula for work problems: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where H is hours and W is work.

Shortcut Trick

By British Flag Theorem: PA² + PC² = PB² + PD²

⇒ 86² + 79² = 49² + PD²

⇒ 7396 + 6241 = 2401 + PD²

⇒ 13637 − 2401 = PD²

⇒ PD = √11236 = 106 cm

∴ The correct option is 1.

Alternate Method

Given:

Rectangle ABCD with internal point P.

PA = 86 cm, PB = 49 cm, PC = 79 cm.

Formula Used:

British Flag Theorem: For any point P inside or outside a rectangle ABCD, the sum of the squares of the distances from P to opposite vertices is equal.

PA² + PC² = PB² + PD²

   A B C D P 86 cm 49 cm 79 cm PD = ?

Calculations:

⇒ PA² + PC² = PB² + PD²

⇒ 86² + 79² = 49² + PD²

⇒ 7396 + 6241 = 2401 + PD²

⇒ 13637 = 2401 + PD²

⇒ PD² = 13637 − 2401

⇒ PD² = 11236

⇒ PD = √11236

⇒ PD = 106 cm

∴ The correct option is A.

Additional Information

British Flag Theorem

This theorem states that for any point P and rectangle ABCD, PA² + PC² = PB² + PD². It holds true even if P is outside the rectangle or on its boundary.

Rectangle Diagonals

The diagonals of a rectangle are equal in length and bisect each other. Diagonal length d = √(l² + w²).

Pythagoras Theorem in Rectangles

Any diagonal divides the rectangle into two congruent right-angled triangles where the diagonal acts as the hypotenuse.

Shortcut Trick

Rate difference per annum = 7% − 3.5% = 3.5%

Total difference for 2 years = 3.5% × 2 = 7%

Required difference in Interest = 7% of 4,80,000

Difference = (7 ÷ 100) × 4,80,000 = 7 × 4800 = 33,600

∴ The correct answer is 33600.

Alternate Method

Given:

Principal (P) = ₹4,80,000 for each bank

Rate of Bank A (R_A) = 3.5% p.a.

Rate of Bank B (R_B) = 7% p.a.

Time (T) = 2 years

Formula Used:

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

Year 0 (P) Year 2 (T) ₹4,80,000  Bank B (7%)  Bank A (3.5%)  Interest Diff

Calculations:

⇒ SI paid to Bank A = (4,80,000 × 3.5 × 2) ÷ 100

⇒ SI paid to Bank A = 4800 × 7 = 33,600

⇒ SI paid to Bank B = (4,80,000 × 7 × 2) ÷ 100

⇒ SI paid to Bank B = 4800 × 14 = 67,200

⇒ Positive Difference = 67,200 − 33,600 = 33,600

∴ The correct answer is 33600.

Additional Information

Difference in Simple Interest

When Principal (P) and Time (T) are constant, the difference in SI is given by ΔSI = (P × ΔR × T) ÷ 100, where ΔR is the difference in rates.

Total Amount in SI

The total amount (A) to be paid back is the sum of Principal and Interest: A = P + SI = P(1 + RT/100).

Effective Annual Rate

The effective interest over T years at rate R% p.a. in simple interest is simply (R × T)% of the Principal.

Volume of layer = 4/3 × π × (R³ − r³)

R = 7.5 cm, r = 7 cm

Volume = 4/3 × 22/7 × (7.5³ − 7³)

Volume = 88/21 × (421.875 − 343) = 88/21 × 78.875

Volume ≈ 330.52 → 331 cm³

∴ The correct answer is 331 cm³.

Alternate Method

Given:

Radius of solid sphere (r) = 7 cm

Radius with paint layer (R) = 7.5 cm

π = 22/7

Formula Used:

Volume of spherical shell = 4/3 × π × (R³ − r³)

r = 7 cm  R = 7.5 cm Painted Layer Volume = Outer − Inner

Calculations:

⇒ Volume = 4/3 × 22/7 × (7.5³ − 7³)

⇒ Volume = 88/21 × (421.875 − 343)

⇒ Volume = 88/21 × 78.875

⇒ Volume = 6941 ÷ 21

⇒ Volume ≈ 330.5238 cm³

Rounding off to the nearest integer, we get 331 cm³.

∴ The correct answer is 331 cm³.

Additional Information

Volume of a Sphere

The amount of space occupied by a sphere is given by V = 4/3 × πr³.

Surface Area of a Sphere

The total surface area of a sphere is given by A = 4πr².

Spherical Shell

A spherical shell is the region between two concentric spheres. Its volume is V = 4/3 × π(R³ − r³), where R is the outer radius and r is the inner radius.

Hemisphere Properties

Volume of hemisphere = 2/3 πr³ and Total Surface Area = 3πr².

Shortcut Trick

Assume base average = 84.

Deviation of 17 students = 17 × (94 − 84) = +170.

Average deviation for all 69 students = 170 ÷ 69 ≈ 2.46.

Class average = 84 + 2.46 = 86.46 ≈ 86.5.

∴ The correct answer is 86.5.

Alternate Method

Given:

Total students = 69

Students in Group 1 (n₁) = 17, Average score (A₁) = 94

Students in Group 2 (n₂) = 69 − 17 = 52, Average score (A₂) = 84

Formula Used:

Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂)

Group 1 n = 17 Avg = 94 Group 2 n = 52 Avg = 84 Total Class (N = 69)

Calculations:

⇒ Total score of Group 1 = 17 × 94 = 1598

⇒ Total score of Group 2 = 52 × 84 = 4368

⇒ Combined total score = 1598 + 4368 = 5966

⇒ Average score of class = 5966 ÷ 69

⇒ Average score ≈ 86.463

⇒ Rounding to one decimal place = 86.5

∴ The correct answer is 86.5.

Additional Information

Weighted Average Formula

The combined average of two or more groups is given by A = (n₁A₁ + n₂A₂ + ...) ÷ (n₁ + n₂ + ...).

Net Deviation Principle

The sum of deviations of all observations from their arithmetic mean is always zero. This is used in the shortcut deviation method.

Rounding Rule

When rounding to one decimal place, if the second decimal digit is 5 or more, the first decimal digit is increased by 1.

Shortcut Trick

Total distance to cover = Sum of lengths of both trains = 299 + 101 = 400 m

Relative speed in m/s = Total distance ÷ Time = 400 ÷ 12 = 100/3 m/s

Relative speed in km/hr = (100/3) × (18/5) = 120 km/hr

In opposite direction, Relative Speed = Speed₁ + Speed₂

Speed₂ = 120 − 71 = 49 km/hr

∴ The correct answer is 49 km/hr.

Alternate Method

Given:

Length of Train 1 (L₁) = 299 m

Length of Train 2 (L₂) = 101 m

Time taken (T) = 12 seconds

Speed of Train 1 (S₁) = 71 km/hr

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Speed = Distance / Time

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

Train 1: 299 m 71 km/hr Train 2: 101 m x km/hr Total Distance = 299 + 101 = 400 m Time = 12 seconds

Calculations:

⇒ Total Distance = L₁ + L₂ = 299 + 101 = 400 m

⇒ Relative Speed = Total Distance / Time

⇒ Relative Speed = 400 / 12 m/s

⇒ Relative Speed in km/hr = (400 / 12) × (18 / 5)

⇒ Relative Speed = (100 / 3) × (18 / 5)

⇒ Relative Speed = 20 × 6 = 120 km/hr

⇒ Since they are moving in opposite directions: S₁ + S₂ = 120

⇒ 71 + S₂ = 120

⇒ S₂ = 120 − 71

⇒ S₂ = 49 km/hr

∴ The correct answer is 49 km/hr.

Additional Information

Relative Speed in Opposite Direction

When two objects move towards each other, their relative speed is the sum of their individual speeds: V_rel = V₁ + V₂.

Relative Speed in Same Direction

When two objects move in the same direction, their relative speed is the difference of their individual speeds: V_rel = |V₁ − V₂|.

Speed Conversion

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

Crossing Objects

When a train crosses another train or a bridge, the total distance covered is the sum of the lengths of both entities.

Shortcut Trick

Let the original number be 100%.

Given: 20% of number + 72 = 100% of number.

⇒ 72 = 100% − 20% = 80%.

⇒ 80% of the number = 72.

⇒ 60% of the number = (72 ÷ 80) × 60 = 0.9 × 60 = 54.

∴ The correct answer is 54.

Alternate Method

Given:

20% of a number added to 72 equals the number itself.

Formula Used:

Value = (Percentage ÷ 100) × Number

20% 80% (Value = 72) Total Number = 100%

Calculations:

⇒ Let the number be x.

⇒ (20/100) × x + 72 = x

⇒ 0.2x + 72 = x

⇒ 72 = x − 0.2x

⇒ 72 = 0.8x

⇒ x = 72 ÷ 0.8

⇒ x = 90

⇒ 60% of x = (60/100) × 90

⇒ 60% of x = 0.6 × 90 = 54

∴ The correct answer is 54.

Additional Information

Percentage to Fraction Conversion

To solve quickly, convert percentages to fractions: 20% = 1/5, 60% = 3/5, 80% = 4/5. Here, (4/5) × x = 72 gives x = 90.

Base Value Concept

Always identify the 100% base. If a part is given, the remaining part (100% − given%) corresponds to the added or subtracted numerical value.

Successive Percentage Formula

For two successive changes of a% and b%, the net change is [a + b + (a × b)/100]%.

Shortcut Trick

If A + B = 90°, then sin A / cos B = 1 and cos A / sin B = 1.

Since 15° + 75° = 90° ⇒ sin 15° / cos 75° = 1.

Since 78° + 12° = 90° ⇒ cos 78° / sin 12° = 1.

Expression = 1 + 1 − 1 = 1.

∴ The correct answer is 1.

Alternate Method

Given:

Expression: (sin 15° / cos 75°) + (cos 78° / sin 12°) − 1

Formula Used:

sin θ = cos(90° − θ)

cos θ = sin(90° − θ)

Calculations:

⇒ sin 15° = cos(90° − 15°) = cos 75°

⇒ cos 78° = sin(90° − 78°) = sin 12°

⇒ Value = (cos 75° / cos 75°) + (sin 12° / sin 12°) − 1

⇒ Value = 1 + 1 − 1

⇒ Value = 1

∴ The correct answer is 1.

Additional Information

Complementary Angle Ratios

Trigonometric ratios of complementary angles satisfy: sin θ = cos(90° − θ), tan θ = cot(90° − θ), and sec θ = cosec(90° − θ).

Sum and Difference of Ratios

When the sum of two angles is 90°, the ratio of their sine and cosine (or tangent and cotangent) is always equal to 1.

Fundamental Identities

The three basic identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Given number: 4651783

According to the question, 

So, the sum of the highest digit and the second from the left and first from the right in the new number thus formed → 8 + 4 + 4 = 16 

Hence, "Option 1" is the correct answer.

Decoding signs:

Given: 12 - 15 ÷ 143 + 98 x 7 = ?

12 - 15 ÷ 143 + 98 × 7 = ? ⇒ 12 × 15 + 143 - 98 ÷ 7 = ?

⇒ 12 × 15 + 143 - 98 ÷ 7 = ?

⇒ 12 × 15 + 143 - 14 = ?

⇒ 180 + 143 - 14 = ?

⇒ 323 - 14 = ?

⇒ 309 = ?

Hence, "Option 4" is the correct answer.

According to the question:

Thus, 'E' is coded as '3'.

Hence, "Option 3" is the correct answer.

The logic followed here is:

Hence, "Option 2" is the correct answer.

Alternate Method

Given: A, B, C, D, W, X and Y are sitting around a circular table facing the centre of the table.

1. Only four people sit between B and A when counted from the right of B.

2. B sits third to the left of X.

3. W sits third to the left of Y.

4. C is an immediate neighbour of both A and X.

The remaining position will be occupied by the only person left i.e. D.

Thus, two people sit between D and C when counted from the right of D.

Hence, "Option 3" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

(I) Some rabbits are cars → Does not follow (As some rabbits are fans and no fan is a car. So it is possible but not definite.)

(II) Some fans are rabbits → Follow (As some rabbits are fans. As some part of rabbit is common with fans, therefore it is true.)

∴ Here, Only conclusion (II) follows.​

Hence, the correct answer is "Option 1".

Given series: (Left) ? * + ? \ & 3 # 9 % 4 1 6 * @ 5 4 8 8 $ \ (Right)

According to the question:

Required condition: Number - Number - Symbol

(Left) ? * + ? \ & 3 # 9 % 4 **1 6 *** @ 5 4 8 8 $ \ (Right)

Thus, there are two such numbers which are immediately preceded by another number and also immediately followed by a symbol.

Hence, "Option 2" is the correct answer.

The logic followed here is:

Option 1) VS-DV

Option 2) LI-SL

Option 3) DA-KD

Option 4) MJ-TM

Thus, "VS-DV" is the odd one out.

Hence, "Option 1" is the correct answer.

Given: O, P, Q, R, W, X and Y are sitting in a straight line facing north.

1. Only three people sit to the left of P.

2. Only R sits to the right of Q.

3. Only three people sit between Q and O.

4) X sits at some place to the left of Y but at some place to the right of W.

Thus, three people sit between W and Y.

Hence, "Option 4" is the correct answer.

Decoding signs:

Given: 14 B 2 D 10 A 5 C 12 = ?

14 B 2 D 10 A 5 C 12 = ? ⇒ 14 ÷ 2 + 10 × 5 - 12 = ?

⇒ 14 ÷ 2 + 10 × 5 - 12 = ?

⇒ 7 + 10 × 5 - 12 = ?

⇒ 7 + 50 - 12 = ?

⇒ 57 - 12 = ?

⇒ 45 = ?

Hence, "Option 2" is the correct answer.

The logic followed here is:

Option 1) HL-OJ

Option 2) FJ-MH

Option 3) DH-KF

Option 4) BF-IE

Thus, "BF-IE" is the odd one out.

Hence, "Option 4" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

(I) Some stories are books → Follow (As some books are novels and all novels are stories. As some part books is common with novels which whole comes in stories, therefore it is true.)

(II) No books are stories → Does not follow (As some books are novels and all novels are stories. As no direct relation is given between books and stories, therefore it is false.)

∴ Here, Only conclusion (I) follows.​

Hence, the correct answer is "Option 2".

Given:

Raghu starts from Point Y and drives 18 km towards west.

He then turns right and drives 26 km, takes a left turn, drives 30 km, turns left and drives 68 km.

He then takes a right turn and drives 31 km.

He takes a right turn, drives 42 km, turns left and drives 16 km to stop at Point Z.

Thus, Raghu should drive 95 km in east direction in order to reach Point Y again.

Hence, "Option 3" is the correct answer.

Given: P, Q, R, S, T and U are sitting in a row facing the north.

1) S and R are sitting at the ends of the line not necessarily in the same order.

2) S is sitting to the immediate left of Q.

3) P is sitting to the immediate right of Q.

4) U is sitting to the immediate right of P and immediate left of T.

Thus, R is sitting to the immediate right of T.

Hence, "Option 3" is the correct answer.

According to the question:

Srinivas rank from the top → 2nd

Srinivas rank from the bottom → 19th

Total students → Srinivas rank from the top + Srinivas rank from the bottom - 1

→ 2 + 19 - 1 = 20

Hence, "Option 3" is the correct answer.

The logic followed here is:

(6, 72, 12) → 72 ÷ 6 = 12​

And,

(5, 60, 12) → 60 ÷ 5 = 12

Similarly,

(3, 36, 12) → 36 ÷ 3 = 12

Hence, "Option 1" is the correct answer.

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

According to the question:

Required condition: Odd Number - Even Number - Even Number

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

Thus, there are two such odd numbers which are immediately preceded by an odd number and also immediately followed by an even number.

Hence, "Option 3" is the correct answer.

According to the question:

Thus, 'A' is coded as '3'.

Hence, "Option 1" is the correct answer.

Given series: (Left) 7 $ 5 @ 3 ! 3 5 6 7 & % $ 3 # 2 7 + ? > < (Right)

According to the question:

All the numbers are dropped → (Left) 7 $ 5 @ 3 ! 3 5 6 7 & % $ 3 # 2 7 + ? > < (Right)

 → (Left) $ @ ! & % $ # + ? > < (Right)

The 7th element from the right → (Left) $ @ ! & % $ # + ? > < (Right) → %

Hence, "Option 2" is the correct answer.