RRB Group D Official Paper (Held On: 27 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is Infinite.

Key Points

• The motion of the ball is governed by Newton’s First Law of Motion, also known as the Law of Inertia. This law states that an object will remain in its state of rest or uniform motion in a straight line unless acted upon by an external unbalanced force.
• In the given scenario, the ball starts from rest at the top of a 5 cm long smooth inclined plane. As it moves down, potential energy is converted into kinetic energy, and the ball reaches the bottom with a specific velocity.
• Once the ball reaches the horizontal smooth surface, there is no frictional force or air resistance (as the problem states to neglect them). Because the surface is smooth, the coefficient of friction is zero.
• On a horizontal plane, the force of gravity acts vertically downwards and is balanced by the normal reaction force acting upwards. Therefore, there is no net force acting on the ball in the horizontal direction of motion.
• According to the equation of motion, if the net force is zero, the acceleration is also zero (based on Newton's Second Law, F = ma). This means the ball will maintain a constant velocity indefinitely. Since it never stops, the distance it travels horizontally is Infinite.

Additional Information

• Galileo’s Experiment on Inertia:
• The concept of Inertia was first extensively studied by Galileo Galilei. He conducted experiments with double inclined planes.
• He observed that if a ball is released from one side, it climbs to the same height on the other side. If the angle of the second slope is reduced, the ball travels a longer distance to reach that same height.
• Galileo concluded that if the second plane were made horizontal (angle of 0 degrees), the ball would continue to move forever in an attempt to reach its original height, provided there is no friction.
• Newton's First Law of Motion:
• It defines Inertia, which is the inherent property of a body to resist changes in its state of motion.
• There are three types of inertia: Inertia of rest, Inertia of motion, and Inertia of direction.
• In this problem, the ball exhibits inertia of motion once it leaves the incline and enters the horizontal path.
• Role of Friction:
• In real-world conditions, surfaces are never perfectly smooth. Frictional forces and air resistance act as external unbalanced forces that eventually bring the ball to rest.
• However, in ideal physics problems where surfaces are described as smooth, we assume the frictional force is 0 Newtons, leading to theoretical infinite motion.

Important Points

• The length of the incline (5 cm) determines the final velocity at the bottom but does not affect the total distance traveled on an infinite smooth horizontal plane.
• The velocity (v) at the bottom of the incline can be calculated using v^2 = u^2 + 2as or conservation of energy (mgh = 1/2 mv^2), but since there is no deceleration on the horizontal part, this velocity remains constant.
• The horizontal distance (d) is calculated by d = velocity × time. As time (t) tends toward infinity in the absence of a stopping force, the distance also becomes Infinite.

The correct answer is Air.

Key Points

• Air is a homogeneous mixture of various gases, primarily consisting of Nitrogen (78%) and Oxygen (21%), along with traces of Argon, Carbon Dioxide, and other noble gases. In this mixture, the solute particles are at the molecular or atomic level (less than 1 nanometer in diameter), and they do not scatter light. Hence, statement 4 is correct as it is a true solution, not a colloid.
• Smoke is a colloidal solution where solid particles (like carbon or ash) are dispersed in a gas medium (air). This specific type of colloid is known as a solid aerosol. Because the particles are large enough to remain suspended but do not settle quickly, it follows the properties of a colloid. Hence, statement 1 is incorrect as it is a colloid.
• Milk is a complex colloidal system where liquid fat globules are dispersed in water. This liquid-in-liquid colloid is specifically termed an emulsion. The presence of proteins like casein acts as an emulsifier to keep the mixture stable. Hence, statement 2 is incorrect as it is a colloid.
• Mist (and Fog) is a colloidal solution where liquid water droplets are dispersed in the atmosphere (gas). It is classified as a liquid aerosol. The droplets are large enough to scatter light, which is why mist appears opaque or translucent. Hence, statement 3 is incorrect as it is a colloid.
• Therefore, the correct answer is Air.

Additional Information

• Definition of a Colloid:
• A colloid is a heterogeneous mixture in which the particle size of the dispersed phase is intermediate between that of a true solution and a suspension.
• The typical particle size ranges from 1 nanometer (nm) to 1000 nanometers.
• Unlike suspensions, colloidal particles do not settle down under the influence of gravity when left undisturbed.
• Types of Colloidal Systems:
• Aerosol: Formed when a solid or liquid is dispersed in a gas (e.g., clouds, smoke, automobile exhaust).
• Emulsion: Formed when a liquid is dispersed in another liquid (e.g., milk, face cream, mayonnaise).
• Foam: Formed when a gas is dispersed in a liquid (e.g., shaving cream, froth) or a solid (e.g., sponge, pumice stone).
• Sol: Formed when a solid is dispersed in a liquid (e.g., Milk of Magnesia, mud).
• Gel: Formed when a liquid is dispersed in a solid (e.g., jelly, cheese, butter).
• The Tyndall Effect:
• This is the phenomenon of scattering of light by colloidal particles.
• When a beam of light passes through a colloid, the path of the light becomes visible. True solutions (like Air or Saltwater) do not exhibit the Tyndall Effect because their particles are too small to scatter light.

Important Points

• Brownian Motion: Colloidal particles exhibit a continuous, zig-zag random motion known as Brownian motion. This motion is caused by the unbalanced bombardment of the particles by the molecules of the dispersion medium, which helps in keeping the colloid stable.
• Stability: Colloids are relatively stable. They can often be separated by centrifugation but not by simple filtration through ordinary filter paper, as the particles are smaller than the pores of the paper.
• Phase Components: Every colloid consists of two parts: the dispersed phase (the substance being scattered) and the dispersion medium (the substance in which the particles are scattered). In Air, there is no such distinction of phases as it is a single-phase homogeneous gas mixture.

The correct answer is The size of atoms is in the range of nanometres.

Key Points

• Atoms are the fundamental building blocks of matter and are incredibly small in size. The radius of an atom is typically measured in nanometres (nm), where 1 nanometre is equal to 1/10^9 metres (or 10⁻⁹ m). Hence, statement 4 is correct.
• Molecules are chemical structures consisting of at least two atoms held together by chemical bonds. Because a molecule is a collection of atoms, it is always larger than the individual atoms that compose it. Hence, statement 1 is incorrect.
• The size of an atom is not constant across all elements. It varies depending on the atomic number, the number of electron shells, and the effective nuclear charge. For instance, a Cesium atom is much larger than a Helium atom. Hence, statement 2 is incorrect.
• An ordinary optical microscope uses visible light to form images, which has a wavelength range of approximately 400 to 700 nanometres. Since atoms are much smaller (around 0.1 to 0.5 nanometres) than the wavelength of light, they cannot be resolved or seen using such devices. Hence, statement 3 is incorrect.
• To "see" or map the surface of atoms, scientists use advanced instruments like the Scanning Tunneling Microscope (STM) or the Atomic Force Microscope (AFM), which were developed in the 1980s.

Additional Information

• Atomic Radius:
• The atomic radius is generally defined as the shortest distance between the atom's nucleus and the outermost shell of electrons.
• In the Periodic Table, the atomic size generally increases as you move down a group because new electron shells are added.
• Conversely, the atomic size generally decreases as you move from left to right across a period because the increasing nuclear charge pulls the electron cloud closer to the nucleus.
• Scale and Examples:
• The smallest atom is Hydrogen, with an atomic radius of approximately 0.037 nanometres (or 37 picometres).
• To put the scale into perspective, more than one million atoms stacked on top of each other would be roughly as thick as a single sheet of paper.
• A single drop of water contains about 2 sextillion (2 x 10²¹) atoms of Oxygen and twice as many Hydrogen atoms.
• Measurement Units:
• While the nanometre (nm) is common, physicists often use the Angstrom (Å) to measure atomic scales. 1 Angstrom is equal to 10⁻¹⁰ metres or 0.1 nanometres.
• For even smaller subatomic distances, the picometre (pm) is used, where 1,000 pm equals 1 nanometre.

The correct answer is Father.

Key Points

• In humans, the determination of the sex of a child is a genetic process governed by the 23rd pair of chromosomes, known as the sex chromosomes.
• Human beings have a total of 46 chromosomes or 23 pairs. Out of these, 22 pairs are autosomes (identical in both males and females), and the last pair defines the biological sex.
• The female genetic makeup consists of two X chromosomes (XX). Consequently, all gametes (eggs) produced by a mother contain only an X chromosome.
• The male genetic makeup consists of one X chromosome and one Y chromosome (XY). This means the father is heterogametic, producing two types of sperm: one carrying the X chromosome and the other carrying the Y chromosome.
• The sex of the child depends entirely on which type of sperm fertilizes the egg. If a sperm with an X chromosome fuses with the egg, the resulting child will be XX (female). If a sperm with a Y chromosome fuses with the egg, the child will be XY (male). Hence, the father is the parent who determines the sex of the child.

Additional Information

• Chromosomal Basis of Sex:
• The mechanism where specific chromosomes determine the sex of an individual is called sex determination. In humans, this is the XY system.
• During meiosis, the chromosome pairs separate so that each gamete receives only one chromosome from each pair.
• Since the ovum (egg) always contributes an X, the zygote's sex is determined at the moment of fertilization by the male gamete.
• Probability of Sex:
• Statistically, there is a 50% chance of having a boy and a 50% chance of having a girl in every pregnancy, as the ratio of X-carrying sperm to Y-carrying sperm is 1:1.
• Other Systems in Nature:
• Not all organisms follow the XY system. For instance, in birds, the system is ZW, where the female (ZW) is the heterogametic sex and determines the sex of the offspring, while the male is ZZ.
• In some reptiles, sex is determined by environmental factors like the incubation temperature of the eggs rather than genetics.

Important Points

• The SRY Gene:
• The Y chromosome contains a specific gene called the SRY gene (Sex-determining Region Y).
• This gene is responsible for the initiation of male sex determination by triggering the development of undifferentiated gonads into testes.
• In the absence of a Y chromosome (and thus the SRY gene), the biological default pathway leads to the development of ovaries and a female phenotype.
• Historical Context:
• The chromosomal theory of sex determination was discovered in the early 20th century (around 1905) by American geneticists Nettie Stevens and Edmund Beecher Wilson.
• Their work shifted the understanding of heredity from vague external factors to specific microscopic structures within the cell nucleus.
• Misconceptions:
• In many cultures historically, mothers were wrongly blamed for not bearing a male child. Scientific evidence clearly shows that the mother's genetic contribution to sex is constant (X), and the father's contribution is the variable factor.

The correct answer is stethoscope.

Key Points

• A stethoscope is an acoustic medical device used by healthcare professionals for auscultation, which involves listening to the internal sounds of a human or animal body.
• The instrument is primarily used to listen to the heartbeat and lung sounds, providing critical information about the rhythm of the heart and the clarity of the airways.
• It was invented in 1816 by the French physician René Laennec at the Necker-Enfants Malades Hospital in Paris. Originally, it was a simple wooden tube.
• Modern stethoscopes consist of a chest piece (comprising a diaphragm and a bell), tubing, and earpieces. The diaphragm is used for high-frequency sounds, while the bell is used for low-frequency sounds.
• When a doctor places the chest piece on a patient, the vibrations of the heart produce sound waves that travel through the hollow tubes to the doctor's ears, allowing them to monitor the heart rate and detect murmurs or gallops.

Additional Information

• Radiography:
• This is an imaging technique that uses ionizing radiation, specifically X-rays, to view the internal form of an object.
• In a medical context, it is used to diagnose bone fractures, joint dislocations, and certain infections like pneumonia.
• It was pioneered by Wilhelm Conrad Röntgen in 1895, who received the first Nobel Prize in Physics for this discovery.
• Reflex Hammer:
• This is a medical instrument used by practitioners to test deep tendon reflexes.
• Testing reflexes is an essential part of a neurological physical examination to assess the integrity of the central nervous system and peripheral nervous system.
• The first modern reflex hammer was designed by John Madison Taylor in 1888.
• Microscope:
• A microscope is a laboratory instrument used to examine objects that are too small to be seen by the naked eye.
• In medicine, it is used in pathology and hematology to observe cells, bacteria, parasites, and viruses in samples like blood or tissue biopsies.
• Antonie van Leeuwenhoek is often referred to as the "Father of Microbiology" for his improvements to the microscope in the 17th century.

Important Points

• Evolution of the Stethoscope: In 1851, Arthur Leared invented the binaural stethoscope (using both ears), and it was further refined in 1852 by George Philip Cammann for commercial production.
• Electronic Stethoscopes: Modern advancements have led to electronic or digital stethoscopes, which can amplify low-level body sounds and convert them into digital signals for recording or visual display (phonocardiogram).
• Auscultation Areas: Doctors typically listen to the heart at four specific locations on the chest: the aortic, pulmonic, tricuspid, and mitral areas to evaluate different heart valves.
• Sphygmomanometer: This device is often used in conjunction with a stethoscope to measure blood pressure by listening to Korotkoff sounds in the brachial artery.

The correct answer is The force becomes one-fourth.

Key Points

• The gravitational force between two bodies is determined by Newton's Universal Law of Gravitation, which states that the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Mathematically, the formula is expressed as F = G (m1m2 / r²), where F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
• According to the Inverse Square Law relationship, if the distance r is doubled (becomes 2r), the new force F' becomes proportional to 1 / (2r)², which simplifies to 1 / 4r².
• This calculation shows that the new force is exactly one-fourth (1/4) of the original force. Hence, the statement that the force becomes one-fourth is correct.
• Conversely, if the distance were halved, the force would increase by four times, illustrating the significant impact of distance on gravitational attraction compared to mass.

Additional Information

• Newton's Universal Law of Gravitation: Formulated by Sir Isaac Newton and published in 1687 in his work "Philosophiæ Naturalis Principia Mathematica". It provides a mathematical description of the gravitational pull that every object in the universe exerts on every other object.
• The Gravitational Constant (G): This is a fundamental physical constant. Its value was first accurately measured by Henry Cavendish in 1798. The modern accepted value is approximately 6.674 x 10⁻¹¹ N·m²/kg².
• Mass and Force Relationship: While distance has an inverse square relationship, the force is directly proportional to the masses. If the mass of one object is doubled, the gravitational force also doubles, provided the distance remains constant.
• Range of Gravity: Gravity is a long-range force that exists between all objects with mass, regardless of how far apart they are, though it becomes negligibly small at extreme astronomical distances.
• Significance in the Solar System: This law explains why planets orbit the Sun and why the Moon orbits the Earth. It also accounts for the phenomenon of tides, which are caused by the gravitational pull of the Moon and the Sun on Earth's oceans.

Important Points

• Weight vs. Gravitational Force: The weight of an object on Earth is actually the gravitational force exerted by the Earth on that object. It is calculated using W = mg, where g is the acceleration due to gravity (approximately 9.8 m/s²).
• Vector Nature: Gravitational force is a vector quantity, meaning it has both magnitude and direction. The direction of the force is always along the line joining the centers of the two masses and is always attractive.
• Universal Applicability: It is called a "Universal" law because it applies to everything from small terrestrial objects to massive celestial bodies like stars and galaxies.
• Center of Mass: For spherical objects with uniform density, the distance r is measured specifically from the center of mass of one object to the center of mass of the other.

The correct answer is Only a and b.

Key Points

• The extraction of metals depends on their position in the reactivity series. Metals at the top of the series, such as Potassium (K), Sodium (Na), Calcium (Ca), Magnesium (Mg), and Aluminium (Al), are highly reactive and cannot be reduced by carbon because their affinity for oxygen is higher than that of carbon. These metals are extracted by the electrolysis of their molten ores (usually chlorides or oxides). Hence, statement a is correct.
• Metals located in the middle of the reactivity series, such as Iron (Fe), Zinc (Zn), and Lead (Pb), are moderately reactive. In nature, these metals are predominantly found as sulphides or carbonates rather than pure oxides or free elements. Hence, statement b is correct.
• During the metallurgical process, it is significantly more difficult to obtain a metal directly from its sulphide or carbonate form. It is much easier to extract a metal from its oxide via reduction. Therefore, ores are first converted into oxides before the actual extraction happens. Hence, statement c is incorrect.
• Consequently, sulphide ores are converted into oxides by roasting, and carbonate ores are converted into oxides by calcination to facilitate easier reduction.
• Hence, the correct answer is Only a and b.

Additional Information

• Roasting:
• It is the process of heating sulphide ores strongly in the presence of excess air.
• For example, Zinc sulphide (Zinc blende) is heated in air to produce Zinc oxide and Sulphur dioxide gas.
• This process helps in removing volatile impurities and preparing the ore for reduction.
• Calcination:
• It is the process of heating carbonate ores strongly in limited air or the absence of air.
• For example, Zinc carbonate (Calamine) decomposes into Zinc oxide and Carbon dioxide when heated.
• This is an essential step for ores like Limestone and Magnesite.
• Reduction of Metal Oxides:
• Once the metal oxide is obtained through roasting or calcination, it is reduced to the metal using reducing agents like Carbon (Coke), Carbon monoxide, or sometimes highly reactive metals like Sodium or Aluminium.
• In the case of Iron, it is reduced in a blast furnace using coke.

Important Points

• Low Reactivity Metals: Metals at the bottom of the series, like Gold (Au) and Platinum (Pt), are found in the native state (uncombined). Metals like Mercury (Hg) and Copper (Cu) are found as sulphides but can be reduced to metal by simply heating in air (auto-reduction).
• Electrolytic Refining: The metals obtained by reduction processes are often impure. They are purified using electrolytic refining, where the impure metal is made the anode and a thin strip of pure metal is made the cathode in a suitable salt solution.
• Aluminium Extraction: Since Aluminium has a very high melting point, it is extracted from Bauxite using the Hall-Heroult process, where Cryolite is added to lower the melting point and increase conductivity.

The correct answer is converges to a point on the principal axis.

Key Points

• A convex lens is also known as a converging lens because of its ability to bring parallel rays of light together at a single point.
• When a beam of light consisting of rays parallel to the principal axis strikes the surface of a convex lens, the rays undergo refraction twice—once when entering the lens and once when exiting it.
• After passing through the lens, these refracted rays converge and intersect at a specific point on the principal axis. Hence, statement 1 is correct.
• This specific point where the parallel rays meet is called the Principal Focus (denoted by F). Since the rays actually meet at this point, the focus of a convex lens is considered a real focus.
• The converging action occurs because the lens is thicker at the center than at the edges, causing the light rays to bend more toward the thicker part of the glass.
• Conversely, a concave lens would cause a parallel beam to diverge away from the axis, making them appear as if they are coming from a virtual focus. Hence, statement 4 is incorrect.
• Only light rays that pass directly through the Optical Center of the lens will pass undeviated. Hence, statement 3 is incorrect.

Additional Information

• Characteristics of Convex Lenses:
• A convex lens is thicker in the middle and tapers off at the edges.
• It can form both real and virtual images depending on the position of the object relative to the focal point.
• When the object is placed at infinity, the image formed is a point-sized, real, and inverted image at the focus.
• Focal Length and Power:
• The distance between the Optical Center (O) and the Principal Focus (F) is known as the focal length (f).
• The Power of a lens is the reciprocal of its focal length (P = 1/f), measured in Dioptres (D). For a convex lens, the power is always positive.
• Applications:
• Human Eye: The natural lens in the human eye is a convex lens that focuses light on the retina.
• Correction of Vision: Convex lenses are used to correct Hypermetropia (farsightedness), where the image of near objects is formed behind the retina.
• Optical Instruments: They are extensively used in microscopes, telescopes, cameras, and magnifying glasses to produce enlarged or focused images.
• Types of Convex Lenses:
• Double Convex (Biconvex): Both surfaces are curved outward.
• Plano-Convex: One surface is flat and the other is curved outward.
• Concavo-Convex (Meniscus): One surface is curved inward and the other outward, with the center being thicker than the edges.

Important Points

• The behavior of light through a lens is governed by Snell's Law of refraction, which relates the angle of incidence to the angle of refraction based on the refractive index of the material.
• The Lens Formula is given by 1/f = 1/v - 1/u, where v is the image distance and u is the object distance.
• The Magnification (m) produced by a lens is the ratio of the height of the image to the height of the object, also expressed as m = v/u.
• A virtual image is formed by a convex lens only when the object is placed between the optical center and the focus; this image is erect and magnified.

The correct answer is DNA copies may have slight differences.

Key Points

• The process of DNA replication is a biochemical reaction. In biology, no chemical reaction is 100% absolutely reliable or accurate every single time.
• When a cell reproduces, it creates a copy of its DNA through these chemical processes. Because of the inherent nature of these reactions, the resulting DNA strands are very similar to the original but may contain minor alterations. Hence, the statement that DNA copies may have slight differences is correct.
• If each DNA copy were totally unrelated, the offspring would not belong to the same species as the parent, which contradicts the fundamental principles of heredity. Hence, option 1 is incorrect.
• During replication, the original DNA acts as a template. It is not destroyed; rather, it provides the structural information for the synthesis of the new strand. Hence, option 2 is incorrect.
• The idea that all DNA copies are identical clones is biologically improbable due to the occurrence of mutations and copying errors. These "errors" are actually the basis for diversity in a population. Hence, option 4 is incorrect.
• The phrase “similar, but may not be identical” acknowledges the existence of biological variations that occur during the transfer of genetic information from one generation to the next.

Additional Information

• DNA (Deoxyribonucleic Acid): It is the genetic material found in the nucleus of almost all living organisms. It carries the blueprints for the body design and functioning of the cell.
• DNA Replication: This is the process by which a double-stranded DNA molecule is copied to produce two identical DNA molecules. It is a critical part of cell division (mitosis and meiosis).
• The Role of Variations: Variations are the differences that exist between individuals of the same species. These variations are necessary for evolution. If an environment changes drastically (e.g., global warming), these slight differences might allow some individuals to survive while others perish.
• Historical Context: The double helix structure of DNA was famously proposed by James Watson and Francis Crick in 1953, which helped scientists understand how DNA could "unzip" and replicate itself.
• Genetic Stability: While variation is important, stability is also required. If the DNA copying process had too many errors, the resulting cellular apparatus would fail to function, leading to the death of the new cell.

Important Points

• Mutation: A mutation is a permanent change in the nucleotide sequence of DNA. These can be caused by copying errors, UV radiation, or chemicals.
• Niche and Adaptation: DNA copying consistency allows organisms to occupy a specific ecological niche. Variations allow them to adapt when that niche undergoes environmental shifts.
• Asexual vs Sexual Reproduction: In asexual reproduction, the variation is minimal because there is only one parent, whereas in sexual reproduction, variations are increased because genetic material from two different individuals is combined.
• Enzymatic Control: The replication process is managed by enzymes like DNA polymerase, which has a proofreading function to minimize errors, though it is not perfect.

The correct answer is Plant cells have large vacuoles, animal cells have small ones.

Key Points

• Vacuoles are membrane-bound organelles found in the cells of plants, animals, fungi, and some protists, acting primarily as storage bubbles.
• In plant cells, there is typically one very large central vacuole that can occupy up to 90% of the cell's total volume. This large size is a defining characteristic of mature plant cells.
• In animal cells, vacuoles are much smaller in size and often temporary. They are distributed throughout the cytoplasm and are frequently involved in transport or short-term storage.
• The large central vacuole in plants is essential for maintaining turgor pressure, which provides structural support and keeps the plant upright by pushing the cytoplasm against the cell wall.
• The statement that plant cells have large vacuoles and animal cells have small ones accurately describes the morphological difference between the two. Hence, statement 3 is correct.

Additional Information

• Tonoplast: The specialized membrane that surrounds the central vacuole in plant cells is known as the tonoplast. It facilitates the active transport of ions and other materials into the vacuole, maintaining a concentration gradient.
• Cell Sap: The fluid contained within the plant vacuole is called cell sap. It is a mixture of water, amino acids, sugars, waste products, and vital mineral salts.
• Anthocyanins: Some plant vacuoles contain pigments like anthocyanins, which are responsible for the red, purple, and blue colors seen in many flowers and fruits, aiding in pollination and seed dispersal.
• Functions in Animals: In animal cells, vacuoles (often called vesicles) are involved in exocytosis (expelling waste) and endocytosis (bringing materials into the cell). They play a key role in the digestive processes of simple organisms.
• Contractile Vacuoles: Some freshwater protists like Amoeba possess contractile vacuoles, which collect and pump out excess water to prevent the cell from bursting, a vital process for osmoregulation.

Important Points

• Structural Difference: Unlike plant cells which have a permanent large central vacuole, the vacuoles in animal cells are often transient and can form or disappear based on the cell's metabolic needs.
• Discovery: Vacuoles were first observed by Antonie van Leeuwenhoek in 1676, though their specific functions in plant turgidity were understood much later.
• Waste Management: In plants, the vacuole acts as a waste disposal site, isolating materials that might be harmful or toxic to the rest of the cell's organelles.
• Growth Mechanism: Plants grow by absorbing water into the central vacuole, which expands the cell with minimal investment in new cytoplasm, allowing for rapid cell elongation.

The correct answer is Two people push a box from opposite sides with equal force.

Key Points

• Balanced forces occur when the resultant force or net force acting on an object is zero. This means the forces acting from all directions cancel each other out perfectly.
• In the case of two people pushing a box from opposite sides with equal force, the forces are equal in magnitude but opposite in direction. Because they act along the same line, they neutralize each other, resulting in no change in the box's state of motion. Hence, statement 3 is correct.
• When a ball is falling freely under gravity, the only significant force acting on it (ignoring air resistance) is the force of gravity. Since there is no upward force to balance it, the ball undergoes acceleration at a rate of approximately 9.8 m/s². Hence, statement 1 is incorrect.
• When a moving car comes to rest upon applying brakes, the frictional force exerted by the brakes creates an unbalanced force in the direction opposite to the motion. This causes negative acceleration or deceleration. Hence, statement 2 is incorrect.
• A person riding a bicycle uphill is subject to the component of gravity pulling them backward and friction. Unless the rider's pedaling force exactly matches these resistive forces to maintain a constant velocity, the forces are typically unbalanced, especially as the rider usually has to exert extra effort to overcome the incline. Hence, statement 4 is incorrect.

Additional Information

• Newton’s First Law of Motion: Often called the Law of Inertia, it states that an object will remain at rest or continue to move at a constant velocity unless acted upon by an unbalanced external force. Balanced forces do not trigger a change in motion.
• Net Force (Fnet): This is the vector sum of all forces acting on an object. For balanced forces, Fnet = 0. For unbalanced forces, Fnet ≠ 0, which leads to acceleration as defined by Newton’s Second Law (F = ma).
• State of Equilibrium: An object under balanced forces is said to be in equilibrium. This can be Static Equilibrium (object is at rest, like a book on a table) or Dynamic Equilibrium (object is moving at a constant speed in a straight line, like a plane flying at constant altitude and velocity).
• Characteristics of Balanced Forces:
• They do not change the speed of an object.
• They do not change the direction of a moving object.
• They may, however, change the shape or size of an object (e.g., squeezing a rubber ball from opposite sides).

Important Points

• Gravity and Normal Force: A common example of balanced forces is an object resting on a horizontal surface. The downward force of gravity is balanced by the upward normal force exerted by the surface.
• Terminal Velocity: When an object falls through a fluid (like air), it eventually reaches a point where the drag force equals the gravitational force. At this terminal velocity, the forces are balanced, and the object stops accelerating, falling at a constant speed.
• Units of Force: The standard SI unit for measuring force is the Newton (N). Force is a vector quantity, meaning it has both magnitude and direction, which is why the direction of "opposite sides" is crucial for forces to be balanced.

The correct answer is Cerebrum.

Key Points

• The Cerebrum is the largest and most prominent part of the human brain, located in the forebrain region. It is the primary center for intelligence, memory, and imagination.
• It specifically controls a person's thinking and learning abilities, along with other complex cognitive functions such as reasoning, problem-solving, speech, and emotions.
• Structurally, the Cerebrum is divided into two halves known as the left and right cerebral hemispheres. These hemispheres are connected by a thick bundle of nerve fibers called the corpus callosum, which allows communication between both sides.
• The outer layer of the cerebrum is called the cerebral cortex (also known as grey matter), which contains a high concentration of neurons. This layer is highly folded into gyri (ridges) and sulci (grooves) to increase the surface area for processing information.
• The Cerebrum is further categorized into four lobes: the frontal lobe (associated with decision-making and personality), the parietal lobe (sensory perception), the temporal lobe (auditory processing), and the occipital lobe (visual processing).

Additional Information

• Hypothalamus:
• Located below the thalamus, it plays a crucial role in maintaining homeostasis.
• It regulates vital body functions such as body temperature, hunger, thirst, sleep-wake cycles, and emotional responses. It also controls the pituitary gland, linking the nervous system to the endocrine system.
• Cerebellum:
• Often referred to as the "little brain," it is located at the back of the skull, beneath the cerebrum.
• Its primary function is to coordinate voluntary muscle movements, maintain posture, and ensure physical balance and equilibrium during activities like walking or running.
• Medulla Oblongata:
• This is the lowest part of the brainstem and connects the brain to the spinal cord.
• It is responsible for regulating involuntary functions essential for survival, including heartbeat, breathing (respiration), blood pressure, and reflexes like swallowing, coughing, and sneezing.

Important Points

• The human brain is broadly divided into three main parts: the Forebrain (comprising the Cerebrum and Diencephalon), the Midbrain, and the Hindbrain (comprising the Pons, Cerebellum, and Medulla).
• While the Cerebrum handles high-level cognitive tasks, the Brainstem (Midbrain, Pons, and Medulla) acts as a relay station for sensory and motor signals and manages life-sustaining autonomic functions.
• Damage to the Cerebrum can lead to significant impairments in cognitive function, memory loss, or paralysis, whereas damage to the Medulla is often fatal because it controls the circulatory and respiratory systems.

The correct answer is To enhance its corrosion resistance and appearance.

Key Points

• Anodising is an electrochemical process that converts the metal surface into a decorative, durable, corrosion-resistant, anodic oxide finish.
• While several non-ferrous metals, such as magnesium and titanium, can be anodized, aluminium is ideally suited to the process due to its specific chemical properties.
• The primary purpose of this process is to enhance corrosion resistance. Aluminium naturally forms a thin layer of oxide upon exposure to air, but anodising creates a much thicker and more robust protective layer that prevents the underlying metal from reacting with the environment. Hence, option 1 is correct.
• Another major objective is to improve the aesthetic appearance of the metal. The structure of the anodic coating is highly porous, which allows for the absorption of dyes and pigments, enabling aluminium to be finished in a wide variety of colors.
• Anodising increases the surface hardness of the aluminium, making it more resistant to wear and tear compared to untreated metal. Hence, statement 2 is incorrect as it makes the surface harder, not softer.
• The process effectively seals the metal, which actually decreases its reactivity with acids and other chemicals by providing a physical barrier. Hence, statement 3 is incorrect.
• Since aluminium oxide is an electrical insulator, an anodized layer does not make the metal a better conductor; in fact, it can act as an insulating barrier. Hence, statement 4 is incorrect.

Additional Information

• The Electrolytic Process: To anodize aluminium, the metal is submerged in an acid electrolyte bath (typically sulphuric acid) and an electric current is passed through the medium. The aluminium part acts as the anode (the positive electrode), which is where the term "anodising" originates.
• Chemical Reaction: Oxygen ions are released from the electrolyte and combine with the aluminium atoms at the surface to form Aluminium Oxide (Al2O3). This layer is fully integrated with the underlying aluminium substrate, so it cannot peel or chip like paint.
• Applications: Because of its durability and beauty, anodized aluminium is widely used in architecture (window frames, roofs), aerospace (aircraft components), automotive parts, and consumer electronics like smartphones and laptops.
• Maintenance: Anodized surfaces are easy to maintain; they generally require only periodic cleaning with mild soap and water to preserve their original appearance for decades.

Important Points

• Hard Anodising: This is a variation of the process (also known as Type III) performed at lower temperatures and higher current densities to produce a much thicker and denser oxide layer, specifically for industrial applications requiring extreme abrasion resistance.
• Sealing Process: After the oxide layer is formed and colored, the pores must be sealed to prevent corrosion and staining. This is usually done by immersing the part in hot deionized water or a nickel acetate solution, which converts the oxide into a hydrated form that swells and closes the pores.
• Environmental Impact: Anodising is considered an environmentally stable process. It does not produce heavy metal toxins and the by-products are generally recyclable or easily neutralized in wastewater treatment plants.
• Date and History: The first industrial anodising process was patented in 1923 by Bengough and Stuart to protect seaplane parts from salt-water corrosion using chromic acid.

The correct answer is Sexual reproduction.

Key Points

• Sexual reproduction is a biological process that involves the fusion of specialized reproductive cells called gametes from two different parents, typically a male and a female.
• During this process, the offspring inherit approximately 50% of their DNA or genetic material from each parent, leading to a unique combination of traits.
• The process of Meiosis is fundamental to sexual reproduction, as it reduces the chromosome number by half to produce haploid gametes (sperm and egg), ensuring the chromosome number remains stable across generations.
• A critical feature of this mode of reproduction is genetic variation, which occurs due to crossing over and independent assortment during meiosis, helping populations adapt to changing environments.
• The fusion of these gametes results in a diploid zygote, which eventually develops into an embryo and then a complete organism that displays features from both parents.

Additional Information

• Asexual Reproduction:
• In contrast to sexual reproduction, asexual reproduction involves only a single parent, and the resulting offspring are genetically identical to the parent.
• Since there is no fusion of gametes, there is no inheritance of features from "both" parents, and the offspring are referred to as clones.
• Binary Fission:
• This is a common method of asexual reproduction found in unicellular organisms such as Amoeba, Bacteria, and Paramecium.
• In Binary fission, the parent cell divides into two equal halves, each growing into a new individual, ensuring that the genetic code remains exactly the same as the original cell.
• Budding:
• Budding is a form of asexual reproduction where a small outgrowth or bud develops on the body of the parent organism due to repeated cell division at one specific site.
• Once the bud matures, it detaches from the parent body to become an independent individual; common examples include Hydra and Yeast.
• Cloning:
• Cloning is the process of producing individuals with identical DNA, either naturally (like identical twins) or artificially through biotechnology.
• The most famous historical example of artificial cloning in animals is Dolly the sheep, who was born in July 1996 and was the first mammal cloned from an adult somatic cell.

Important Points

• Heredity and Variation: Sexual reproduction is the primary driver of biodiversity within a species, as it constantly shuffles the genetic deck through recombination.
• Evolutionary Advantage: While asexual reproduction is faster and requires less energy, sexual reproduction provides a survival advantage in unstable environments because genetic diversity increases the likelihood that some individuals will survive disease or climate changes.
• Gametes: In most animals, the male gamete is the sperm (small and motile) and the female gamete is the ovum or egg (large and non-motile), both of which are haploid.

The correct answer is They are replaced by other atoms or groups of atoms.

Key Points

• A substitution reaction is a fundamental type of chemical reaction where one functional group or atom in a chemical compound is replaced by another atom or group of atoms.
• In the case of alkanes, which are saturated hydrocarbons, the carbon atoms are already bonded to the maximum number of hydrogen atoms through single covalent bonds.
• During a substitution reaction involving an alkane, one or more hydrogen atoms are removed and replaced by atoms of another element, most commonly halogens (Chlorine, Bromine, etc.). Hence, statement 1 is correct.
• These reactions do not result in the growth of the carbon chain length through simple addition, which is a characteristic of addition reactions found in unsaturated hydrocarbons like alkenes. Hence, statement 2 is incorrect.
• The hydrogen atoms are not simply removed to form double bonds; the formation of double bonds from alkanes is typically classified as a dehydrogenation or elimination reaction, which usually requires high temperatures or specific catalysts. Hence, statement 3 is incorrect.
• While alkanes do react with oxygen to form water and carbon dioxide, this process is known as combustion (an oxidation reaction), not a substitution reaction. Hence, statement 4 is incorrect.

Additional Information

• Nature of Alkanes: Alkanes are often called paraffins (Latin: parum affinis = little affinity) because they are relatively unreactive under normal conditions due to the stability of C-C and C-H sigma bonds.
• Halogenation of Alkanes: This is the most common substitution reaction for alkanes. It typically occurs in the presence of ultraviolet (UV) light or diffused sunlight, which provides the energy needed to break the bonds of the halogen molecules.
• Reaction Mechanism: The substitution reaction in alkanes follows a free-radical mechanism. This process involves three distinct steps: initiation (creation of radicals), propagation (reaction of radicals with alkanes), and termination (radicals combining to form stable molecules).
• Example: When Methane (CH₄) reacts with Chlorine (Cl₂) in sunlight, a hydrogen atom is replaced by a chlorine atom to form Chloromethane (CH₃Cl) and Hydrogen Chloride (HCl). If chlorine is in excess, the reaction can continue until all four hydrogen atoms are replaced, resulting in Carbon Tetrachloride (CCl₄).

Important Points

• Reactivity Order: The reactivity of halogens toward alkanes in substitution reactions follows the order: Fluorine > Chlorine > Bromine > Iodine. Fluorination is often too explosive to control, while iodination is very slow and reversible.
• Homologous Series: Alkanes form a homologous series with the general formula CnH₂n+₂. As the molecular mass increases in the series, the boiling point and melting point also increase due to stronger Van der Waals forces.
• Industrial Use: Substitution reactions are vital in the chemical industry for producing alkyl halides, which serve as precursors for manufacturing polymers, refrigerants, and pharmaceuticals.

The correct answer is sharing of electrons.

Key Points

• Covalent bonds are a type of chemical link that involves the sharing of electron pairs between atoms. These electron pairs are also known as bonding pairs or shared pairs.
• The stable balance of attractive and repulsive forces between atoms, when they share electrons, is known as covalent bonding. This allows each atom involved to reach the equivalent of a full outer shell, corresponding to a stable electronic configuration.
• This type of bonding primarily occurs between non-metal atoms. Since non-metals typically have high electronegativity, they prefer to share electrons rather than lose them entirely to another atom.
• The formation of these bonds is driven by the Octet Rule, which states that atoms are most stable when they have eight electrons in their valence shell, similar to the electron configuration of noble gases.
• Depending on the number of shared pairs, covalent bonds can be classified as single bonds (sharing one pair), double bonds (sharing two pairs), or triple bonds (sharing three pairs).

Additional Information

• Ionic Bonding: Unlike covalent bonding, this involves the complete transfer of electrons from one atom to another, usually from a metal to a non-metal. This results in the formation of oppositely charged ions (cations and anions) that attract each other.
• Polar Covalent Bonds: When electrons are shared unequally between two atoms due to differences in electronegativity, the bond is polar. A common example is the water molecule (H2O), where oxygen attracts electrons more strongly than hydrogen.
• Non-polar Covalent Bonds: These are formed when electrons are shared equally between atoms, typically occurring between two atoms of the same element, such as in Oxygen (O2), Nitrogen (N2), or Hydrogen (H2).
• Properties: Covalent compounds generally exhibit lower melting and boiling points compared to ionic compounds. They are typically poor conductors of electricity because they do not contain free-moving ions or electrons in their molecular form.
• Coordinate Covalent Bond: This is a special subtype where both electrons in the shared pair originate from the same atom. It is also frequently referred to as a dative bond.

Important Points

• Lewis Structures: Proposed by Gilbert N. Lewis in 1916, these diagrams represent the valence electrons of atoms within a molecule and help visualize how sharing occurs.
• Bond Strength and Length: There is an inverse relationship between bond length and strength. Triple bonds are the shortest and strongest, followed by double bonds, while single bonds are the longest and weakest among covalent types.
• VSEPR Theory: The Valence Shell Electron Pair Repulsion theory is used to predict the three-dimensional geometry of molecules based on the repulsion between shared and unshared electron pairs around a central atom.
• Electronegativity Difference: If the difference in electronegativity between two atoms is less than 1.7, the bond is generally considered covalent; if it is greater than 1.7, the bond is typically ionic.

The correct answer is HCl only releases H+ ions when dissolved in water.

Key Points

• Acids are defined by their ability to produce Hydrogen ions (H+) when dissolved in a solvent like water. In the case of Hydrogen Chloride (HCl), the dry gas consists of covalent molecules that do not contain free ions.
• When dry HCl gas is passed over dry litmus paper, there is no water present to facilitate the ionization process. Consequently, no H+ ions are released, and the litmus paper does not change its color.
• In an aqueous solution, HCl molecules react with water (H2O) to form Hydronium ions (H3O+) and Chloride ions (Cl-). This process is known as dissociation. Hence, statement 3 is the correct explanation.
• The acidic properties of a substance are solely manifested through these H+ ions. Litmus paper, which is a plant-based indicator, requires the presence of these ions to undergo a chemical change from blue to red.
• Water reacts with litmus, not HCl: This is incorrect because water alone is neutral (pH 7) and does not change the color of litmus paper; it is the HCl that provides the acidic nature once dissolved. Hence, statement 1 is incorrect.
• Litmus paper only works in the presence of light: The reaction of litmus is a chemical acid-base reaction, not a photochemical one, meaning it does not depend on light. Hence, statement 2 is incorrect.
• Dry HCl gas is non-acidic: While dry HCl does not show acidic behavior, it is technically still an acidic gas by chemical nature (it is a proton donor). The lack of moisture simply prevents the manifestation of its acidity. Hence, statement 4 is partially misleading and not the best scientific explanation.

Additional Information

• The Arrhenius Theory: This scientific theory proposed by Svante Arrhenius in 1884 states that acids are substances that dissociate in aqueous solutions to yield hydrogen ions.
• Litmus: It is a natural indicator extracted from Lichens (specifically species like Roccella tinctoria). It turns red in acidic conditions (pH below 4.5) and blue in alkaline conditions (pH above 8.3).
• Hydrogen Chloride (HCl): It is a colorless gas at room temperature. When dissolved in water, it forms Hydrochloric Acid, which is a strong mineral acid widely used in industries for pickling steel and regulating pH.
• Polarity of Water: Water is a polar solvent. The electrostatic attraction between the partial charges of water molecules and the HCl molecules is what pulls the H and Cl apart, allowing ionization to occur.

Important Points

• Hydronium Ion: It is important to note that H+ ions do not exist alone in water. They quickly combine with water molecules to form Hydronium ions (H3O+). The equation is represented as: HCl + H2O → H3O+ + Cl-.
• Moisture Requirement: If you use moist litmus paper with dry HCl gas, the gas will dissolve in the water present on the paper, forming ions and causing the color to change to red immediately.
• Safety: HCl gas is highly corrosive and toxic. It can react with the moisture in the human respiratory system to form acid, which can cause severe irritation or burns.

The correct answer is Reduce, Reuse, and Recycle.

Key Points

• The principle of Reduce, Reuse, and Recycle, often referred to as the 3Rs, is the foundation of environmental conservation and sustainable living. It aims to minimize waste and the depletion of natural resources. Hence, statement 2 is correct.
• The "Use and discard" approach is a linear consumption model that leads to massive waste generation and environmental degradation. It encourages the "throwaway culture" which fills up landfills rapidly. Hence, statement 1 is incorrect.
• The mantra of "Produce more and consume more" is linked to over-exploitation of resources. This principle prioritizes industrial growth over ecological balance, leading to a higher carbon footprint and habitat loss. Hence, statement 3 is incorrect.
• The idea to "Extract as much as possible" promotes the exhaustive removal of minerals, forest products, and water without considering the regeneration rate of the Earth. This leads to ecosystem collapse. Hence, statement 4 is incorrect.
• Implementing the 3Rs helps in energy conservation because manufacturing products from recycled materials typically requires significantly less energy than producing them from raw materials.

Additional Information

• Reduce:
• This is the most effective way to help the environment. It involves buying fewer products and using them for a longer duration.
• Examples include reducing the use of single-use plastics, opting for minimal packaging, and decreasing water and electricity consumption at home.
• Reuse:
• Instead of throwing items away, finding new ways to use them extends their life cycle.
• Common practices include using refillable glass bottles, donating old clothes, and repurposing containers for storage. This prevents the need for new resource extraction.
• Recycle:
• This involves the processing of waste materials into new products. In India, the Solid Waste Management Rules (2016) mandate the segregation of waste at the source to facilitate effective recycling.
• Materials like paper, glass, aluminum, and certain plastics can be recycled to save raw materials and reduce pollution.
• Circular Economy:
• The 3Rs are central to the concept of a Circular Economy, where waste is "designed out" of the system. This model is being adopted globally to achieve United Nations Sustainable Development Goals (SDGs), specifically SDG 12 (Responsible Consumption and Production).

Important Points

• Impact on Climate Change: By practicing Reduce, Reuse, and Recycle, we can lower greenhouse gas emissions. For instance, recycling aluminum saves about 95% of the energy required to make the same amount of aluminum from its virgin source, bauxite ore.
• Resource Management: The Earth's resources are finite. Principles of sustainability help in maintaining biodiversity and ensuring that future generations have access to necessary resources.
• Economic Benefits: Sustainable living is not just good for nature but also for the economy. It creates jobs in the recycling and upcycling industries and reduces the cost of waste management for municipalities.

The correct answer is It covers and protects organs and cavities of the body.

Key Points

• Epithelial tissue is the simplest tissue in the animal body and serves as a primary protective layer.
• Its main function is to cover the external surface of the body and line the internal organs and body cavities. Hence, statement 1 is correct.
• It acts as a barrier to keep different body systems separate, ensuring that the internal environment of various organs remains distinct.
• The cells of epithelial tissue are tightly packed and form a continuous sheet. They have almost no intercellular spaces between them, which enhances their protective capability.
• Anything entering or leaving the body must cross at least one layer of the epithelium. Because of this, the permeability of the cells of various epithelia plays a critical role in regulating the exchange of materials between the body and the external environment.
• Common examples of locations where epithelial tissue is found include the skin, the lining of the mouth, the lining of blood vessels, lung alveoli, and kidney tubules.

Additional Information

• Types of Epithelial Tissue:
• Simple Squamous Epithelium: These cells are extremely thin and flat, forming a delicate lining. They are found in sites where transport of substances occurs through a selectively permeable surface, such as the oesophagus and the lining of the mouth.
• Stratified Squamous Epithelium: Found in the skin, these cells are arranged in many layers to prevent wear and tear.
• Ciliated Columnar Epithelium: Found in the respiratory tract, these cells have hair-like projections called cilia that move to clear mucus forward.
• Cuboidal Epithelium: These cube-shaped cells provide mechanical support and are found in kidney tubules and the ducts of salivary glands. Hence, statement 2 is incorrect as mechanical support is a secondary function specific to cuboidal types, not the primary general function of all epithelial tissues.
• Glandular Epithelium: Sometimes a portion of epithelial tissue folds inward to form a multicellular gland for secretion.
• Other Animal Tissues:
• Connective Tissue: This tissue connects different organs and provides structural support. For example, tendons connect muscles to bones, and ligaments connect bones to bones. Hence, statement 3 is incorrect.
• Fluid Connective Tissue: Blood is a type of connective tissue that has a fluid matrix called plasma. It is responsible for transporting gases, digested food, hormones, and waste materials to different parts of the body. Hence, statement 4 is incorrect.
• Muscular Tissue: Responsible for movement in our body, consisting of elongated cells called muscle fibres.
• Nervous Tissue: Specialized for being stimulated and then transmitting the stimulus very rapidly from one place to another within the body.

Important Points

• All epithelium is usually separated from the underlying tissue by an extracellular fibrous basement membrane.
• The skin, which protects the body, is also made of squamous epithelium arranged in patterns of layers.
• In the intestine, where absorption and secretion occur, tall epithelial cells called columnar epithelium are present to facilitate movement across the epithelial barrier.
• The study of tissues is known as Histology.

The correct answer is 1 kWh.

Key Points

• The electrical energy consumed by an appliance can be calculated using the formula: Energy = Power x Time.
• In this problem, the power (P) of the electric heater is given as 1000 W and the time (t) is given as 1 hour.
• Multiplying these values, we get: 1000 Watts x 1 hour = 1000 Watt-hours (Wh).
• By definition, 1000 Watt-hours is equal to 1 kilowatt-hour (1 kWh), which is the standard unit for measuring energy consumption. Hence, the correct answer is 1 kWh.
• If we were to convert this into the SI unit of energy (Joules), 1 kWh would be equal to 1000 Watts x 3600 seconds, which results in 3,600,000 Joules.

Additional Information

• Kilowatt-hour (kWh):
• The kilowatt-hour is the commercial unit of electrical energy used by electricity boards to calculate bills.
• One unit of electricity consumed in a household refers to 1 kWh of energy used.
• It represents the amount of energy dissipated by a 1000-watt appliance working continuously for one hour.
• Power and its Units:
• Power is defined as the rate at which work is done or energy is consumed.
• The SI unit of power is the Watt (W), named after James Watt. 1 Watt is equivalent to 1 Joule per second (1 J/s).
• For high-power devices like heaters, motors, and engines, power is often measured in kilowatts (kW) or Horsepower (hp), where 1 hp is approximately 746 Watts.
• Joule (J):
• The Joule is the SI unit of energy. It is a very small unit compared to the kilowatt-hour.
• To convert kWh to Joules, you multiply by 3.6 million (3,600,000) because there are 3600 seconds in an hour and 1000 Watts in a kilowatt.
• Practical Examples:
• An LED bulb rated at 10 W would take 100 hours to consume 1 kWh of energy.
• In contrast, a heavy-duty air conditioner rated at 2000 W (2 kW) would consume 1 kWh in just 30 minutes.
• Understanding these ratings helps in managing household energy efficiency and reducing electricity costs.

The correct answer is +3/4.

Key Points

• Magnification (m) produced by a spherical mirror is the ratio of the height of the image to the height of the object, which is also related to the object distance (u) and image distance (v).
• According to the New Cartesian Sign Convention, the object distance (u) is always taken as negative because the object is placed in front of the mirror. In this case, u = -40 cm.
• For a convex mirror, the image is always virtual, erect, and formed behind the mirror. Therefore, the image distance (v) is taken as positive. In this case, v = +30 cm.
• The formula for magnification (m) for a mirror is given by m = -v / u. Substituting the given values into the formula: m = -(30) / (-40).
• The negative signs in the denominator and the numerator (from the formula) cancel out, resulting in m = 30 / 40, which simplifies to +3/4 or +0.75. Hence, the calculated magnification is +3/4.
• The positive sign in the magnification value indicates that the image formed is virtual and erect, which is a characteristic property of all images formed by a convex mirror.

Additional Information

• Convex Mirror Properties:
• A convex mirror is also known as a diverging mirror because it diverges light rays that strike its reflective surface.
• The focal length (f) and the radius of curvature (R) of a convex mirror are always considered positive.
• Regardless of the position of the object, a convex mirror always forms a diminished, virtual, and erect image between the pole and the focus.
• Uses of Convex Mirrors:
• They are widely used as rear-view mirrors in vehicles because they provide a wider field of view and always form an erect image.
• Used in security mirrors in shops and hallways to see around corners and monitor large areas.
• Sign Convention Rules:
• Distances measured in the direction of incident light are positive.
• Distances measured against the direction of incident light are negative.
• Heights measured upward and perpendicular to the principal axis are positive, while those downward are negative.

Important Points

• Magnitude of Magnification:
• If |m| > 1, the image is enlarged (larger than the object).
• If |m| < 1, the image is diminished (smaller than the object). In this problem, 0.75 < 1, confirming the image is diminished.
• If |m| = 1, the image is the same size as the object, which typically happens with a plane mirror or when an object is at the Center of Curvature of a concave mirror.
• Nature of Image from Sign:
• A positive sign (+) for magnification indicates a virtual and erect image.
• A negative sign (-) for magnification indicates a real and inverted image.
• Since a convex mirror cannot form a real image of a real object, the magnification will always be positive and less than 1.

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

Key Points

• Loudness and Amplitude: The loudness of a sound is determined by the amplitude of the sound wave. A sound wave with a larger amplitude results in a louder sound, whereas a wave with a smaller amplitude produces a soft or feeble sound. Because a loud sound requires a greater displacement of particles, it possesses a higher amplitude than a soft sound. Hence, statement (i) is incorrect.
• Energy in Sound Waves: Sound is a form of mechanical energy that propagates through a medium. The energy carried by a sound wave is directly proportional to the square of its amplitude. Since loud sounds have larger amplitudes, they carry significantly more energy compared to soft sounds. Hence, statement (ii) is correct.
• Response of the Ear: Loudness is a physiological sensation rather than just a physical measurement. It is the measure of the response of the human ear to the intensity of sound. Even if two sounds have the same physical intensity, our ears might perceive one as louder than the other if it falls within a frequency range where the ear is more sensitive. Hence, statement (iii) is correct.
• Decibel Scale: The level of loudness is expressed in a unit called the decibel (dB). On this scale, a sound of 80 dB or above is physically painful to the human ear, while normal conversation typically occurs at around 60 dB.
• Proportionality: The loudness of sound is proportional to the square of the amplitude of the vibration producing the sound. For example, if the amplitude becomes twice as large, the loudness increases by a factor of four.

Additional Information

• Intensity vs. Loudness: While often used interchangeably, they are different. Intensity is an objective physical quantity representing the sound energy passing per unit area per unit time (measured in Watts per square meter). Loudness is a subjective sensation that depends on the sensitivity of the listener's ear.
• Pitch and Frequency: The pitch of a sound depends on its frequency. A high-frequency sound (like a whistle) has a high pitch, while a low-frequency sound (like a drum) has a low pitch. Loudness and pitch are independent characteristics of sound.
• Quality or Timbre: This is the characteristic of sound that enables us to distinguish between two sounds of the same loudness and pitch produced by different sources, such as a violin and a flute.
• Human Auditory Range: The human ear can typically hear sounds in the frequency range of 20 Hz to 20,000 Hz. Sounds below this are infrasonic, and sounds above this are ultrasonic.

Important Points

• Factors Affecting Loudness:
• Surface Area: A larger vibrating surface area (like a large bell) produces a louder sound than a smaller one.
• Distance from Source: As the distance between the listener and the source increases, the loudness decreases because the sound energy spreads out over a larger spherical area.
• Density of Medium: Loudness also depends on the density of the medium through which the sound travels; denser media generally transmit sound energy more efficiently.
• Threshold of Hearing: The quietest sound a healthy human ear can detect is assigned a value of 0 dB. This corresponds to an intensity of 10 to the power of minus 12 Watts per square meter.
• Noise Pollution: Persistent exposure to sounds above 85 to 90 dB can lead to permanent hearing loss and other health issues like hypertension and stress.

The correct answer is A zigzag line with an arrow which is drawn across it.

Key Points

• A rheostat is an electrical instrument used to control current by varying the resistance in a circuit without changing the voltage source.
• The standard circuit symbol for a rheostat is a zigzag line (which represents resistance) with a diagonal arrow drawn across it or an arrow pointing down onto the zigzag. Hence, option 4 is correct.
• It functions as a variable resistor and is typically constructed using a resistive wire (like manganin or constantan) wrapped around an insulating ceramic core, with a sliding contact.
• By moving the slider (the arrow in the symbol), the effective length of the wire through which the current passes changes, thereby altering the total resistance of the circuit according to the formula R = ρ(L/A).
• In Ohm's Law experiments, a rheostat is frequently used to adjust the current flowing through a conductor to observe the corresponding change in potential difference.

Additional Information

• Ammeter:
• Represented by an 'A' inside a circle.
• It is a device used to measure the electric current in a circuit in Amperes.
• It has low resistance and is always connected in series with the circuit component. Hence, statement 1 is incorrect.
• Voltmeter:
• Represented by a circle with a 'V' inside it.
• It is used to measure the potential difference (voltage) between two points in an electrical circuit.
• It has high resistance and is always connected in parallel across the points where the voltage is to be measured. Hence, statement 2 is incorrect.
• Fixed Resistor:
• Represented by a simple zigzag line.
• It provides a specific, unchanging amount of electrical resistance to the flow of current.
• The unit of measurement for resistance is the Ohm (Ω). Hence, statement 3 is incorrect.
• Galvanometer:
• Represented by a 'G' inside a circle.
• It is an electromechanical instrument used for detecting and indicating the presence and direction of small electric currents.

Important Points

• Potentiometer vs. Rheostat: While both are variable resistors, a potentiometer is typically used as a voltage divider using three terminals, whereas a rheostat is used to control current using only two terminals.
• Applications: Rheostats are used in motor speed control, light dimmers, and older volume control knobs in radio sets.
• Ohm’s Law: The relationship V = IR dictates that if voltage (V) is constant, increasing the resistance (R) using a rheostat will decrease the current (I).
• Safety: When using a rheostat in laboratory experiments, it is vital to ensure it can handle the wattage or heat generated by the current to prevent damage to the component.

The correct answer is The constituents can be separated fairly easily by physical methods..

Key Points

• A compound is a substance composed of two or more different elements that are chemically bonded together in a fixed proportion by mass.
• The constituents of a compound cannot be separated by physical methods such as filtration, evaporation, or distillation. They can only be separated into their constituent elements by chemical or electrochemical reactions. Hence, statement 1 is incorrect (and thus the correct answer).
• In a compound, the composition of the elements is always fixed. For example, in Water (H2O), the ratio of the mass of hydrogen to the mass of oxygen is always 1:8, regardless of the source. Hence, statement 2 is correct.
• A compound possesses totally different properties from its constituent elements. For instance, Sodium (Na) is a highly reactive metal and Chlorine (Cl) is a poisonous gas, but their compound, Sodium Chloride (NaCl), is a harmless table salt used in food. Hence, statement 3 is correct.
• Compounds are formed only when elements react chemically with each other. During this process, chemical bonds are formed or broken, leading to the creation of a new substance. Hence, statement 4 is correct.

Additional Information

• Chemical Substances:
• Elements: Pure substances consisting of only one type of atom (e.g., Gold, Oxygen).
• Compounds: Substances formed by the chemical combination of two or more elements (e.g., Carbon Dioxide, Glucose).
• Mixtures vs. Compounds:
• In a mixture, substances are physically blended but not chemically combined. They retain their individual properties and can be separated by physical methods like magnetization or sieving.
• In a compound, the identity of the individual elements is lost, and a new chemical identity is formed.
• Homogeneity: Compounds are always homogeneous, meaning they have a uniform composition and properties throughout the sample.
• Energy Changes: The formation of a compound is usually accompanied by a significant energy change, typically in the form of heat or light being absorbed or released.

Important Points

• Law of Definite Proportions: This law, also known as Proust's Law, states that a chemical compound always contains exactly the same proportion of elements by mass.
• Electrolysis: This is a common electrochemical method used to separate compounds. For example, electrolysis of water produces Hydrogen and Oxygen gases.
• Melting and Boiling Points: Compounds have sharp and fixed melting and boiling points, which is a characteristic feature that distinguishes them from mixtures.
• Representation: Every compound is represented by a chemical formula (like H2SO4 for Sulphuric Acid), which indicates the fixed number of atoms of each element present in one molecule of that compound.

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

Key Points

• Work Done Definition: In physics, work is said to be done when a force acts upon an object to cause a displacement. It is mathematically expressed as W = Fs cosθ, where F is the force, s is the displacement, and θ is the angle between the force and displacement vectors.
• Analysis of Statement (i): The work done by a force can be zero under three conditions: if the force applied is zero, if the displacement is zero, or if the force acts perpendicularly (at an angle of 90 degrees) to the direction of motion (since cos 90 = 0). Examples include a person standing still with a heavy load or a satellite moving in a circular orbit where the gravitational force is perpendicular to the velocity. Hence, statement (i) is incorrect.
• Analysis of Statement (ii): If the force and the displacement are in the same direction, the angle θ is 0 degrees. Because the value of cos 0 is 1, the work done is positive. This is common when pushing an object across a floor in the direction of the push. Hence, statement (ii) is correct.
• Analysis of Statement (iii): The Standard International (S.I.) unit of work is the Joule (J). One Joule is defined as the amount of work done when a force of 1 Newton displaces an object by 1 meter in the direction of the force. Hence, statement (iii) is correct.
• Hence, the correct answer is Both (ii) and (iii).

Additional Information

• Nature of Work:
• Work is a scalar quantity, meaning it has only magnitude and no direction, even though it is the dot product of two vector quantities (force and displacement).
• Negative Work: Work is considered negative when the force acts in the opposite direction of displacement (angle θ = 180 degrees). A primary example is the force of friction acting against a moving object.
• Units and Conversions:
• In the CGS system, the unit of work is the erg.
• The relationship between the Joule and the Erg is 1 Joule = 10^7 ergs.
• The dimensional formula for work is [M1 L2 T-2], which is identical to the dimensional formula for energy and torque.
• Types of Work in Different Contexts:
• Mechanical Work: Moving an object against a force.
• Thermodynamic Work: Often represented as PΔV (pressure times change in volume) in gases.
• Electrical Work: Done when a charge is moved through a potential difference.

Important Points

• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy. If the work done is positive, the kinetic energy increases; if negative, the kinetic energy decreases.
• Zero Work Examples:
• A coolie carrying a luggage on his head and walking on a horizontal platform does zero work against gravity because the force (upward) is perpendicular to the displacement (horizontal).
• The centripetal force acting on a body in uniform circular motion does no work because it is always directed towards the center, perpendicular to the instantaneous displacement.
• Constant vs. Variable Force: While the question specifies a constant force, it is important to note that if the force varies with position, the work done is calculated as the integral of force over the displacement (the area under a Force-Displacement graph).

Shortcut Trick

Use the multiplying factor method for successive changes.

Net Factor = (1 − 0.35) × (1 − 0.55) × (1 − 0.35) × (1 + 0.65)

Net Factor = 0.65 × 0.45 × 0.65 × 1.65 ≈ 0.3137

Percentage change = (0.3137 − 1) × 100 = −68.63%

∴ The correct answer is Decreases by 68.63%.

Alternate Method

Given: Successive changes of −35%, −55%, −35% and +65%.

Formula: Final Price = Initial Price × (100 ± r₁)/100 × (100 ± r₂)/100 ...

⇒ Let the original price be 100 units.

⇒ Price after 1st month = 100 × (65/100) = 65

⇒ Price after 2nd month = 65 × (45/100) = 29.25

⇒ Price after 3rd month = 29.25 × (65/100) = 19.0125

⇒ Price after 4th month = 19.0125 × (165/100) = 31.370625

⇒ Net change = 31.370625 − 100 = −68.629375

⇒ Percentage decrease ≈ 68.63%

∴ The correct answer is Decreases by 68.63%.

Additional Information

Successive Percentage Formula

For two successive changes a% and b%, the net change is (a + b + ab/100)%.

Percentage Increase/Decrease

Percentage change = (Final Value − Initial Value) / Initial Value × 100.

Multiplying Factor (MF)

For an x% increase, MF = (1 + x/100). For an x% decrease, MF = (1 − x/100).

Shortcut Trick

Larger number = (Sum + Difference) ÷ 2 = (74 + 12) ÷ 2 = 43

Smaller number = (Sum − Difference) ÷ 2 = (74 − 12) ÷ 2 = 31

Required Ratio (Smaller : Larger) = 31 : 43

∴ The correct answer is 31 : 43.

Alternate Method

Given: Sum of numbers = 74, Difference = 12

Formula: If x + y = S and x − y = D, then x = (S+D)/2 and y = (S−D)/2

⇒ Let the numbers be x (larger) and y (smaller).

⇒ x + y = 74

⇒ x − y = 12

⇒ Adding both equations: 2x = 86 ⇒ x = 43

⇒ Subtracting both equations: 2y = 62 ⇒ y = 31

⇒ Ratio y : x = 31 : 43

∴ The correct answer is 31 : 43.

Additional Information

Linear Equations in Two Variables

When the sum (S) and difference (D) are known, the values are (S+D)/2 and (S−D)/2.

Ratio and Proportion

A ratio compares two quantities by division; always ensure the order (smaller to larger) matches the question.

Shortcut Trick

For a cyclic quadrilateral with sides a, b, c, and d, the area is given by Brahmagupta's Formula:

Area = √((s − a)(s − b)(s − c)(s − d))

where s is the semi-perimeter: s = (a + b + c + d) ÷ 2

Given: a = 10 cm, b = 14 cm, c = 8 cm, d = 12 cm

⇒ s = (10 + 14 + 8 + 12) ÷ 2 = 22 cm

⇒ Area = √((22 − 10)(22 − 14)(22 − 8)(22 − 12)) = √(12 × 8 × 14 × 10) = √(13440) = 8√210 cm²

∴ The correct answer is 8√210 cm².

Alternate Method

Given:

Sides of the cyclic quadrilateral ABCD:

⇒ AB = a = 10 cm

⇒ BC = b = 14 cm

⇒ CD = c = 8 cm

⇒ DA = d = 12 cm

Formula Used:

Area of a cyclic quadrilateral = √((s − a)(s − b)(s − c)(s − d))

where s = (a + b + c + d) ÷ 2

Calculations:

Step 1: Find the semi-perimeter (s) of the cyclic quadrilateral:

⇒ s = (10 + 14 + 8 + 12) ÷ 2

⇒ s = 44 ÷ 2

⇒ s = 22 cm

Step 2: Substitute the values of s, a, b, c, and d into Brahmagupta's formula:

⇒ Area = √((22 − 10) × (22 − 14) × (22 − 8) × (22 − 12))

⇒ Area = √(12 × 8 × 14 × 10)

⇒ Area = √(96 × 140)

⇒ Area = √(13440)

Step 3: Simplify the square root:

⇒ Area = √(64 × 210)

⇒ Area = 8√210 cm²

Verification of Options:

⇒ Option (1): 8√222 cm² = √(64 × 222) = √(14208) ≠ √(13440) (Incorrect)

⇒ Option (2): 10√210 cm² = √(100 × 210) = √(21000) ≠ √(13440) (Incorrect)

⇒ Option (3): 8√210 cm² = √(64 × 210) = √(13440) (Correct)

⇒ Option (4): 10√222 cm² = √(100 × 222) = √(22200) ≠ √(13440) (Incorrect)

∴ The correct answer is 8√210 cm².

Additional Information

Brahmagupta's Formula

This formula is a generalization of Heron's formula for the area of a triangle and is specifically used to find the area of any cyclic quadrilateral given only its side lengths.

Ptolemy's Theorem

For any cyclic quadrilateral ABCD, the product of the diagonals is equal to the sum of the products of the opposite sides: AC × BD = AB × CD + BC × DA.

Opposite Angles Property

In a cyclic quadrilateral, the sum of opposite angles is always supplementary, i.e., ∠A + ∠C = 180° and ∠B + ∠D = 180°.

Shortcut Trick

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

Here, a = 9% and b = 6%.

Equivalent discount = (9 + 6 − (9 × 6) ÷ 100)%.

Equivalent discount = (15 − 54 ÷ 100)% = (15 − 0.54)% = 14.46%.

∴ The correct answer is 14.46%.

Alternate Method

Given → First discount (d₁) = 9%, Second discount (d₂) = 6%. Formulas: Single Equivalent Discount = 100 − [100 × (100 − d₁)/100 × (100 − d₂)/100]

⇒ Let the Marked Price (MP) be 100. ⇒ Price after 9% discount = 100 − 9 = 91. ⇒ Price after 6% discount on 91 = 91 − (6% of 91) = 91 − 5.46 = 85.54. ⇒ Total discount = 100 − 85.54 = 14.46. ⇒ Single equivalent discount % = (14.46 ÷ 100) × 100 = 14.46%. ∴ The correct answer is 14.46%.

Additional Information

Successive Discounts

When a series of discounts are allowed one after another, they are called successive discounts. The net discount is always less than the sum of individual discount rates.

Effective Discount for 3 values

For three successive discounts a%, b%, and c%, first calculate the equivalent of a and b, then use that result with c.

Shortcut Trick

Combined efficiency (P+Q) = 1/15, P efficiency = 1/60.

Efficiency of Q = (1/15) − (1/60) = (4 − 1)/60 = 3/60 = 1/20.

Time for Q to fill 1/4th of cistern = (1/4) ÷ (1/20) = 20 ÷ 4 = 5 hours.

∴ The correct answer is 5 hours.

Alternate Method

⇒ Total capacity of cistern = LCM of 15 and 60 = 60 units.

⇒ Efficiency of Q = (Efficiency of P + Q) − (Efficiency of P)

⇒ Efficiency of Q = 4 − 1 = 3 units/hour.

⇒ One-fourth of cistern capacity = 60 × (1/4) = 15 units.

⇒ Time taken by Q = Work ÷ Efficiency = 15 ÷ 3 = 5 hours.

∴ The correct answer is 5 hours.

Additional Information

Combined Efficiency Rule

If two pipes fill a tank in T₁ and T₂ hours respectively, their combined work rate is 1/T₁ + 1/T₂.

Pipes and Cisterns Logic

Inlet pipes have positive efficiency, while outlet (leakage) pipes have negative efficiency. Total Work = Time × Efficiency.

Shortcut Trick

For Compound Interest at 6% for 2 years, the Effective Interest Rate = x + y + (xy/100)

Effective Rate = 6 + 6 + (6 × 6)/100 = 12.36%

Total Amount = 100% (Principal) + 12.36% (Interest) = 112.36% of Principal

Amount = 7500 × 112.36/100 = 75 × 112.36 = ₹8,427

∴ The correct answer is ₹8,427.

Alternate Method

Given: Principal (P) = ₹7,500, Rate (R) = 6% per annum, Time (n) = 2 years.

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

⇒ A = 7500 × (1 + 6/100)²

⇒ A = 7500 × (106/100) × (106/100)

⇒ A = 7500 × (1.06) × (1.06)

⇒ A = 7500 × 1.1236

⇒ A = ₹8,427

∴ The correct answer is ₹8,427.

Additional Information

CI vs SI Difference

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

Half-Yearly Compounding

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

Effective Rate for 3 Years

The effective rate for 3 years can be calculated by applying the successive formula [x + y + (xy/100)] twice or using 3R.3R²R³ shortcut.

Shortcut Trick

Total Cost Price (CP) of 24 kg = 3,600 ⇒ CP per kg = 3,600 ÷ 24 = 150.

First 12 kg (half) gain: 12 × 150 × 20% = +360.

Remaining 12 kg (half) loss: 12 × 150 × 10% = −180.

Net Profit/Loss = 360 − 180 = 180 (Positive value indicates Gain).

∴ The correct answer is Gain ₹180.

Alternate Method

Given → Total CP = 3,600 | 1st half gain = 20% | 2nd half loss = 10%. Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100 | SP = CP × (100 + P%) ÷ 100.

⇒ Total Cost Price = 3,600. CP of 12 kg = 1,800. ⇒ SP of first 12 kg = 1,800 × (120/100) = 2,160. ⇒ SP of remaining 12 kg = 1,800 × (90/100) = 1,620. ⇒ Total Selling Price = 2,160 + 1,620 = 3,780. ⇒ Net Gain = Total SP − Total CP = 3,780 − 3,600 = 180.

∴ The correct answer is Gain ₹180.

Additional Information

Net Gain/Loss on Equal Parts

When two equal parts are sold at x% gain and y% loss, the net percentage change is (x − y) ÷ 2.

Selling Price Calculation

SP = CP × (100 ± Change%) ÷ 100. This is the basic multiplier for profit or loss adjustments.

Shortcut Trick

Let the total votes be 100%.

Dinesh got 46.5%, so Rizwan got 100% − 46.5% = 53.5%.

Difference in percentage = 53.5% − 46.5% = 7%.

Given 7% ≡ 3150 votes.

Rizwan's votes (53.5%) = (3150 ÷ 7) × 53.5 = 450 × 53.5 = 24075.

∴ The correct answer is 24,075 votes.

Alternate Method

Given: Votes for Dinesh = 46.5% | Margin = 3150 votes | All votes are valid.

⇒ Total Votes = 100%

⇒ Votes obtained by Rizwan = 100% − 46.5% = 53.5%

⇒ Difference between Rizwan and Dinesh = 53.5% − 46.5% = 7%

⇒ According to the question, 7% of Total Votes = 3150

⇒ 1% of Total Votes = 3150 ÷ 7 = 450

⇒ Rizwan's share = 53.5 × 450

⇒ Rizwan's share = 24075

∴ The correct answer is 24,075 votes.

Additional Information

Percentage Change

Percentage change is calculated as (Difference ÷ Original Value) × 100.

Valid and Invalid Votes

Total Cast Votes = Valid Votes + Invalid Votes. Winners and losers are usually calculated based on Valid Votes.

Winning Margin

Margin = Votes of Winner − Votes of Loser. It represents the number of votes by which the winner defeated the opponent.

Shortcut Trick

For third proportional to a and b, the formula is c = b²/a.

Given: a = 50, b = Z, and c = 18.

⇒ 18 = Z² ÷ 50

⇒ Z² = 18 × 50 = 900

⇒ Z = √900 = 30

∴ The correct answer is 30.

Alternate Method

Given:

First number (a) = 50

Second number (b) = Z

Third proportional (c) = 18

Formula Used:

In continued proportion a, b, c ⇒ a/b = b/c

Calculations:

⇒ 50 / Z = Z / 18

⇒ Z × Z = 50 × 18

⇒ Z² = 900

⇒ Z = √900

⇒ Z = 30

∴ The correct answer is 30.

Additional Information

Mean Proportional

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

Fourth Proportional

If a : b :: c : d, then d is the fourth proportional and d = (b × c) ÷ a.

Continued Proportion

Three quantities a, b, c are in continued proportion if the ratio of first to second equals second to third (a/b = b/c).

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 300 + 300 = 600 m

Total Time = 100 + 50 = 150 sec

Average Speed = 600 ÷ 150 = 4 m/sec

∴ The correct answer is 4 m/sec.

Alternate Method

Given:

Distance of first lap = 300 m, Time taken = 100 sec

Distance of second lap = 300 m, Time taken = 50 sec

Formula Used:

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Total Distance = 300 + 300 = 600 m

⇒ Total Time = 100 + 50 = 150 sec

⇒ Average Speed = 600 ÷ 150

⇒ Average Speed = 4 m/sec

∴ The correct answer is 4 m/sec.

Additional Information

Average Speed (Equal Distances)

When two equal distances are covered at speeds x and y, the average speed is 2xy ÷ (x + y).

Unit Conversion

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

Shortcut Trick

Sum of ages 5 years ago → 67 − (5 × 2) = 57 years.

Age ratio 5 years ago (Elder : Younger) = 2 : 1.

Elder sister's age 5 years ago = (2/3) × 57 = 38 years.

Current age of elder sister = 38 + 5 = 43 years.

∴ The correct answer is 43 years.

Alternate Method

Given: Sum of present ages = 67 years. Ratio 5 years ago = 2 : 1.

Formula: Age after 'n' years = Age 'n' years ago + (n × 2 for two people).

⇒ Let ages 5 years ago be 2x and x.

⇒ Present ages = (2x + 5) and (x + 5).

⇒ (2x + 5) + (x + 5) = 67

⇒ 3x + 10 = 67

⇒ 3x = 57 ⇒ x = 19.

⇒ Elder sister's current age = 2x + 5 = (2 × 19) + 5 = 38 + 5 = 43.

∴ The correct answer is 43 years.

Additional Information

Age Difference Constant

The difference between the ages of two people always remains constant, regardless of the number of years passed.

Sum of Ages Change

If there are 'n' people, the sum of their ages increases or decreases by (n × number of years).

Ratio Method

When a constant number of years is added to both parts of a ratio, the difference between ratio units should be balanced if the age difference is known.

Shortcut Trick

Given sec θ = Hypotenuse / Base = 17 / 12.

Using Pythagoras Theorem: Perpendicular² = Hypotenuse² − Base².

P² = 17² − 12² = 289 − 144 = 145 ⇒ P = √145.

sin θ = Perpendicular / Hypotenuse = √145 / 17.

∴ The correct answer is √145/17.

Alternate Method

Given: sec θ = 17/12

Formula Used: sin²θ + cos²θ = 1 OR sec θ = 1 / cos θ

⇒ cos θ = 1 / sec θ = 12 / 17

⇒ sin²θ = 1 − cos²θ

⇒ sin²θ = 1 − (12/17)²

⇒ sin²θ = 1 − 144 / 289

⇒ sin²θ = (289 − 144) / 289 = 145 / 289

⇒ sin θ = √(145 / 289) = √145 / 17

∴ The correct answer is √145/17.

Additional Information

Trigonometric Identities

Fundamental relations include sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Right Angled Triangle Ratios

In a right triangle: sin θ = P/H, cos θ = B/H, tan θ = P/B, where P is perpendicular, B is base, and H is hypotenuse.

Pythagoras Theorem

For any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: H² = P² + B².

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 430 + 470 + 620 = 1520 m

Total Time = 10 + 11 + 17 = 38 minutes

Average Speed = 1520 ÷ 38 = 40 m/minutes

∴ The correct answer is 40 m/minutes.

Alternate Method

Given:

Distance 1 (d₁) = 430 m, Time 1 (t₁) = 10 minutes

Distance 2 (d₂) = 470 m, Time 2 (t₂) = 11 minutes

Distance 3 (d₃) = 620 m, Time 3 (t₃) = 17 minutes

Formula Used:

Average Speed = (d₁ + d₂ + d₃) ÷ (t₁ + t₂ + t₃)

Calculations:

⇒ Total Distance = 430 + 470 + 620 = 1520 m

⇒ Total Time = 10 + 11 + 17 = 38 minutes

⇒ Average Speed = 1520 ÷ 38

⇒ Average Speed = 40 m/minutes

∴ The correct answer is 40 m/minutes.

Additional Information

Average Speed for Equal Distances

If a person covers two equal distances at speeds x and y, the average speed is given by (2xy) ÷ (x + y).

Relative Speed

Speed of two bodies moving in the same direction is (u − v) and in opposite directions is (u + v).

Unit Conversion

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

Shortcut Trick

Deviation = New Score − Old Average = 157 − 90 = 67 runs.

Increase in Average = Deviation ÷ Total Innings = 67 ÷ 7 ≈ 9.57 runs.

Rounding 9.57 to the nearest whole number gives 10 runs.

∴ The correct answer is 10 runs.

Alternate Method

Given:

Average for 6 innings = 90 runs

Runs in 7^th inning = 157 runs

Formula Used:

Average = Total Runs ÷ Number of Innings

Calculations:

⇒ Total runs in 6 innings = 6 × 90 = 540 runs

⇒ Total runs in 7 innings = 540 + 157 = 697 runs

⇒ New average = 697 ÷ 7 = 99.5714... runs

⇒ Increase in average = 99.57... − 90 = 9.57... runs

⇒ Rounding 9.57 to the nearest whole number ≈ 10

∴ The correct answer is 10 runs.

Additional Information

Average Change Formula

When a new value is added, New Average = Old Average + (Extra Runs ÷ New Total Innings).

Weighted Average

Weighted Avg = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂), where n is the number of items and A is the average.

Batting Average in Cricket

Batting Average = (Total Runs Scored) ÷ (Number of times Out). In simple problems, innings played is often treated as times out.

Shortcut Trick

Check options for Sum = 88 and LCM = 870.

Option (1): 30 + 58 = 88. LCM of 30 and 58: 30 = 2 × 3 × 5; 58 = 2 × 29. LCM = 2 × 3 × 5 × 29 = 870.

Option (2): 28 + 60 = 88. LCM ends in 0 but 28 is not a factor of 870.

Option (3): 23 + 65 = 88. 23 is prime, 870 is not divisible by 23.

∴ The correct answer is 30, 58.

Alternate Method

Given: Sum of numbers (x + y) = 88, LCM(x, y) = 870.

Formula: Product of two numbers = HCF × LCM.

⇒ Let the numbers be x and y.

⇒ x + y = 88

⇒ LCM(x, y) = 870

⇒ Prime factors of 870 = 2 × 3 × 5 × 29

⇒ Since sum is 88, one number must be 2 × 29 = 58 (as 29 is a large prime factor).

⇒ If one number is 58, then the other number = 88 − 58 = 30.

⇒ Verification: LCM(30, 58) = LCM(2 × 3 × 5, 2 × 29) = 2 × 3 × 5 × 29 = 870.

∴ The correct answer is 30, 58.

Additional Information

Product Rule

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

HCF of Sum and LCM

The HCF of two numbers also divides their sum and their LCM.

Prime Factorization Method

LCM is the product of the highest powers of all prime factors involved in the numbers.

Shortcut Trick

Identify square roots by looking at unit digits and nearest tens.

√6889 → ends in 9 (3 or 7), between 80² and 90² → 83.

√4761 → ends in 1 (1 or 9), between 60² and 70² → 69.

8√x = 83 + 69 = 152 ⇒ √x = 19 ⇒ x = 361.

∴ The correct answer is 361.

Alternate Method

Given: 8(x)^1/2 = √6889 + √4761

Step 1: Calculate √6889

⇒ 80² = 6400 and 90² = 8100. Since 6889 ends in 9, the square root must be 83 or 87.

⇒ 85² = 7225. Since 6889 < 7225, √6889 = 83.

Step 2: Calculate √4761

⇒ 60² = 3600 and 70² = 4900. Since 4761 ends in 1, the square root must be 61 or 69.

⇒ 65² = 4225. Since 4761 > 4225, √4761 = 69.

Step 3: Substitute values back into the equation

⇒ 8√x = 83 + 69

⇒ 8√x = 152

⇒ √x = 152 ÷ 8

⇒ √x = 19

Step 4: Squaring both sides to find x

⇒ x = 19²

⇒ x = 361

∴ The correct answer is 361.

Additional Information

Square Root Extraction

For a perfect square ending in 9, the unit digit of the root is 3 or 7; for 1, it is 1 or 9.

Squaring trick for 5

A number (n5)² can be calculated as [n × (n+1)] followed by 25. e.g., 65² = (6×7)25 = 4225.

BODMAS Rule

Always solve operations in the order: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.

Shortcut Trick

Let Total Capacity be LCM(9, 12, 18) = 36 units.

Efficiencies: A = 4, B = 3, C = −2. Combined efficiency (A+B−C) = 4 + 3 − 2 = 5.

Work in first 4 hours = 5 × 4 = 20 units. Remaining work = 36 − 20 = 16 units.

Time for A+B to finish = 16 ÷ 7 hours. Total time = 4 + 16/7 = (28 + 16)/7 = 44/7 hours.

∴ The correct answer is 44/7 hr.

Alternate Method

Given: Filling time Pipe A = 9 hr, Pipe B = 12 hr, Emptying time Pipe C = 18 hr.

⇒ Total capacity of tank = 36 units (LCM of 9, 12, 18).

⇒ Efficiency of (A + B + C) = 4 + 3 − 2 = 5 units/hr.

⇒ Water filled in 4 hours = 5 × 4 = 20 units.

⇒ Remaining capacity to be filled = 36 − 20 = 16 units.

⇒ Time taken by (A + B) to fill remaining part = 16 ÷ (4 + 3) = 16/7 hours.

⇒ Total time taken = 4 + 16/7 = 44/7 hours.

∴ The correct answer is 44/7 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency) while outlet pipes remove water (negative efficiency). Net work per hour is the algebraic sum of efficiencies.

Time and Work Relation

Total Work = Efficiency × Time. If a pipe can fill a tank in 'x' hours, its 1-hour work is 1/x of the total capacity.

Shortcut Trick

When multiple identical spheres are melted to form a single sphere, the volume remains constant.

Relationship: R³ = n × r³, where R is the new radius, n is the count, and r is the old radius.

R³ = 8 × 10³

R³ = 8 × 1000 = 8000 ⇒ R = 20 cm

∴ The correct answer is 20 cm.

Alternate Method

Given:

Number of spherical balls (n) = 8

Radius of each small ball (r) = 10 cm

Formula Used:

Volume of a Sphere = 4/3 × π × radius³

Calculations:

⇒ Total volume of 8 small balls = 8 × (4/3 × π × 10³)

⇒ Volume of the new single large ball = 4/3 × π × R³

⇒ According to the principle of conservation of volume:

⇒ 4/3 × π × R³ = 8 × 4/3 × π × 1000

⇒ R³ = 8 × 1000

⇒ R³ = 8000

⇒ R = √³8000

⇒ R = 20 cm

∴ The correct answer is 20 cm.

Additional Information

Volume Conservation

When a solid is melted and recast into another shape, the total volume remains identical unless specified otherwise.

Surface Area of Sphere

The Total Surface Area (TSA) of a sphere is given by 4 × π × r². Note that TSA changes during recasting.

Radius Ratio

If n spheres are melted into 1, the ratio of radii is R/r = n^1/3. For n=8, R/r = 8^1/3 = 2.

Shortcut Trick

Assume a base mean of 50 and find deviations: +28, +16, +39, +5, +23, −5, −27, −4, −21.

Calculate Net Deviation: 28 + 16 + 39 + 5 + 23 − 5 − 27 − 4 − 21 = +54.

Average Deviation = 54 ÷ 9 = +6.

Final Mean = Assumed Mean + Average Deviation = 50 + 6 = 56.

∴ The correct answer is 56.

Alternate Method

Given observations: 78, 66, 89, 55, 73, 45, 23, 46, 29

Number of observations (n) = 9

Formula Used: Arithmetic Mean = (Sum of all observations) ÷ (Total number of observations)

⇒ Total Sum = 78 + 66 + 89 + 55 + 73 + 45 + 23 + 46 + 29

⇒ Total Sum = 504

⇒ Arithmetic Mean = 504 ÷ 9

⇒ Arithmetic Mean = 56

∴ The correct answer is 56.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of all observations from their arithmetic mean is always zero: Σ(x_i − Mean) = 0.

Effect of Operations

If each observation is increased, decreased, multiplied, or divided by a constant k, the mean also changes by the same operation with k.

Combined Mean Formula

For two groups with sizes n₁, n₂ and means x₁, x₂, the combined mean is (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Shortcut Trick

Total multiplier for consecutive discounts of 2%, 8%, and 15%.

Effective Multiplier = 0.98 × 0.92 × 0.85 = 0.76636.

Equivalent Discount% = (1 − 0.76636) × 100 = 23.364%.

Rounding to two decimal places, we get 23.36%.

∴ The correct answer is 23.36%.

Alternate Method

Given → Discounts are 2%, 8%, and 15%. Formulas: Successive Discount = a + b − (a × b) ÷ 100.

⇒ First, combine 2% and 8%: 2 + 8 − (2 × 8) ÷ 100 = 10 − 0.16 = 9.84%. ⇒ Now, combine 9.84% and 15%: 9.84 + 15 − (9.84 × 15) ÷ 100. ⇒ Total = 24.84 − (147.6 ÷ 100) = 24.84 − 1.476. ⇒ Total Discount = 23.364%.

∴ The correct answer is 23.36%.

Additional Information

Single Equivalent Discount

It is the total reduction offered on the Marked Price through multiple stages. For three discounts a, b, and c, use the formula iteratively or the Multiplier Method.

Marked Price and Selling Price

The Selling Price (SP) after discount is: SP = MP × (100 − D₁)/100 × (100 − D₂)/100...

Shortcut Trick

Combine Ratio: A : B : C = (4 × 9) : (15 × 9) : (15 × 3) = 36 : 135 : 45.

Simplified Ratio: 12 : 45 : 15 ⇒ 4 : 15 : 5.

Total units = 4 + 15 + 5 = 24 units ⇒ ₹21,800.

B's share = (15 ÷ 24) × 21,800 = (5 ÷ 8) × 21,800 = 5 × 2725 = ₹13,625.

∴ The correct answer is ₹13,625.

Alternate Method

Given: Total Profit = ₹21,800, A : B = 4 : 15, B : C = 9 : 3.

Formula: Combined Ratio A : B : C = (A × B₁) : (B₁ × B₂) : (B₂ × C).

⇒ To equalize B, multiply A : B by 3 and B : C by 5.

⇒ A : B = 12 : 45.

⇒ B : C = 45 : 15.

⇒ A : B : C = 12 : 45 : 15 = 4 : 15 : 5.

⇒ Sum of ratio parts = 4 + 15 + 5 = 24.

⇒ B's share = (15 ÷ 24) × 21,800.

⇒ B's share = 0.625 × 21,800 = ₹13,625.

∴ The correct answer is ₹13,625.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (a/b × c/d), the result is the compound ratio (ac/bd).

Partnership Profit Distribution

Profit is distributed in the ratio of (Capital × Time). If time is constant, profit ratio equals the investment ratio.

Proportion Rule

If a : b = c : d, then the product of extremes equals the product of means (a × d = b × c).

Shortcut Trick

Use the direct formula for reduction in consumption: [r / (100 + r)] × 100%

Here, increase rate r = 20%.

Reduction = [20 / (100 + 20)] × 100 = (20 / 120) × 100 = 1/6 × 100

Calculation: 100 / 6 = 16(2/3)%

∴ The correct answer is 16(2/3)%.

Alternate Method

Given:

Initial Price of milk = 100 units

Increased Price = 100 + 20% of 100 = 120 units

Formula Used:

Expenditure = Price × Consumption

To keep Expenditure constant, the new consumption must be brought back to the base of the new price.

⇒ Required reduction in consumption = New Price − Old Price = 120 − 100 = 20 units

⇒ Percentage reduction = (Reduction / New Price) × 100

⇒ Percentage reduction = (20 / 120) × 100

⇒ Percentage reduction = (1 / 6) × 100

⇒ Percentage reduction = 16.66% or 16(2/3)%

∴ The correct answer is 16(2/3)%.

Additional Information

Price-Consumption Relationship

When Expenditure is constant, Price is inversely proportional to Consumption (P ∝ 1/C).

Price Decrease Case

If price decreases by r%, the increase in consumption to keep expenditure constant is [r / (100 − r)] × 100%.

Fraction Conversion

20% increase is 1/5. For constant product, if one factor increases by 1/n, the other must decrease by 1/(n+1). Here, 1/(5+1) = 1/6 = 16(2/3)%.

Shortcut Trick

The biggest even number less than 30 is 28.

Check products from options: (1) 33, (2) 21, (3) 26, (4) 10.

The largest even product less than 28 (or specifically less than 30) among options is 26.

∴ The correct answer is 2 and 13.

Alternate Method

Given: Product must be an even number less than 30.

Property: To get an even product from two prime numbers, one of the numbers must be the only even prime, which is 2 (since Odd × Odd = Odd and Even × Any = Even).

⇒ Target: Largest product P < 30 where P is even.

⇒ Option 1: 3 × 11 = 33 (Odd, > 30) ×

⇒ Option 2: 3 × 7 = 21 (Odd) ×

⇒ Option 3: 2 × 13 = 26 (Even, < 30) ✓

⇒ Option 4: 2 × 5 = 10 (Even, < 30) × [26 is larger than 10]

∴ The correct answer is 2 and 13.

Additional Information

Even Prime Number

The number 2 is the only even prime number. All other prime numbers are odd.

Product Parity Rules

Odd × Odd = Odd; Even × Odd = Even; Even × Even = Even.

Prime Number Definition

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Shortcut Trick

Calculate the semi-perimeter (s) of the triangle: s = (35 + 44 + 75) ÷ 2 = 77 cm.

Find the Area using Heron's Formula: Area = √(77 × (77−35) × (77−44) × (77−75)) = √(77 × 42 × 33 × 2).

Area = √(7 × 11 × 7 × 6 × 3 × 11 × 2) = 7 × 11 × 6 = 462 cm².

Altitude (h) corresponding to the 44 cm base = (2 × Area) ÷ base = (2 × 462) ÷ 44 = 21 cm.

∴ The correct answer is 21 cm.

Alternate Method

Given: Sides of the triangle are a = 35 cm, b = 44 cm, and c = 75 cm.

Formula Used:

1. Semi-perimeter (s) = (a + b + c) ÷ 2
2. Area = √(s(s − a)(s − b)(s − c))
3. Area = (1 ÷ 2) × base × altitude

Calculations:

⇒ s = (35 + 44 + 75) ÷ 2

⇒ s = 154 ÷ 2 = 77 cm

⇒ Area = √(77 × (77 − 35) × (77 − 44) × (77 − 75))

⇒ Area = √(77 × 42 × 33 × 2)

⇒ Area = √((7 × 11) × (7 × 6) × (11 × 3) × 2)

⇒ Area = √(7² × 11² × 6²) = 7 × 11 × 6 = 462 cm²

⇒ 462 = (1 ÷ 2) × 44 × altitude

⇒ 462 = 22 × altitude

⇒ altitude = 462 ÷ 22 = 21 cm

∴ The correct answer is 21 cm.

Additional Information

Heron's Formula for Area

Used to find the area of a triangle when all three sides are known: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter.

Relationship between Area and Altitude

For any side taken as the base, the area is always 1/2 × base × corresponding height. Thus, height = (2 × Area) ÷ base.

Triangle Inequality Theorem

A triangle is valid only if the sum of any two sides is greater than the third side (e.g., 35 + 44 = 79, which is > 75).

Shortcut Trick

The given expression follows the algebraic identity (a² − b²) / (a + b) = (a − b).

Here, let a = 100.5 and b = 79.2.

The solution is simply the difference between the two numbers: 100.5 − 79.2 = 21.3.

∴ The correct answer is 21.3.

Alternate Method

Given:

Expression: ((100.5)² − (79.2)²) / (100.5 + 79.2)

Formula Used:

Algebraic Identity: a² − b² = (a + b)(a − b)

Calculations:

⇒ Let a = 100.5 and b = 79.2.

⇒ Numerator = (100.5)² − (79.2)² = (100.5 + 79.2) × (100.5 − 79.2)

⇒ Substituting this into the main expression:

⇒ [(100.5 + 79.2) × (100.5 − 79.2)] / (100.5 + 79.2)

⇒ 100.5 − 79.2 (Cancelling the common term 100.5 + 79.2)

⇒ 21.3

The calculated value corresponds to option (4).

∴ The correct answer is 21.3.

Additional Information

Difference of Squares

The identity a² − b² = (a + b)(a − b) is used to simplify expressions involving squares of two numbers.

Square of a Sum

The formula (a + b)² = a² + 2ab + b² expands the square of the sum of two variables.

Square of a Difference

The formula (a − b)² = a² − 2ab + b² expands the square of the difference between two variables.

Fraction Simplification

If the numerator and denominator share a common factor, they can be cancelled: (x × y) / x = y, provided x ≠ 0.

Logic followed here is:

Logic: Each letter in the first word is shifted backward by 3 positions in the English alphabetical series to obtain the corresponding letter of the second word.

XBNO is related to UYKL →

PAQJ is related to MXNG →

The pattern observed is: (Positional value of letter - 3) = Resulting letter.

Similarly, let's find the related word for FWOM:

Therefore, the word related to 'FWOM' is CTLJ.

Hence, the correct answer is "Option 2".

Logic followed here is:

The series follows a pattern where the difference between consecutive terms increases by 1 at each step.

Next term in series = 74 + 11 = 85

 

Therefore, the number that should come in place of the question mark is 85.

Hence, the correct answer is "Option 3".

Given Series:

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

We need to find the even numbers which are:

→ Immediately preceded by an odd number

→ Immediately followed by an even number

Required pattern: Odd → Even → Even

Checking the series:

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

The even numbers satisfying the condition are:

2 in (1 2 4)

4 in (1 4 2)

Therefore, the number of such even numbers is 2.

Hence, the correct answer is "Option 2".

Logic followed here is: Each letter in a group is moving forward by 10 positions in the English alphabetical order to obtain the corresponding letter of the next group.

Therefore, the term that comes in place of the question mark is ZWA.

Hence, the correct answer is "Option 2".

Given:

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

Only two people sit to the left of C.

Only three people sit between C and G.

B sits second to the right of C.

Only two people sit between B and A.

E sits somewhere to the right of F but somewhere to the left of D.

Find how many people sit to the right of C.

To the right of C are: E, B, D and G

Total people to the right of C = 4

Hence, the correct answer is "Option 3".

Given:

Town X is to the north of Town Y.

Town Z is to the west of Town Y.

Town W is to the east of Town X.

Town Z is to the north of Town V.

Now find the position of W with respect to Z.

W is to the east of Z and also north of Z.

Therefore, W is in the North-East direction with respect to Z.

Hence, the correct answer is "Option 3".

Given Mapping:

Given Equation:

(20 Q 4) P 7 S 5 P 9 R (40 Q 8) P 4 = ?

Replace the letters with symbols:

⇒ (20 ÷ 4) × 7 + 5 × 9 − (40 ÷ 8) × 4

⇒ (5 × 7) + (5 × 9) − (5 × 4)

⇒ 35 + 45 − 20

⇒ 80 − 20

⇒ 60

Therefore, the value that comes in place of the question mark is 60.

Hence, the correct answer is "Option 3".

Logic followed here is: The series follows a pattern where each cluster's letters are shifted backwards by 3 positions in the English alphabetical order, and the numbers increase by a constant value of 11.

 

Therefore, the cluster that replaces the question mark is MJK 102.

Hence, the correct answer is "Option 2".

Given:

“ROUT” is coded as “1625”

“AUTO” is coded as “1546”

Therefore, the code for “A” is 4.

Hence, the correct answer is "Option 1".

Given Coding:

 

Given Expression:

T + O × R - N & M: T is is the son of O. O is the wife of R. R is the brother of N. N is the father of M.

So, T is M’s father’s brother’s son.

Therefore, T is related to M as Father's brother's son.

Hence, the correct answer is "Option 1".

Logic followed here is:

The relationship between the letter clusters is established by applying a specific arithmetic sequence (-5, +4, -3, +2) to the positional values of the letters in the English alphabet.

For

DQXG is related to YUUI →

 

GLQC is related to BPNE →

The pattern observed is: -5, +4, -3, +2 applied to the positional values.

Similarly, let's find the code for TKWD:

 

Therefore, 'TKWD' is related to 'OOTF'.

Hence, the correct answer is "Option 1".

Checking Option 1: JMO

Checking Option 2: RUW

 

Checking Option 3: CGH

Checking Option 4: YBD

 

 

 

The pattern for JMO, RUW, and YBD is (+3, +2). However, for CGH, the pattern is (+4, +1).

Therefore, the letter-cluster 'CGH' does not belong to the group.

Hence, the correct answer is "Option 3".