RRB Group D Official Paper (Held On: 23 Dec, 2025 Shift 3) (Previous Year Paper)

The correct answer is Copper.

Key Points

• An element is a pure substance that consists entirely of one type of atom and cannot be broken down into simpler substances by chemical means.
• Copper is a chemical element represented by the symbol Cu and carries the atomic number 29. It exists as a pure substance in its metallic form, making it a classic example of an element found in the periodic table. Hence, statement 1 is correct.
• Salt, commonly known as table salt, is a chemical compound called Sodium Chloride with the formula NaCl. It is formed by the chemical bonding of Sodium (Na) and Chlorine (Cl) in a fixed 1:1 ratio. Hence, statement 2 is incorrect.
• Air is not a single substance but a homogeneous mixture of various gases. It primarily consists of Nitrogen (78%) and Oxygen (21%), along with traces of Argon, Carbon Dioxide, and water vapor. These gases are physically mixed but not chemically bonded. Hence, statement 3 is incorrect.
• Water is a compound composed of two different elements: Hydrogen and Oxygen. Its chemical formula is H2O, indicating that two atoms of hydrogen are chemically bonded to one atom of oxygen. Hence, statement 4 is incorrect.
• Hence, the correct answer is Copper.

Additional Information

• Characteristics of Elements:
• There are 118 confirmed elements known to science, of which 94 occur naturally on Earth.
• Elements are categorized into Metals, Non-metals, and Metalloids based on their physical and chemical properties.
• Copper is classified as a transition metal, located in Group 11 and Period 4 of the periodic table.
• Compounds vs. Mixtures:
• Compounds (like Water or Salt) have a fixed composition and properties that differ significantly from their constituent elements. They can only be separated by chemical methods.
• Mixtures (like Air) do not have a fixed ratio and retain the individual properties of their components. They can be separated by physical methods such as filtration or distillation.
• Properties and Uses of Copper:
• Copper is highly valued for its excellent electrical conductivity and thermal conductivity, ranking second only to Silver among pure metals.
• It is widely used in electrical wiring, telecommunications, electronics, and plumbing.
• Historically, it was the first metal to be smelted from its ore around 5000 BC and the first metal to be cast into a shape in a mold around 4000 BC.

Important Points

• Atomic Structure: The identity of an element is determined by the number of protons in its nucleus, known as the atomic number. For Copper, every atom contains exactly 29 protons.
• State of Matter: At standard temperature and pressure (STP), Copper is a reddish-orange solid, Water is a liquid, and Air consists of gases.
• Abundance: While Copper is an essential trace mineral for all living organisms, it is found in the Earth's crust at a concentration of about 50 parts per million (ppm).
• Common Alloys: Because it is an element, Copper can be mixed with other elements to form alloys. For example, Brass is an alloy of Copper and Zinc, while Bronze is primarily Copper and Tin.

The correct answer is 6.67 x 10^-7 N.

Key Points

• Newton's Law of Universal Gravitation states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The mathematical formula for this force is F = G (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance.
• In this problem, the given values are: mass m1 = 100 kg, mass m2 = 100 kg, and distance r = 1 m.
• The Universal Gravitational Constant (G) is provided as 6.67 x 10^-11 Nm^2/kg^2.
• By substituting the values into the formula: F = (6.67 x 10^-11) * (100 * 100) / (1)^2.
• Solving the calculation: 100 * 100 = 10,000 (which is 10^4). Therefore, F = 6.67 x 10^-11 * 10^4.
• Adding the exponents (-11 + 4 = -7), we get the final magnitude of force as 6.67 x 10^-7 Newtons.

Additional Information

• Sir Isaac Newton: He formulated the Law of Universal Gravitation and published it in his work Philosophiae Naturalis Principia Mathematica in 1687.
• Inverse Square Law: The force of gravity follows an inverse-square relationship. If the distance between two objects is doubled, the gravitational force becomes one-fourth (1/4) of its original strength.
• Universal Constant (G): Unlike the acceleration due to gravity (g), which changes depending on location (like on Earth vs. the Moon), the value of G remains constant throughout the universe.
• Henry Cavendish: The value of G was first measured accurately in a laboratory by the English scientist Henry Cavendish in 1798 using a torsion balance.
• Always Attractive: Gravitational force is always attractive in nature and acts along the line joining the centers of the two bodies. It is one of the four fundamental forces of nature.

Important Points

• Weakness of Gravity: Despite being the force that keeps planets in orbit, gravity is the weakest of the fundamental forces. It only becomes significant when dealing with massive bodies like stars, planets, or moons.
• Mass vs. Weight: Mass is the amount of matter in an object and is measured in kilograms (kg), while weight is the gravitational force acting on that mass and is measured in Newtons (N).
• Medium Independence: The gravitational force between two masses does not depend on the medium (water, air, or glass) placed between them. It remains the same regardless of the environment.
• Range: The range of gravitational force is infinite, meaning every object in the universe exerts some amount of gravitational pull on every other object, no matter how far apart they are.

The correct answer is The current reduces to one-half of its former value.

Key Points

• Ohm's Law states that the electric current (I) flowing through a metallic conductor is directly proportional to the potential difference (V) applied across its terminals, provided that physical conditions like temperature and pressure remain constant.
• The relationship is mathematically expressed as V = IR, where R is the resistance of the conductor.
• When the resistance (R) remains constant, the current can be calculated using the formula I = V / R. This implies that the current is linearly dependent on the voltage.
• If the potential difference is reduced to one-half (1/2) of its original value, the new current (I') becomes (V/2) / R, which is equivalent to 1/2 * (V/R). Hence, the statement that the current reduces to one-half of its former value is correct.
• This proportionality means that any factor by which the voltage is scaled will result in the current being scaled by that same factor, as long as the metallic wire obeys Ohm's Law and its resistance does not change.

Additional Information

• Electric Current (I):
• It is defined as the rate of flow of electric charge through a cross-section of a conductor.
• The SI unit of current is the Ampere (A), named after André-Marie Ampère.
• It is measured using an instrument called an ammeter, which is always connected in series in an electrical circuit to ensure the full current flows through it.
• Potential Difference (V):
• Also known as voltage, it is the work done to move a unit charge from one point to another in an electric field.
• The SI unit is the Volt (V), named after Alessandro Volta.
• It is measured using a voltmeter, which is connected in parallel across the points where the potential difference is to be determined.
• Resistance (R):
• It is the property of a material to oppose the flow of electric current.
• The SI unit is the Ohm (Ω). One Ohm is defined as the resistance of a conductor such that a potential difference of 1 Volt causes a current of 1 Ampere to flow.
• The resistance of a wire depends on its length (L), cross-sectional area (A), and the resistivity (ρ) of the material.

Important Points

• Discovery of Ohm's Law: The law was determined experimentally by the German physicist Georg Simon Ohm and published in his 1827 treatise, "The Galvanic Circuit Investigated Mathematically."
• Ohmic and Non-Ohmic Conductors:
• Materials that strictly follow V = IR are called Ohmic conductors (e.g., copper, silver, and most metals at constant temperature).
• Materials that do not follow this linear relationship are Non-Ohmic, such as diodes, transistors, and semiconductors, where the resistance changes with the applied voltage or current.
• V-I Graph: For a conductor obeying Ohm's Law, a graph of Voltage (V) versus Current (I) is a straight line passing through the origin (0,0). The slope of this line represents the resistance (R) of the conductor.
• Power Dissipation: In a circuit where the potential difference is halved and resistance is constant, the power (P = V²/R) dissipated by the wire actually reduces to one-fourth (1/4) of its original value.

The correct answer is x=2, y=1, z=1.

Key Points

• The provided chemical equation involves the reaction between Calcium Carbonate (CaCO3) and Hydrochloric Acid (HCl) to produce Calcium Chloride (CaCl2), Carbon Dioxide (CO2), and Water (H2O).
• To balance the equation, we apply the Law of Conservation of Mass, which states that the number of atoms of each element must be equal on both the reactant and product sides.
• Balancing Calcium (Ca): There is 1 atom of Ca on the reactant side. To balance this, the product side must also have 1 atom of Ca. Since Ca is in CaCl2, the coefficient y must be 1. Hence, the statement for y=1 is correct.
• Balancing Oxygen (O): There are 3 Oxygen atoms in CaCO3 on the left. On the right, there are 2 atoms in CO2 and z atoms in H2O. The equation 3 = 2 + z gives z = 1. Hence, the statement for z=1 is correct.
• Balancing Hydrogen (H): There are x atoms of Hydrogen on the left in HCl. On the right, there are 2 atoms in H2O (since z=1, 2 × 1 = 2). Therefore, x must be 2. Hence, the statement for x=2 is correct.
• Balancing Chlorine (Cl): On the left, we have x = 2 atoms of Chlorine. On the right, there are 2y atoms in CaCl2. Since y = 1, 2(1) = 2. This confirms that the Chlorine atoms are balanced.
• The final balanced chemical equation is CaCO3 + 2 HCl -> CaCl2 + CO2 + H2O.

Additional Information

• Calcium Carbonate (CaCO3): This is a white, insoluble solid widely found in nature as limestone, marble, and chalk. It is a major component of eggshells and seashells.
• Hydrochloric Acid (HCl): A strong, colorless, highly pungent solution of hydrogen chloride in water. It is a primary gastric acid found in the human stomach and is used industrially for metal cleaning and pH regulation.
• Reaction Products: The reaction results in Calcium Chloride (CaCl2), which is a salt commonly used for de-icing roads and as a firming agent in food. The Carbon Dioxide (CO2) gas produced causes visible effervescence (fizzing).
• Carbon Dioxide Test: The identity of the evolved gas can be confirmed using the limewater test. When CO2 is bubbled through Calcium Hydroxide (limewater), it forms a white precipitate of CaCO3, turning the solution milky.
• Reaction Type: This is an acid-carbonate reaction. General chemistry dictates that an acid reacting with a metal carbonate will always yield a salt, water, and carbon dioxide gas.

Important Points

• Stoichiometry: According to the balanced equation, 1 mole of Calcium Carbonate reacts with 2 moles of Hydrochloric Acid. This 1:2 molar ratio is essential for calculating reactant quantities in a laboratory setting.
• Conservation of Mass: Formulated by Antoine Lavoisier in 1789, this principle is the foundation for all chemical equation balancing, ensuring that no matter is lost during the process.
• State Symbols: In a formal setting, the reaction is written as CaCO3 (s) + 2 HCl (aq) -> CaCl2 (aq) + CO2 (g) + H2O (l), indicating the physical states of the reactants and products.
• Practical Application: This reaction is the fundamental mechanism behind antacid tablets (containing CaCO3) neutralizing stomach acid (HCl) to relieve heartburn and indigestion.

The correct answer is Dogs can regrow lost body parts and form whole new individuals.

Key Points

• Regeneration is a process by which some organisms can regrow a lost body part or even form a whole new individual from a small fragment of their body.
• In complex animals such as mammals (including humans and dogs), the ability to regenerate is strictly limited to the repair of tissues and specific organs rather than the creation of entire new individuals or complex limbs. Hence, statement 1 is correct.
• Dogs are highly complex multicellular organisms with specialized organ systems; they do not possess the biological capability to regrow lost limbs or regenerate a whole new dog from a body part. Hence, statement 2 is incorrect and is the correct answer.
• Fragmentation is a form of asexual reproduction seen in simple multicellular organisms like Spirogyra, where the body simply breaks into smaller pieces upon maturation, and each piece grows into a new filament. Hence, statement 3 is correct.
• Regeneration is carried out by specialised cells that proliferate to make a large number of cells; from this mass of cells, different cells undergo changes to become various cell types and tissues through cell division and differentiation. Hence, statement 4 is correct.

Additional Information

• Organisms with High Regenerative Capacity:
• Simple animals like Hydra and Planaria (flatworms) exhibit remarkable regenerative powers. If they are cut into any number of pieces, each piece can grow into a complete organism.
• This process is not the same as reproduction, as most organisms would not normally depend on being cut up to be able to reproduce.
• Mechanism of Regeneration:
• It involves morphallaxis (remodeling of existing tissues) or epimorphosis (growth of new tissue via a blastema).
• In Planaria, the process is driven by pluripotent stem cells called neoblasts, which can transform into any cell type required by the organism.
• Difference between Fragmentation and Regeneration:
• Fragmentation occurs in organisms with relatively simple body designs where every cell/part has the potential to become a new individual spontaneously.
• Regeneration is a more complex mechanism where specialized cells respond to an injury to restore lost anatomy.

Important Points

• Regeneration in Vertebrates:
• While Dogs cannot regenerate limbs, some Amphibians like Salamanders and Axolotls can regrow entire limbs, tails, and even parts of their hearts and brains.
• Lizards exhibit autotomy, where they can shed their tail to escape a predator and subsequently regrow it, although the new tail is often different in structure (cartilage instead of bone).
• Human Limitations:
• In Humans, the Liver is the only internal organ capable of significant natural regeneration; as little as 25% of the original liver mass can regenerate back to its full size within weeks.
• Other human tissues, like skin and blood, are constantly being renewed through cell division, but this is categorized as physiological turnover rather than whole-body regeneration.

The correct answer is scattering of shorter wavelength of sunlight by air molecules.

Key Points

• The blue color of the sky is a result of a phenomenon known as Rayleigh scattering. As Sunlight reaches the Earth’s atmosphere, it is scattered in all directions by the gases and particles present in the air.
• Light travels as waves of different wavelengths. Blue light travels in shorter, smaller waves, whereas red light travels in longer, larger waves.
• According to the Rayleigh Law of Scattering, the intensity of scattered light is inversely proportional to the fourth power of the wavelength of light. This means that shorter wavelengths (like blue and violet) are scattered much more strongly than longer wavelengths (like red and yellow) when they strike the molecules of nitrogen and oxygen in the atmosphere. Hence, statement 2 is correct.
• While violet light has an even shorter wavelength than blue light and is scattered even more, the sky appears blue to us because the Sun emits much more blue light than violet, and the human eye is significantly more sensitive to blue than to violet.
• Refraction of light is the bending of light as it passes from one medium to another. While it causes phenomena like twinkling of stars, it is not responsible for the blue color of the sky. Hence, statement 1 is incorrect.
• Absorption would involve the air molecules taking in the light energy and not re-emitting it, which would make the sky appear dark rather than bright blue. Hence, statement 3 is incorrect.
• Reflection usually occurs on larger surfaces or particles (like mirrors or water droplets) and does not explain the uniform color distribution of the clear sky. Hence, statement 4 is incorrect.

Additional Information

• The Visible Spectrum:
• The visible light spectrum consists of colors ranging from Violet (shortest wavelength, approx 400 nm) to Red (longest wavelength, approx 700 nm).
• In the vacuum of space, where there is no atmosphere to scatter light, the sky appears completely black, and the Sun looks like a bright white disk.
• Red Appearance of Sunset and Sunrise:
• During sunrise and sunset, sunlight has to travel through a much thicker layer of the atmosphere to reach our eyes.
• By the time the light reaches us, most of the shorter wavelengths (blue and violet) have been scattered away, leaving primarily the longer wavelengths like red and orange to pass through.
• Tyndall Effect:
• The Tyndall effect is the scattering of light by much larger colloidal particles or very fine suspensions (like dust or smoke in a room).
• This is different from Rayleigh scattering, which occurs when the scattering particles (molecules) are much smaller than the wavelength of the light.
• Atmospheric Composition:
• The Earth's atmosphere is composed of approximately 78% Nitrogen and 21% Oxygen. These gas molecules are the primary agents for the Rayleigh scattering that creates the blue sky.
• If the Earth had no atmosphere, the sky would look as dark as it does on the Moon.

Important Points

• Lord Rayleigh: The phenomenon was theoretically explained by the British physicist Lord Rayleigh in the year 1871. He formulated the mathematical relationship showing how scattering varies with wavelength.
• Scattering Intensity: The relationship is expressed as Intensity (I) ∝ 1 / λ⁴, where λ is the wavelength. This confirms that shorter wavelengths are scattered with exponentially greater intensity.
• Sky on Other Planets: On Mars, the sky can appear pinkish-red during the day due to the presence of iron-rich dust in its thin atmosphere, which scatters light differently than Earth's gas-heavy atmosphere.

The correct answer is In the electrical circuit, the voltmeter is connected in series with the component.

Key Points

• In an electrical circuit, a voltmeter is always connected in parallel with the component across which the potential difference is to be measured. This is because a voltmeter has a very high resistance so that it takes a negligible current from the circuit. If it were connected in series, it would significantly increase the total resistance of the circuit and reduce the current flow. Hence, statement 1 is incorrect.
• A bulb glows only when there is a continuous and unbroken path for the flow of electricity, which is known as a closed circuit. In an open circuit, the path is broken, preventing the movement of electrons. Hence, statement 2 is correct.
• A switch is a device used to either break the circuit or complete it. When the switch is in the "open" position, it creates an air gap that acts as an insulator, ensuring that no current flows through the circuit. Hence, statement 3 is correct.
• By convention, the direction of electric current is taken as the direction of the flow of positive charges. Therefore, it flows from the positive terminal of the battery to the negative terminal through the external circuit. This is known as conventional current. Hence, statement 4 is correct.
• Hence, the incorrect statement (and the correct answer) is that the voltmeter is connected in series.

Additional Information

• Voltmeter and Ammeter:
• A voltmeter is used to measure voltage or potential difference between two points. It must have infinite resistance ideally to avoid drawing current from the main branch.
• An ammeter is used to measure electric current in a circuit. Unlike a voltmeter, it is always connected in series and must have a very low resistance so that it does not change the current it is supposed to measure.
• Flow of Electrons vs. Conventional Current:
• While conventional current is defined as flowing from positive to negative, the actual physical flow of electrons (negative charges) is from the negative terminal to the positive terminal.
• This distinction is historical, as the concept of current was established before the discovery of the electron by J.J. Thomson in 1897.
• Circuit Components:
• A simple circuit consists of a source (like a cell or battery), conductors (connecting wires), a load (like a bulb or resistor), and a control device (switch).
• Ohm's Law, formulated by Georg Simon Ohm in 1827, states that the current through a conductor is directly proportional to the voltage across it, provided physical conditions remain constant (V = IR).

Important Points

• Resistance: It is the property of a conductor to resist the flow of charges through it. Its SI unit is the Ohm (Ω).
• Galvanometer: This is an instrument used to detect the presence and direction of a small electric current in a circuit. It can be converted into a voltmeter by connecting a high resistance in series or into an ammeter by connecting a low resistance (shunt) in parallel.
• Safety Devices: A fuse is a safety device connected in series with the circuit. It has a low melting point and breaks the circuit if the current exceeds a safe limit, preventing short circuits or overloading.

The correct answer is To reflect sound uniformly in all directions.

Key Points

• The ceilings of concert halls, conference halls, and cinema halls are designed with a curved shape so that sound, after reflection from the ceiling, reaches all corners of the hall.
• This design ensures the uniform distribution of sound waves, allowing the audience to hear the performance clearly regardless of their seating position. Hence, statement 3 is correct.
• A curved ceiling acts like a large concave reflector. When sound waves hit the curved surface, they are reflected back toward the floor in a spreading pattern, covering a larger area than a flat ceiling would.
• The curvature does not stop reverberation; in fact, controlled reverberation is often desirable in music halls for a richer sound. Hence, statement 1 is incorrect.
• The speed of sound is a physical property of the medium (air) and is affected by temperature and humidity, not by the shape of the ceiling. Hence, statement 2 is incorrect.
• While certain materials are used to absorb sound to prevent echoes, the specific purpose of the curved shape itself is reflection and distribution rather than absorption. Hence, statement 4 is incorrect.

Additional Information

• Soundboard: In many large auditoriums, a curved soundboard is placed behind the speaker or the stage. The source of sound is placed at the focus of the board, so the reflected sound travels as a parallel beam toward the audience.
• Reverberation: This is the persistence of sound in a hall due to repeated reflections. If the reverberation time is too long, sound overlaps and becomes distorted. To control this, sound-absorbent materials like compressed fiberboard or heavy draperies are used on walls.
• Multiple Reflection of Sound: This principle is utilized in various instruments and devices. For example, in a stethoscope, the sound of the patient's heartbeat reaches the doctor's ears by multiple reflections inside the tube.
• Echo: An echo is the sound heard after reflection from a distant obstacle. To hear a distinct echo, the time interval between the original sound and the reflected one must be at least 0.1 seconds. At a standard temperature of 22 degrees Celsius, the minimum distance required for an echo is 17.2 meters.
• Acoustics: The branch of science that deals with the production, control, transmission, reception, and effects of sound is known as Acoustics. Proper acoustic engineering is vital for the construction of theaters and stadiums.

Important Points

• Reflection Laws: Sound follows the same laws of reflection as light, where the angle of incidence is equal to the angle of reflection, and all rays lie in the same plane.
• Intensity: Curved surfaces help in maintaining the intensity of the sound over long distances by preventing the sound energy from scattering into the upper atmosphere of the building.
• Shape: While ceilings are often parabolic or spherical in curve to optimize focus, the general cylindrical or curved design is the most common standard for architectural acoustics.

The correct answer is Both air and water.

Key Points

• Rusting is a specific type of corrosion that occurs when iron or iron-based alloys are exposed to oxygen and moisture for a prolonged period.
• For rusting to occur, the presence of both air (oxygen) and water (moisture) is absolutely essential. If either of these components is missing, the chemical reaction cannot proceed. Hence, the statement "Both air and water" is correct.
• The process is an oxidation reaction where iron reacts with oxygen in the presence of water to form hydrated iron (III) oxide, which is commonly known as rust.
• The chemical equation representing this process is: 4Fe + 3O2 + 6H2O → 2Fe2O3.3H2O. This shows that iron, oxygen, and water are the primary reactants.
• If iron is kept in dry air (air without moisture), it does not rust. Similarly, if iron is kept in boiled distilled water (water without dissolved air/oxygen) and sealed, it does not rust. Hence, the options "Only air", "Only water", and "Dry air only" are incorrect.

Additional Information

• Chemical Nature of Rust: Rust is a reddish-brown flaky substance. Unlike the protective oxide layer on aluminum, rust is porous and crumbles easily, exposing fresh layers of iron to further oxidation, eventually leading to the structural failure of the object.
• Factors Accelerating Rusting:
• Salinity: The presence of salt in water (like seawater) increases the speed of rusting because ions in the saltwater conduct electricity better, accelerating the electrochemical reaction.
• Acidity: Low pH levels (acidic environments) increase the rate of corrosion.
• Impure Iron: Pure iron rusts slowly, but impurities in the metal can create small electrochemical cells that speed up the process.
• Prevention Methods:
• Galvanization: This is the process of applying a protective zinc coating to iron or steel. Zinc is more reactive than iron and acts as a sacrificial anode.
• Alloying: Mixing iron with elements like chromium and nickel creates stainless steel, which is highly resistant to rusting.
• Barrier Protection: Painting, oiling, or greasing the surface of iron prevents direct contact with air and moisture.

Important Points

• Redox Reaction: Rusting is a redox reaction where iron acts as a reducing agent (loses electrons) and oxygen acts as an oxidizing agent (gains electrons).
• Economic Impact: Rusting causes significant economic loss globally every year by damaging infrastructure such as bridges, ships, pipelines, and automobiles.
• Environmental Factor: High humidity (above 50%) in coastal areas makes iron objects much more susceptible to rapid rusting compared to arid regions.

The correct answer is Work = force × displacement.

Key Points

• Work is defined in physics as the measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement.
• The mathematical formula for work is Work (W) = Force (F) × Displacement (s). This formula applies when the force is constant and acting in the same direction as the displacement. Hence, statement 4 is correct.
• In Statement 1, work is defined as displacement divided by force. This is mathematically incorrect as it does not represent any standard physical quantity in this context. Hence, statement 1 is incorrect.
• Statement 2 suggests that work equals force divided by displacement. This ratio (Force/Displacement) is actually related to the stiffness or spring constant in Hooke's Law, not work. Hence, statement 2 is incorrect.
• Statement 3 proposes that work is the sum of force and displacement. Physical quantities with different dimensions (Newton and Meter) cannot be added together. Hence, statement 3 is incorrect.
• The SI unit of work is the Joule (J), which is equivalent to 1 Newton-meter (N·m). It is named after the English physicist James Prescott Joule.

Additional Information

• Nature of Work: Work is a scalar quantity, meaning it has magnitude but no direction, even though both force and displacement are vector quantities.
• Angle Dependency: If the force is applied at an angle (θ) to the displacement, the formula becomes W = Fs cos(θ).
• If θ = 0°, work is maximum and positive.
• If θ = 90° (force is perpendicular to displacement), the work done is zero.
• If θ = 180° (force is opposite to displacement), the work done is negative.
• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy. This is a fundamental concept in classical mechanics.
• Power: Power is the rate at which work is performed. It is calculated as Power = Work / Time. The SI unit for power is the Watt (W), where 1 Watt = 1 Joule per second.
• Examples of Zero Work: A person holding a heavy load and standing still does no work because the displacement is zero. Similarly, a satellite orbiting Earth in a circular path does no work because the gravitational force is perpendicular to its velocity.

Important Points

• Dimensions: The dimensional formula for work is [M^1 L^2 T^-2], which is the same as the dimensions for energy and torque.
• CGS Unit: In the Centimetre-Gram-Second system, the unit of work is the erg. One Joule is equal to 10^7 ergs.
• Conservative Forces: Work done by conservative forces (like gravity or electrostatic force) depends only on the initial and final positions of the object, not the path taken.
• Non-Conservative Forces: Work done by non-conservative forces (like friction or air resistance) depends on the actual path traveled. Friction always does negative work as it opposes motion.

The correct answer is Active participation of local communities in protecting forests.

Key Points

• The Chipko Andolan, also known as the Chipko Movement, was a non-violent social and ecological movement by rural villagers, particularly women, in India during the 1970s.
• The movement originated in the Reni village of Chamoli district, Uttarakhand (which was then part of Uttar Pradesh) in 1973.
• It was aimed at protecting trees and forests from being cut down by government-backed logging contractors. The term "Chipko" translates to "to hug" or "to stick" in Hindi, reflecting the protestors' primary tactic of hugging trees to prevent them from being felled.
• This movement is a classic example of the active participation of local communities in environmental conservation, proving that decentralized, grassroots efforts can successfully challenge industrial exploitation of natural resources.
• Prominent leaders of this movement included Sunderlal Bahuguna, who gave the slogan "Ecology is permanent economy," and Chandi Prasad Bhatt, who was a pioneer in organizing the local community through the Dasholi Gram Swarajya Sangh.

Additional Information

• Historical Roots (Bishnoi Movement):
• The Chipko movement found its inspiration in a much older event from 1730 in Rajasthan, where Amrita Devi and 363 other members of the Bishnoi community sacrificed their lives to protect Khejri trees from the King's soldiers.
• Role of Women (Eco-feminism):
• The movement is often celebrated as a landmark in Eco-feminism. Women like Gaura Devi played a decisive role; in March 1974, she led 27 women to confront loggers when the village men were away, successfully protecting the Reni forest.
• Legislative Impact:
• As a direct result of the movement's widespread appeal and success, the then Prime Minister Indira Gandhi enforced a 15-year ban on felling green trees in the Himalayan region in 1980 to allow the forest cover to regenerate.
• Awards and Recognition:
• The Chipko Movement was awarded the Right Livelihood Award (often called the Alternative Nobel Prize) in 1987 for its dedication to the conservation, restoration, and ecologically responsible use of India's natural resources.
• Regional Variations (Appiko Movement):
• Inspired by the success in the north, the Appiko Movement was launched in September 1983 in the Uttara Kannada district of Karnataka to protect the Western Ghats from commercial deforestation.

Important Points

• Significance: The movement shifted the perspective of forest management from revenue generation to ecological stability. It highlighted that local communities depend on forests for fuel, fodder, and water, and are thus the best custodians of these resources.
• Non-Violence: Much like the Satyagraha methods of Mahatma Gandhi, the movement relied entirely on peaceful resistance and moral persuasion rather than physical conflict.
• Forest Policy: The movement influenced the Forest Conservation Act of 1980 and the National Forest Policy of 1988, which began to acknowledge the role of tribal and local communities in forest protection.

The correct answer is It transmits information rapidly from one body part to another.

Key Points

• Nervous tissue is a specialized tissue found in the nervous system that is responsible for the rapid transmission of signals throughout the body.
• It is composed of neurons (nerve cells) and neuroglial cells (support cells) that work together to detect stimuli and coordinate physiological responses.
• The brain, spinal cord, and peripheral nerves are the primary locations where this tissue is found.
• It functions by converting external or internal stimuli into electrical impulses (nerve impulses) and conducting them to different parts of the body. Hence, statement 4 is correct.
• Statement 1 is incorrect because providing rigidity and strength is the primary function of skeletal connective tissue, such as bones.
• Statement 2 is incorrect because protecting internal organs from mechanical injury is primarily done by epithelial tissue and structural connective tissues like the rib cage or adipose cushions.
• Statement 3 is incorrect because storing energy in the form of fat globules is the specific function of adipose tissue.

Additional Information

• Structure of a Neuron:
• A neuron consists of three main parts: the cell body (soma or cyton), dendrites, and an axon.
• The cell body contains the nucleus and cytoplasm, managing the metabolic activities of the cell.
• Dendrites are short, branched extensions that receive incoming signals from other neurons or receptors.
• The axon is a single, long fiber that carries the nerve impulse away from the cell body toward a synapse or an effector organ (like a muscle).
• Types of Nervous Systems:
• The Central Nervous System (CNS) consists of the brain and the spinal cord, acting as the control center for processing information.
• The Peripheral Nervous System (PNS) consists of cranial and spinal nerves that connect the CNS to the rest of the body parts.
• Neuroglia (Glial Cells):
• These are non-neuronal cells that do not produce electrical impulses.
• Their functions include providing structural support, insulation (myelin sheath), and nutrients to the neurons.
• They outnumber neurons in the nervous system and are crucial for homeostasis.

Important Points

• Communication Method: Nervous tissue communicates via electrochemical signals. When a neuron is stimulated, an action potential travels down the axon.
• Synapse: The functional junction between two neurons is called a synapse, where chemical messengers called neurotransmitters (like acetylcholine or dopamine) bridge the gap.
• Myelination: Many axons are covered by a myelin sheath, which acts as an electrical insulator and significantly increases the speed of nerve impulse transmission.
• Irritability and Conductivity: These are the two most distinctive features of nervous tissue. Irritability is the ability to respond to a stimulus, while conductivity is the ability to transmit that response.

The correct answer is Potassium.

Key Points

• Potassium (K) is placed at the very top of the reactivity series among the given options, making it the most reactive. It is an alkali metal belonging to Group 1 of the Periodic Table with an atomic number of 19. It reacts violently with water and oxygen even at room temperature. Hence, statement 4 is correct.
• Zinc (Zn) is a transition metal with an atomic number of 30. In the reactivity series, it is positioned below magnesium and aluminum but above iron. It reacts with steam and dilute acids but is significantly less reactive than potassium. Hence, statement 2 is incorrect.
• Iron (Fe) has an atomic number of 26 and is located further down the series than zinc. It does not react with cold water but reacts slowly with steam to form iron oxide and hydrogen gas. It is commonly found in the form of ores like haematite and magnetite. Hence, statement 3 is incorrect.
• Copper (Cu) is a noble-like metal with an atomic number of 29. It is placed near the bottom of the reactivity series, specifically below hydrogen. Due to its low reactivity, it does not react with water or dilute hydrochloric acid, making it useful for electrical wiring and plumbing. Hence, statement 1 is incorrect.
• The overall order of reactivity for these four metals from most to least reactive is Potassium (K) > Zinc (Zn) > Iron (Fe) > Copper (Cu).

Additional Information

• The Reactivity Series:
• It is an arrangement of metals in descending order of their chemical reactivity. This series helps in predicting the behavior of metals in displacement reactions, where a more reactive metal displaces a less reactive metal from its salt solution.
• Highly reactive metals like Potassium (K), Sodium (Na), and Calcium (Ca) are found at the top, while least reactive metals like Silver (Ag), Gold (Au), and Platinum (Pt) are at the bottom.
• Storage of Highly Reactive Metals:
• Metals like Potassium and Sodium are so reactive that they catch fire if exposed to moist air. To prevent accidental fires, they are stored submerged in kerosene oil.
• This safety protocol is essential in laboratories and industrial settings to handle these Group 1 elements.
• Reaction with Acids:
• Metals above Hydrogen (H) in the series can displace hydrogen from dilute acids (like HCl or H2SO4) to form metal salts and hydrogen gas.
• Metals like Copper, Mercury, and Gold do not react with dilute acids under standard conditions because they cannot lose electrons as easily as hydrogen.
• Extraction Processes:
• The position of a metal in the reactivity series determines its extraction method. Metals at the top are extracted via electrolysis of their molten chlorides or oxides.
• Metals in the middle (like Zinc and Iron) are reduced using carbon or carbon monoxide in a blast furnace.

Important Points

• Standard Reactivity Sequence: The general sequence from highest to lowest reactivity is Potassium (K) > Sodium (Na) > Lithium (Li) > Calcium (Ca) > Magnesium (Mg) > Aluminium (Al) > Zinc (Zn) > Iron (Fe) > Tin (Sn) > Lead (Pb) > [Hydrogen (H)] > Copper (Cu) > Mercury (Hg) > Silver (Ag) > Gold (Au).
• Chemical Nature: Reactivity is linked to the ease with which an atom loses electrons to form positive ions (cations). Potassium has only one electron in its outermost shell, which it loses very easily, explaining its high electropositivity.
• Displacement Example: If a piece of Iron is placed in a Copper Sulphate (CuSO4) solution, the iron will displace the copper because Iron is more reactive than Copper, resulting in Iron Sulphate (FeSO4) and Copper metal.

The correct answer is Castor.

Key Points

• Seed dispersal by the bursting of fruits is a mechanical process known as Autochory or explosive dehiscence.
• In plants like Castor (Ricinus communis) and Balsam, the fruits act as the primary agent for scattering seeds.
• As the Castor fruit matures and undergoes dehydration, the internal tissues experience a buildup of mechanical tension.
• When the fruit dries completely, it bursts open with a sudden jerk, effectively scattering the seeds several feet away from the parent plant. Hence, the correct option is Castor.
• The fruit of the Castor plant is specifically called a regma, which is a type of schizocarpic capsule that splits into three one-seeded parts called cocci during the bursting process.
• This mechanism ensures that the young seedlings do not have to compete with the parent plant for essential resources such as sunlight, water, and minerals.

Additional Information

• Dispersal by Animals (Zoochory):
• Certain plants produce seeds with specialized structures like hooks, spines, or stiff hairs to facilitate movement.
• Xanthium seeds are equipped with numerous hooked spines that easily attach to the fur of animals or the clothing of humans.
• Similarly, Urena fruits have hooked bristles that help them cling to passing organisms, allowing the seeds to be transported to distant locations.
• Dispersal by Wind (Anemochory):
• Lightweight seeds often use wind as a medium for transport.
• In the Sunflower, the seeds (achenes) are often dispersed by the wind. Some members of this family possess a pappus, a modified calyx that acts like a parachute to catch air currents.
• Other examples include Dandelion and Maple (which has winged seeds).
• Dispersal by Water (Hydrochory):
• Plants found near aquatic environments, such as the Coconut palm, produce fruits with a fibrous or spongy outer coat.
• This adaptation allows the fruit to float on water surfaces, enabling it to travel across rivers or oceans to reach new habitats.

Important Points

• Ecological Significance: Seed dispersal is vital for the survival and evolution of plant species. It prevents overcrowding in a single geographical area.
• Variety in Bursting Mechanisms: While Castor uses dehydration to trigger bursting, some plants like Squirting Cucumber use turgor pressure (liquid pressure) to eject seeds.
• Seed Characteristics: The seeds of Castor are rich in oil and contain a toxic protein called ricin, which serves as a defense mechanism against herbivores in their natural environment.
• Global Distribution: Due to efficient dispersal and human cultivation for industrial oil, Castor plants are now found in many tropical and subtropical regions across the world.

The correct answer is forces always occurring in pairs.

Key Points

• Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This principle fundamentally defines the nature of forces as interactions between objects.
• This law implies that forces never exist in isolation; they always occur in pairs. If Object A exerts a force on Object B, then Object B simultaneously exerts a force of equal magnitude and opposite direction back on Object A.
• The two forces in these pairs are often referred to as action and reaction forces. It is important to note that these forces act on different objects, which is why they do not cancel each other out to produce a zero net force on a single body.
• The interaction is simultaneous; there is no time delay between the action and the reaction. Both forces appear and disappear at the exact same moment.
• Option (1) describes the Law of Inertia, option (3) describes the Law of Acceleration, and option (4) is a specific application of Universal Gravitation. Therefore, the statement regarding pairs is the only one describing the Third Law.

Additional Information

• Sir Isaac Newton: He formulated the three laws of motion, which were first published in his work Philosophiæ Naturalis Principia Mathematica in the year 1687. These laws form the basis of classical mechanics.
• Real-world Examples:
• Recoil of a gun: When a bullet is fired (action), the gun pushes the bullet forward, and the bullet pushes the gun backward (reaction), causing recoil.
• Swimming: A swimmer pushes the water backward with their hands, and the water pushes the swimmer forward with an equal force.
• Propulsion: In a rocket, hot gases are expelled downward (action), and these gases exert an upward force on the rocket (reaction), allowing it to lift off.
• Newton's First Law: Also known as the Law of Inertia, it states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force. This relates to an object resisting a change in motion.
• Newton's Second Law: This law provides a quantitative description of force. It states that the force acting on an object is equal to the mass of that object times its acceleration (F = ma).
• Distinction between Third Law and Equilibrium: While equilibrium involves multiple forces acting on the same object that sum to zero, Newton's Third Law involves forces acting on two different objects. For example, a book resting on a table involves gravity and normal force (equilibrium on the book), but the Third Law pair is the book pushing the table down and the table pushing the book up.

Important Points

• Nature of Force: In physics, a force is not a property of an object but a result of an interaction. This means you cannot "have" force; you can only "exert" or "experience" it through contact or field interactions like gravity or magnetism.
• Vector Quantity: Force is a vector quantity, meaning it has both magnitude and direction. The Third Law emphasizes the directional relationship (exactly 180 degrees apart) of the force pairs.
• Universal Application: This law applies to all types of forces, including frictional forces, tension, gravitational pull, and electrostatic forces. Even when the Earth pulls an apple down, the apple pulls the Earth up with the same amount of force.

p data-end="57" data-start="0">The correct answer is precipitation reaction.

 

Key Points

• A precipitation reaction is a type of chemical reaction where two aqueous solutions react to form an insoluble solid, known as a precipitate, which settles out of the solution.
• In the reaction between Sodium Sulphate (Na2SO4) and Barium Chloride (BaCl2), both substances are initially in an aqueous state. When mixed, they undergo a chemical change.
• The chemical equation for this process is: Na2SO4 (aq) + BaCl2 (aq) -> BaSO4 (s) + 2NaCl (aq). The (s) denotes that Barium Sulphate is formed as a solid.
• This reaction is primarily a double displacement reaction because the Sulphate ions (SO4 2-) and Chloride ions (Cl-) exchange their respective cations (Sodium and Barium).
• The white precipitate formed in this specific reaction is Barium Sulphate, which is insoluble in water. This makes it a classic example of a precipitation reaction.
• The other product, Sodium Chloride, remains dissolved in the water as an aqueous solution.

Additional Information

• Combination Reaction:
• A reaction where two or more reactants combine to form a single product.
• Example: The burning of magnesium ribbon in oxygen to form magnesium oxide (2Mg + O2 -> 2MgO).
• In the given question, there are two products formed, so it cannot be a combination reaction.
• Decomposition Reaction:
• A reaction in which a single compound breaks down into two or more simpler substances.
• This usually requires energy in the form of heat, light, or electricity.
• Example: Calcium Carbonate decomposing into Calcium Oxide and Carbon Dioxide (CaCO3 -> CaO + CO2).
• Displacement Reaction (Single Displacement):
• A reaction in which a more reactive element displaces a less reactive element from its compound.
• Example: Iron reacting with Copper Sulphate solution (Fe + CuSO4 -> FeSO4 + Cu), where iron displaces copper.
• In the given reaction, two ions are exchanging places, not just one element displacing another.
• Double Displacement Reaction:
• A reaction where there is an exchange of ions between the reactants to form new compounds.
• The reaction between Sodium Sulphate and Barium Chloride is a prime example of this, as the Sodium and Barium cations switch their anion partners.

Important Points

• Observation: The most distinct feature of the Sodium Sulphate-Barium Chloride reaction is the instantaneous appearance of a milky white solid.
• Applications: Precipitation reactions are widely used in analytical chemistry to identify specific ions present in a solution and in water treatment to remove heavy metals or contaminants.
• Barium Sulphate is specifically used in medical imaging (Barium meal) because it is radio-opaque and insoluble, making it safe for X-ray examinations of the digestive system.

The correct answer is Acceleration is positive if it is in the direction of velocity, and negative if it is opposite..

Key Points

• Acceleration is a vector quantity defined as the rate of change of velocity of an object with respect to time. Because it is a vector, it has both magnitude and direction.
• The sign of acceleration (positive or negative) is determined by the direction of the acceleration vector relative to the velocity vector and the chosen coordinate system.
• When an object is speeding up, the acceleration acts in the same direction as the velocity. In standard physical conventions, if the velocity is moving in the positive direction, this acceleration is considered positive. Hence, statement 4 is correct.
• If the acceleration is directed opposite to the direction of motion (velocity), the object slows down. This is known as negative acceleration or retardation. Hence, statement 1 is incorrect.
• Acceleration is not always positive; it can be zero (constant velocity), positive (increasing speed in a positive direction), or negative (decreasing speed or increasing speed in a negative direction). Hence, statement 2 is incorrect.
• If acceleration were negative while being in the same direction as velocity, it would imply the entire coordinate system is flipped, but fundamentally, matching directions result in an increase in speed, which is the definition of positive acceleration in that frame of reference. Hence, statement 3 is incorrect.
• Hence, the correct answer is Acceleration is positive if it is in the direction of velocity, and negative if it is opposite.

Additional Information

• Mathematical Formula: The average acceleration (a) is calculated using the formula a = (v - u) / t, where v is the final velocity, u is the initial velocity, and t is the time interval.
• SI Units: The Standard International (SI) unit for acceleration is meters per second squared (m/s2). It describes how many meters per second the velocity changes every second.
• Retardation/Deceleration: These terms specifically refer to acceleration that opposes velocity, leading to a decrease in speed. For example, when a driver applies brakes to a car, the acceleration is negative relative to the car's forward velocity.
• Uniform vs. Non-Uniform Acceleration:
• Uniform Acceleration: This occurs when the velocity changes by equal amounts in equal time intervals. An example is an object in free fall, which experiences a constant gravitational acceleration of approximately 9.8 m/s2 on Earth.
• Non-Uniform Acceleration: This occurs when the velocity changes at a variable rate over time, such as a bus moving through heavy traffic with frequent starts and stops.
• Directional Context: It is important to note that a negative acceleration does not always mean slowing down. If an object is moving in the negative direction (velocity is negative) and the acceleration is also negative, the object is actually speeding up in that negative direction.

Important Points

• Velocity-Time Graphs: On a velocity-time (v-t) graph, the slope represents the acceleration. A positive slope indicates positive acceleration, while a negative slope indicates negative acceleration.
• Instantaneous Acceleration: This refers to the acceleration of an object at a specific moment in time, calculated as the derivative of velocity with respect to time (dv/dt).
• Newton's Second Law: Acceleration is directly proportional to the net force acting on an object and inversely proportional to its mass, represented by the law F = ma, established by Isaac Newton in 1687.

The correct answer is the sum of the atomic masses of all atoms in a formula unit of a compound.

Key Points

• The formula unit mass of a substance is the sum of the atomic masses of all atoms in a formula unit of a compound. Hence, statement 3 is correct.
• It is calculated by adding the atomic masses of all the atoms present in the chemical formula of the substance. This is conceptually similar to molecular mass, but the terminology "formula unit" is preferred for ionic compounds.
• The atomic number (Statement 1) represents only the number of protons in an atom and does not account for the neutrons, which contribute significantly to the total mass. Hence, statement 1 is incorrect.
• The number of protons (Statement 2) is equivalent to the atomic number and does not define the mass of the entire formula unit. Hence, statement 2 is incorrect.
• The number of electrons (Statement 4) has a negligible contribution to the total mass of an atom compared to protons and neutrons, and thus their sum does not constitute the formula unit mass. Hence, statement 4 is incorrect.
• The unit used to express formula unit mass is the unified atomic mass unit, represented by the symbol u.

Additional Information

• Ionic Compounds and Formula Units:
• Unlike covalent compounds, ionic compounds do not exist as individual molecules. Instead, they form a three-dimensional crystal lattice structure consisting of many positive and negative ions.
• The formula unit represents the simplest whole-number ratio of these ions. For instance, in Sodium Chloride, the ratio of Na+ to Cl- ions is 1:1, so the formula unit is NaCl.
• Calculation Examples:
• Sodium Chloride (NaCl): The atomic mass of Sodium (Na) is 23 u and Chlorine (Cl) is 35.5 u. The formula unit mass is 23 + 35.5 = 58.5 u.
• Calcium Chloride (CaCl2): The atomic mass of Calcium (Ca) is 40 u and Chlorine (Cl) is 35.5 u. The mass is 40 + (2 x 35.5) = 40 + 71 = 111 u.
• Zinc Oxide (ZnO): The atomic mass of Zinc (Zn) is 65 u and Oxygen (O) is 16 u. The formula unit mass is 65 + 16 = 81 u.
• The Unified Atomic Mass Unit (u):
• This is the standard unit used to measure the relative mass of atoms. It is defined as exactly 1/12th of the mass of a single Carbon-12 atom.
• Previously, it was denoted as amu, but it was changed to u (unified mass) as per IUPAC recommendations.

Important Points

• Difference between Molecular Mass and Formula Unit Mass:
• Molecular mass is used specifically for substances that exist as discrete molecules (e.g., H2O, CH4).
• Formula unit mass is used for substances that consist of ions or giant covalent structures where discrete molecules are not present.
• Numerically, both are calculated using the same method: summing the atomic masses of all atoms in the given chemical formula.
• Molar Mass Connection:
• The molar mass of a substance is numerically equal to its formula unit mass but is expressed in grams per mole (g/mol).
• One mole of any substance contains 6.022 x 10^23 formula units, a constant known as Avogadro's Number.

The correct answer is It transports food, carbon dioxide and waste materials..

Key Points

• Blood Plasma is a pale yellow liquid component of blood that holds the blood cells in whole blood in suspension. It makes up about 55% of the body's total blood volume.
• The primary function of plasma is to transport nutrients (food), hormones, and proteins to the parts of the body that need them. It also carries waste products like urea and carbon dioxide from the cells to the excretory organs. Hence, statement 3 is correct.
• Red Blood Cells (RBCs), or erythrocytes, contain a protein called hemoglobin, which binds to oxygen. The primary role of carrying oxygen from the lungs to all body parts is performed by RBCs, not the plasma itself. Hence, statement 1 is incorrect.
• Maintaining an oxygen-rich blood supply to tissues is the physiological responsibility of the circulatory system driven by the heart and mediated by hemoglobin in red blood cells. Hence, statement 2 is incorrect.
• While plasma contains clotting factors (like fibrinogen), the primary cellular component that helps in clotting blood during injuries to prevent excessive bleeding is the platelets (thrombocytes). Hence, statement 4 is incorrect.

Additional Information

• Composition of Plasma:
• Water: Approximately 90-92% of plasma is water, which acts as a solvent for transporting various substances.
• Plasma Proteins: These include Albumin (maintains osmotic pressure), Globulins (involved in immunity/antibodies), and Fibrinogen (essential for blood coagulation).
• Electrolytes: Ions like Sodium (Na+), Chloride (Cl-), and Bicarbonate (HCO3-) help maintain pH balance and osmotic pressure.
• Functions in Homeostasis:
• Plasma plays a critical role in maintaining a constant body temperature by absorbing and distributing heat throughout the body.
• It helps in maintaining the acid-base balance (pH) of the body, which is vital for various biochemical reactions.
• Transport Mechanism:
• Carbon dioxide is transported in the plasma mainly in the form of bicarbonate ions (about 70%), while a small amount is dissolved directly in the plasma.
• Nutrients absorbed from the digestive tract, such as glucose, amino acids, and lipids, are dissolved in the plasma to be delivered to tissues.

Important Points

• Serum vs. Plasma: Serum is the liquid part of blood that remains after blood clotting has occurred. It is essentially plasma without the clotting factors (like fibrinogen).
• Blood pH: The pH of human blood is slightly alkaline, typically ranging between 7.35 and 7.45. Plasma acts as a buffer to keep this range stable.
• Discovery: The technique for plasma storage and its use for transfusions was significantly advanced by Dr. Charles Drew in the 1940s.

The correct answer is Image is virtual, erect and smaller than the object.

Key Points

• A concave lens is also known as a diverging lens because it diverges the light rays that pass through it. It is thinner at the center than at the edges.
• For a concave lens, the focal length (f) is always taken as negative. In this case, f = -15 cm.
• The object distance (u) is always negative according to the sign convention. Here, u = -10 cm.
• Using the Lens Formula, 1/v - 1/u = 1/f, where v is the image distance:
• 1/v = 1/f + 1/u
• 1/v = 1/(-15) + 1/(-10)
• 1/v = -1/15 - 1/10
• 1/v = (-2 - 3) / 30 = -5/30 = -1/6
• Therefore, v = -6 cm.
• The negative sign of v indicates that the image is virtual and formed on the same side as the object. All virtual images formed by a single lens are erect.
• The magnification (m) is calculated as m = v/u. Substituting the values: m = -6 / -10 = 0.6. Since the magnitude of m is less than 1, the image is smaller than the object (diminished).
• Statement 1: Image is real, inverted and smaller than the object. Hence, statement 1 is incorrect.
• Statement 2: Image is virtual, erect and larger than the object. Hence, statement 2 is incorrect.
• Statement 3: Image is virtual, erect and smaller than the object. Hence, statement 3 is correct.
• Statement 4: Image is real, inverted and larger than the object. Hence, statement 4 is incorrect.

Additional Information

• Nature of Images in Concave Lenses:
• Regardless of the position of the object, a concave lens always produces a virtual, erect, and diminished image.
• The image is always formed between the optical center and the principal focus (F1) on the same side as the object.
• Magnification (m):
• If m is positive, the image is virtual and erect.
• If m is negative, the image is real and inverted.
• If |m| < 1, the image is diminished. If |m| > 1, the image is enlarged.
• Sign Convention (New Cartesian):
• The optical center (O) is taken as the origin.
• Distances measured in the direction of incident light are positive; opposite to it are negative.
• Height measured upwards and perpendicular to the principal axis is positive.

Important Points

• Power of a Lens: The power (P) is the reciprocal of the focal length in meters (P = 1/f). For a concave lens, since f is negative, the power is also negative.
• Applications of Concave Lenses:
• Used in spectacles to correct myopia (short-sightedness).
• Used in peepholes of house doors to provide a wide-angle view of people outside.
• Used in flashlights to spread the light beam over a wider area.
• Comparison with Convex Lenses: Unlike concave lenses, convex lenses can form both real and virtual images, which can be enlarged, diminished, or same-sized depending on the object's distance from the lens.

The correct answer is It changes a liquid to gas from the surface below boiling point.

Key Points

• Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. It is fundamentally different from boiling because it occurs only at the surface and can happen at any temperature below the boiling point of the liquid.
• Statement 1 suggests that it changes solids directly into vapors. This process is actually known as sublimation (e.g., dry ice or camphor) and does not involve the liquid phase at all. Hence, statement 1 is incorrect.
• Statement 2 claims it takes place only at the boiling temperature. This is the definition of boiling, which is a bulk phenomenon where the entire mass of the liquid transitions to gas. Evaporation, conversely, happens continuously even at room temperature. Hence, statement 2 is incorrect.
• Statement 3 defines the process correctly as the transition from liquid to gas from the surface at temperatures below the boiling point. In this process, particles at the surface gain enough kinetic energy to overcome the intermolecular forces of attraction. Hence, statement 3 is correct.
• Statement 4 mentions that the liquid must freeze first. Freezing is the transition from liquid to solid, which is the opposite direction of the energy change required for gas formation. Hence, statement 4 is incorrect.

Additional Information

• Factors Affecting Evaporation:
• Surface Area: Since evaporation is a surface phenomenon, increasing the surface area (e.g., spreading out wet clothes) increases the rate of evaporation.
• Temperature: An increase in temperature provides more kinetic energy to the particles, allowing a greater number of them to escape into the vapor state.
• Humidity: It is the amount of water vapor present in the air. If the humidity is high, the air cannot hold much more moisture, thus decreasing the rate of evaporation.
• Wind Speed: Higher wind speeds move water vapor particles away from the surface, making room for more molecules to evaporate, thus increasing the rate.
• Cooling Effect:
• Evaporation causes cooling. During the process, the particles of the liquid absorb energy (the Latent Heat of Vaporization) from the surroundings to regain the energy lost during evaporation.
• Common examples include sweating in humans to regulate body temperature and the cooling of water stored in earthen pots (matkas) during summer.
• Comparison with Boiling:
• While boiling is a fast process that occurs throughout the bulk of the liquid at a fixed temperature, evaporation is a slow and silent process occurring only at the surface at variable temperatures.

Important Points

• The Latent Heat of Vaporization is the heat energy required to change 1 kg of a liquid into gas at atmospheric pressure at its boiling point.
• In the context of the Water Cycle, evaporation from oceans, seas, and lakes is the primary pathway for water to move from the liquid state back into the atmosphere as water vapor.
• Vapor Pressure: Evaporation continues until the partial pressure of the vapor in the air reaches the equilibrium vapor pressure of the liquid.

The correct answer is Highly ductile and malleable.

Key Points

• Metals are characterized by their physical property of being ductile, which means they can be drawn into thin wires without breaking. Gold is considered the most ductile metal; a single gram can be stretched into a wire nearly 2 kilometers long. Hence, statement 1 is correct.
• Malleability is another defining physical property of metals, allowing them to be hammered or rolled into thin sheets. Metals like silver and aluminium exhibit high malleability, which is why they are used in making foils and decorative leaves. Hence, statement 1 is also supported by this fact.
• Most metals are hard and possess high tensile strength, meaning they can support heavy loads without snapping. In contrast, it is the non-metals that are generally soft or brittle and break when force is applied. Hence, statement 2 is incorrect.
• Metals are excellent conductors of heat and electricity due to the presence of free electrons. While silver and copper are the best conductors, materials that are poor conductors or brittle are typically classified as non-metals or metalloids. Hence, statement 3 is incorrect.
• At room temperature, almost all metals exist in a solid state because of the strong metallic bonding between atoms. The only exception is mercury (Hg), which is a liquid. Non-metals like hydrogen, nitrogen, and oxygen are the elements that typically form gases at room temperature. Hence, statement 4 is incorrect.

Additional Information

• Metallic Lustre: Metals in their pure state have a shining surface. This property is known as metallic lustre. Metals like gold, silver, and platinum are highly valued for their ability to retain this shine, making them ideal for jewelry.
• Sonority: Metals are sonorous, meaning they produce a ringing sound when struck by a hard object. This property is utilized in making church bells, musical instruments, and school bells.
• Melting and Boiling Points: Metals generally have high melting points and boiling points. For example, Tungsten (W) has the highest melting point (approximately 3422°C), which is why it is used in bulb filaments. However, metals like Gallium and Cesium have very low melting points and can melt if kept on a palm.
• Density: Most metals have a high density, making them heavy. Osmium is known to be the densest naturally occurring element. Conversely, alkali metals like lithium, sodium, and potassium have low densities and are soft enough to be cut with a knife.
• Alloy Formation: Metals can be mixed with other metals or non-metals to form alloys to enhance their properties. For instance, Steel (an alloy of iron and carbon) is much stronger than pure iron and is resistant to corrosion.

Important Points

• Exceptions in Metals: While most metals are hard solids, Mercury is a liquid at room temperature. Sodium and Potassium are so soft that they can be cut with a knife, which is a rare physical trait for metals.
• Chemical Reactivity: Metals typically react with oxygen to form basic oxides (e.g., Magnesium Oxide). They are electropositive elements because they lose electrons to form cations during chemical reactions.
• Periodic Table Placement: On the Modern Periodic Table, metals are located on the left side and in the center. The s-block, d-block, and f-block consist entirely of metals, while the p-block contains a mix of metals, non-metals, and metalloids.

The correct answer is Father’s sperm.

Key Points

• In humans, the biological sex of an individual is determined by the chromosomal makeup established at the moment of fertilization.
• Human cells contain 23 pairs of chromosomes, for a total of 46. The first 22 pairs are known as autosomes, which carry genes for general body traits and are identical in both males and females.
• The 23rd pair consists of the sex chromosomes (allosomes). Females possess two X chromosomes (XX), while males possess one X and one Y chromosome (XY).
• During gametogenesis, females produce ova (eggs) that only carry an X chromosome. Males, being the heterogametic sex, produce two types of sperm: approximately 50% carry an X chromosome and 50% carry a Y chromosome.
• If a sperm carrying an X chromosome fertilizes the egg, the zygote will be XX (female). If a sperm carrying a Y chromosome fertilizes the egg, the zygote will be XY (male). Hence, the correct answer is Father’s sperm.

Additional Information

• Sex Chromosomes (Allosomes):
• These are distinct from autosomes and are responsible for the development of primary and secondary sexual characteristics. The Y chromosome is significantly smaller than the X chromosome and contains fewer genes.
• Mitochondrial DNA (mtDNA):
• Unlike nuclear DNA, mitochondrial DNA is inherited maternally. This is because the mitochondria in a developing embryo are derived almost entirely from the cytoplasm of the mother's ovum, as the sperm contributes only its nuclear genetic material during fusion. It does not determine the sex of the child.
• Autosomes:
• Humans have 44 autosomes. Any variation in the number of autosomes (such as Trisomy 21 or Down Syndrome) leads to genetic disorders but does not change the biological sex of the offspring.
• Spermatogenesis vs. Oogenesis:
• Spermatogenesis in the male testes results in four viable sperm from one primary spermatocyte, maintaining the X and Y ratio. Oogenesis in the female ovaries results in one functional ovum and polar bodies, all containing X chromosomes.

Important Points

• The SRY Gene: The Sex-determining Region Y (SRY) gene is located on the short arm of the Y chromosome. This gene acts as the "master switch" for testis-determining factor (TDF), which triggers the development of male gonads in an embryo.
• Probability: Since the segregation of X and Y chromosomes during male meiosis occurs randomly, there is a theoretical 50% probability for a child to be born male or female in every single pregnancy.
• Genetic Inheritance: The concept that the father determines the sex was scientifically solidified in the early 20th century (around 1905) through the independent work of Nettie Stevens and Edmund Beecher Wilson, who discovered the role of X and Y chromosomes in insects and humans.
• Biological Determination: It is important to note that the mother’s ovum is passive regarding sex determination, as it can only provide an X chromosome; the outcome depends entirely on whether the paternal gamete introduces a second X or a Y.

The correct answer is Red Sindhi.

Key Points

• Red Sindhi is a premier indigenous dairy breed of cattle that originated in the Sindh province (now in Pakistan), but is extensively found and reared across India.
• This breed is world-renowned for its high resistance to diseases and its remarkable ability to tolerate tropical heat and high humidity, which are common in South Asian environments. Hence, the correct answer is Red Sindhi.
• Physically, the Red Sindhi is characterized by its deep reddish-brown color, ranging from light red to dark brown. They possess a large udder and are considered one of the best milch breeds among the Zebu cattle (humped cattle).
• The average milk yield for this breed is approximately 1100 to 2600 kg per lactation. Because of its hardy nature and strong immune system, it is frequently used in crossbreeding programs globally to improve the disease resistance of high-yielding exotic breeds.
• Unlike exotic breeds, Red Sindhi cattle are less susceptible to ectoparasites like ticks and the various vector-borne diseases they carry, making them economically viable for small-scale farmers in developing nations.

Additional Information

• Jersey:
• This is an exotic breed originating from the Island of Jersey, United Kingdom.
• While it is famous for having the highest fat content (about 5.3%) in its milk, it is highly sensitive to heat stress and lacks the natural immunity to tropical diseases found in Indian breeds.
• Brown Swiss:
• Originating from Switzerland, this exotic breed is one of the oldest in the world.
• It is a dual-purpose breed used for both milk and meat production. However, it requires a temperate climate to thrive and is prone to many tropical ailments.
• Aseel:
• It is important to note that Aseel is not a breed of cattle; it is a famous indigenous breed of poultry (chicken).
• Native to the Andhra Pradesh and Chhattisgarh regions, it is known for its stamina, aggressive behavior (used in cockfighting), and high-quality meat.
• Classification of Indian Cattle:
• Milch Breeds: High milk producers like Sahiwal, Gir, and Red Sindhi.
• Draught Breeds: Used for agricultural labor and transport, such as Amritmahal and Hallikar.
• Dual-purpose Breeds: Serving both milk and labor needs, such as Haryana and Tharparkar.

Important Points

• Zebu Cattle (Bos indicus): These are humped cattle found in India and Africa. They possess a dewlap and a hump, which help in thermoregulation (cooling the body) and provide natural resistance to various pathogens.
• National Dairy Research Institute (NDRI): Located in Karnal, Haryana, this research institute plays a vital role in the genetic improvement and conservation of indigenous breeds like Red Sindhi.
• Disease resistance in indigenous breeds: Breeds like Red Sindhi and Sahiwal have evolved over centuries to develop innate immunity against Foot and Mouth Disease (FMD) and Rinderpest, compared to their European counterparts.

The correct answer is Robert Brown.

Key Points

• Robert Brown, a prominent Scottish botanist, is credited with the discovery and description of the cell nucleus in the year 1831.
• While examining the epidermis of orchids under a microscope, he noticed an opaque area within the cells, which he termed the nucleus (derived from the Latin word for "kernel" or "nut").
• This discovery was a fundamental milestone in the field of cytology (cell biology), as it challenged the previous notion that the cell wall was the only significant part of a cell.
• His observations helped scientists understand that cells have internal organization and specialized organelles responsible for various biological processes.
• Brown's work paved the way for the later formulation of the Cell Theory, as it highlighted a consistent internal structure across different types of plant cells.

Additional Information

• Robert Hooke:
• In 1665, he discovered the cell for the first time while observing a thin slice of cork.
• He used a primitive compound microscope and observed box-like structures that reminded him of small rooms occupied by monks, hence the name "cell".
• Hooke’s discovery was limited to dead plant cells, specifically the cell walls of the cork tissue.
• Anton van Leeuwenhoek:
• In 1674, he was the first to observe living cells using his own high-quality single-lens microscopes.
• He discovered bacteria, protozoa, spermatozoa, and red blood cells, referring to them as "animalcules."
• His work expanded the understanding of microscopic life and the diversity of cellular organisms.
• Rudolf Virchow:
• In 1855, he made a crucial addition to the Cell Theory by proposing the concept of "Omnis cellula e cellula".
• This phrase translates to "all cells arise from pre-existing cells," refuting the theory of spontaneous generation.
• His contribution shifted the focus of pathology to the cellular level, suggesting that diseases arise in individual cells.
• Matthias Schleiden and Theodore Schwann:
• In 1838 and 1839, respectively, they proposed that all plants and animals are composed of cells.
• Together, they established the first two tenets of the Cell Theory: that the cell is the basic unit of life and that all living things are made of cells.

Important Points

• The nucleus is often described as the control center or brain of the cell because it houses the genetic material (DNA).
• It is surrounded by a nuclear envelope, which is a double-layered membrane that regulates the transport of materials via nuclear pores.
• The primary functions of the nucleus include DNA replication, transcription (production of RNA), and maintaining the integrity of genes.
• In eukaryotic cells, the nucleus contains the nucleolus, which is the site of ribosome biogenesis.
• While most cells have one nucleus, certain cells like human erythrocytes (mature red blood cells) are enucleated (lack a nucleus), while others like skeletal muscle cells are multinucleated.

Shortcut Trick

Total Distance = Sum of lengths = 259 + 431 = 690 m

Relative Speed (Opposite) = 52 + 56 = 108 km/h

Convert to m/s: 108 × (5/18) = 30 m/s

Time = Distance ÷ Relative Speed = 690 ÷ 30 = 23 sec

∴ The correct answer is 23 sec.

Alternate Method

Given:

Length of first train (L₁) = 259 m

Length of second train (L₂) = 431 m

Speed of first train (S₁) = 52 km/h

Speed of second train (S₂) = 56 km/h

Formula Used:

Time = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 259 + 431 = 690 m

⇒ Relative Speed = 52 + 56 = 108 km/h

⇒ Relative Speed in m/s = 108 × (5/18) = 30 m/s

⇒ Time taken = 690 ÷ 30

⇒ Time = 23 sec

∴ The correct answer is 23 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). In the same direction, they are subtracted (S₁ − S₂).

Unit Conversion

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

Train Crossing Logic

When a train crosses another train, the total distance covered is always the sum of the lengths of both trains.

Shortcut Trick

Let the total votes be 100%.

Candidate 1 = 42% | Candidate 2 = 100% − 42% = 58%.

Margin = 58% − 42% = 16%.

Given 16% = 416 votes.

100% = (416 ÷ 16) × 100 = 26 × 100 = 2600.

∴ The correct answer is 2600 votes.

Alternate Method

Given: Votes of losing candidate = 42%, Defeat margin = 416 votes.

Formula: Total Votes = (Margin × 100) ÷ (Difference in % share).

⇒ Let Total Votes = V.

⇒ Winning Candidate Share = 100% − 42% = 58%.

⇒ Difference in share = 58% − 42% = 16%.

⇒ 16% of V = 416.

⇒ (16 ÷ 100) × V = 416.

⇒ V = (416 × 100) ÷ 16.

⇒ V = 26 × 100 = 2600.

∴ The correct answer is 2600 votes.

Additional Information

Percentage Change

To calculate the percentage change between two values: [(New Value − Old Value) ÷ Old Value] × 100.

Invalid Votes Concept

If invalid votes are present: Total Valid Votes = Total Cast Votes − Invalid Votes. The winner/loser percentages are usually calculated on valid votes.

Base Change Rule

If a candidate gets x% of the total votes, the other gets (100 − x)% provided there are only two candidates and no invalid votes.

Shortcut Trick

Using the Multiplying Factor method for successive changes:

Final Price = 100 × 0.60 × 0.75 × 0.55 × 1.55

Final Price = 100 × 0.383625 = 38.3625

Net change = 38.3625 − 100 = −61.6375%

∴ The correct answer is Decreases by 61.64%.

Alternate Method

Given: Original Price = 100 units. Changes: −40%, −25%, −45%, +55%.

⇒ Price after 1st month = 100 − (40% of 100) = 60

⇒ Price after 2nd month = 60 − (25% of 60) = 60 − 15 = 45

⇒ Price after 3rd month = 45 − (45% of 45) = 45 − 20.25 = 24.75

⇒ Price after 4th month = 24.75 + (55% of 24.75) = 24.75 + 13.6125 = 38.3625

⇒ Percentage Change = (38.3625 − 100) / 100 × 100 = −61.6375%

⇒ Approximately 61.64% decrease.

∴ The correct answer is Decreases by 61.64%.

Additional Information

Successive Percentage Formula

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

Multiplying Factor (MF)

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

Order of Operations

In successive percentage changes, the order in which increases or decreases are applied does not affect the final result.

Shortcut Trick

Check LCM of 6 and 9 → LCM(6, 9) = 18. All numbers must be multiples of 18.

Numbers divisible by 12 are also divisible by 4 (since 12 = 3 × 4).

For 126: Last two digits (26) are not divisible by 4, so it is not divisible by 12.

For 72, 108, 252: All are divisible by 12 (72/12=6, 108/12=9, 252/12=21).

∴ The correct answer is 126.

Alternate Method

Given: Numbers 126, 252, 72, 108.

Conditions: Divisible by 6 AND 9 (meaning divisible by 18) AND NOT divisible by 12.

⇒ 126 ÷ 6 = 21 (Yes)

⇒ 126 ÷ 9 = 14 (Yes)

⇒ 126 ÷ 12 = 10.5 (No - Not Divisible)

⇒ 252 ÷ 12 = 21 (Divisible)

⇒ 72 ÷ 12 = 6 (Divisible)

⇒ 108 ÷ 12 = 9 (Divisible)

∴ The correct answer is 126.

Additional Information

Divisibility Rule of 6

A number is divisible by 6 if it is divisible by both 2 (even number) and 3 (sum of digits is a multiple of 3).

Divisibility Rule of 9

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

Divisibility Rule of 12

A number is divisible by 12 if it is divisible by both 3 and 4 (last two digits divisible by 4).

Shortcut Trick

Combined efficiency of A, B, and C = (1/3 + 1/15 + 1/10) of work per day.

LCM of 3, 15, 10 is 30. Combined efficiency = (10 + 2 + 3)/30 = 15/30 = 1/2 work per day.

Time for single work = 2 days. For twice the work = 2 × 2 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

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

Formula Used: Efficiency = Total Work ÷ Time taken

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

⇒ Total Work to be done = 2 × 30 = 60 units.

⇒ Required Time = Total Work ÷ Combined Efficiency = 60 ÷ 15 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Total Work (LCM Method)

The most efficient way to solve Time & Work problems is to assume Total Work as the LCM of given days, making efficiency an integer.

Work & Time Relationship

Time taken is directly proportional to the amount of Work and inversely proportional to Efficiency (T = W/E).

Pipes & Cisterns

Same logic applies where Filling Pipe has positive efficiency and Leaking/Emptying Pipe has negative efficiency.

Shortcut Trick

Convert Rate 5% to fraction: 5/100 = 1/20.

For 2 years, the ratio of Principal to Amount is 20² : 21² = 400 : 441.

Given 400 units = ₹2,000 ⇒ 1 unit = ₹5.

Amount = 441 units × ₹5 = ₹2,205.

∴ The correct answer is ₹2,205.

Alternate Method

Given: Principal (P) = ₹2,000, Rate (R) = 5%, Time (n) = 2 years.

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

Calculations:

⇒ A = 2000 × (1 + 5/100)²

⇒ A = 2000 × (1 + 1/20)²

⇒ A = 2000 × (21/20) × (21/20)

⇒ A = 2000 × (441/400)

⇒ A = 5 × 441

⇒ A = ₹2,205

∴ The correct answer is ₹2,205.

Additional Information

Effective Rate of Interest

For 2 years, the effective Compound Interest rate can be calculated as R + R + (R×R)/100.

Difference between CI and SI

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

Compounding Half-Yearly

If interest is compounded half-yearly, the Rate is halved (R/2) and the Time is doubled (2n).

Shortcut Trick

Assume Emma's age is 1 unit.

Saloni's age = 2 × 1 unit + 7 = 2 units + 7.

Total age = 1 unit + (2 units + 7) = 3 units + 7 = 25 years.

3 units = 25 − 7 = 18 → 1 unit = 6 years.

∴ The correct answer is 6 years.

Alternate Method

Given:

Sum of Saloni and Emma's ages = 25 years.

Saloni's age = 2 × (Emma's age) + 7.

Let Emma's age be x years.

Then Saloni's age = 2x + 7.

⇒ x + (2x + 7) = 25

⇒ 3x + 7 = 25

⇒ 3x = 25 − 7

⇒ 3x = 18

⇒ x = 18 ÷ 3

⇒ x = 6

∴ The correct answer is 6 years.

Additional Information

Linear Equations in One Variable

An equation of the form ax + b = c, where x is the unknown variable to be solved by isolating it.

Age Relation Problems

Always define the independent age (usually the younger person) as 'x' to simplify the mathematical expression for the older person.

Verification Method

Plug the result back: if Emma is 6, Saloni is (2×6)+7 = 19. Sum = 6 + 19 = 25. The answer is consistent with the data.

Shortcut Trick

Selling Price (SP) = Marked Price (MP) × (100 − Discount%) ÷ 100

SP = 6,420 × (100 − 30) ÷ 100

SP = 6,420 × 0.7

SP = 4,494

∴ The correct answer is ₹4,494.

Alternate Method

Given: Marked Price (MP) = ₹6,420 and Discount = 30%. Formula: Selling Price (SP) = MP − Discount Amount | Discount Amount = (Discount% ÷ 100) × MP.

⇒ Discount Amount = (30 ÷ 100) × 6,420 ⇒ Discount Amount = 0.3 × 6,420 = 1,926 ⇒ Selling Price = 6,420 − 1,926 ⇒ Selling Price = 4,494

∴ The correct answer is ₹4,494.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP to arrive at the Selling Price (SP).

Relationship Formula

SP = MP × (100 − D%) / 100. If multiple discounts are given, they are applied successively.

Shortcut Trick

Ratio of articles P : Q : R = 3 : 4 : 5

Cost of one set = (3 × 280) + (4 × 180) + (5 × 100) = 840 + 720 + 500 = ₹2060

Number of sets = Total expenditure ÷ Cost of one set = 26780 ÷ 2060 = 13

Articles of type Q = 4 × 13 = 52

∴ The correct answer is 52.

Alternate Method

Given: Prices of P, Q, R are ₹280, ₹180, ₹100. Ratio of quantity = 3 : 4 : 5. Total = ₹26,780.

Formula Used: Total Cost = ∑(Price × Quantity)

⇒ Let the quantities be 3x, 4x, and 5x respectively.

⇒ Total Cost = (280 × 3x) + (180 × 4x) + (100 × 5x) = 26780

⇒ 840x + 720x + 500x = 26780

⇒ 2060x = 26780

⇒ x = 26780 ÷ 2060 = 13

⇒ Number of articles of type Q = 4x = 4 × 13 = 52

∴ The correct answer is 52.

Additional Information

Ratio and Proportion

When items are bought in a ratio a : b : c, the actual quantities are ax, bx, and cx where x is the common multiplier.

Weighted Average Logic

Total value is the sum of products of individual values and their respective weights (quantities). Total = ∑(v_i × w_i).

Shortcut Trick

((3²)⁶)⁵ ⇒ multiply powers

= 3^2×6×5 = 3⁶⁰

Now 9 = 3²

⇒ 3⁶⁰ = (3²)³⁰ = 9³⁰

∴ Exponent = 30

Alternate Method

Given:

((3²)⁶)⁵

Concept Used:

Laws of exponents

Calculation:

⇒ (3²)⁶ = 3¹²

⇒ (3¹²)⁵ = 3⁶⁰

⇒ 9 = 3²

⇒ 3⁶⁰ = (3²)³⁰ = 9³⁰

∴ The correct answer is 30.

Additional Information

Power Rule

(a^m)ⁿ = a^mn

Base Conversion

Express numbers in same base to compare powers

Exponent Simplification

Multiply exponents when powers are raised again

Shortcut Trick

Use the options substitution method for rapid verification.

From Option (1): Price of mango = 3, Price of apple = 8.

Check first condition: (10 × 3) + (13 × 8) = 30 + 104 = 134. (True)

Check second condition: (15 × 3) − (4 × 8) = 45 − 32 = 13. (True)

Since both conditions are satisfied, Option (1) is the correct pair.

∴ The correct answer is 3, 8.

Alternate Method

Given → Let cost of one mango be m and one apple be a. Formulas: Total Cost = (Quantity₁ × Price₁) + (Quantity₂ × Price₂)

⇒ Multiply Eq 1 by 4 and Eq 2 by 13 to eliminate 'a'. ⇒ (40m + 52a = 536) + (195m − 52a = 169) ⇒ 235m = 705 ⇒ m = 705 ÷ 235 = 3 ⇒ Substitute m = 3 in Eq 1: 10(3) + 13a = 134 ⇒ 30 + 13a = 134 ⇒ 13a = 104 ⇒ a = 8

∴ The correct answer is 3, 8.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c can be solved using the Elimination Method or Substitution Method to find the unique values of variables.

Verification of Roots

Always plug the values back into the original constraints to ensure both logical conditions are satisfied simultaneously.

Shortcut Trick

Substitute standard trigonometric values directly.

sin 30° = 1/2, cos 60° = 1/2, cot 45° = 1.

Expression = (1/2 + 1/2) ÷ 1 = 1 ÷ 1 = 1.

∴ The correct answer is 1.

Alternate Method

Given: sin 30°, cos 60°, cot 45°

Formula Used: Values of trigonometric ratios at specific angles.

⇒ sin 30° = 1/2

⇒ cos 60° = 1/2

⇒ cot 45° = 1

⇒ Value = (sin 30° + cos 60°) ÷ cot 45°

⇒ Value = (1/2 + 1/2) ÷ 1

⇒ Value = 1 ÷ 1 = 1

∴ The correct answer is 1.

Additional Information

Complementary Angles

For any angle θ, sin θ = cos(90° − θ). Therefore, sin 30° = cos(90° − 30°) = cos 60°.

Reciprocal Identities

cot θ is the reciprocal of tan θ. Since tan 45° = 1, cot 45° = 1/1 = 1.

Pythagorean Identity

sin²θ + cos²θ = 1 is the fundamental identity for all trigonometric calculations.

Shortcut Trick

Total sum of 8 numbers = 121 × 8 = 968.

Sum of 7 numbers = 35 + 35 + 49 + 83 + 102 + 80 + 20 = 404.

Eighth number = Total Sum − Sum of 7 numbers.

8th number = 968 − 404 = 564.

∴ The correct answer is 564.

Alternate Method

Given:

Average of 8 numbers = 121

Value of 7 numbers = 35, 35, 49, 83, 102, 80, 20

Formula Used:

Average = Sum of all observations ÷ Total number of observations

Calculations:

⇒ Sum of 8 numbers = 121 × 8 = 968

⇒ Sum of 7 numbers = 35 + 35 + 49 + 83 + 102 + 80 + 20

⇒ Sum of 7 numbers = 404

⇒ Let the eighth number be x.

⇒ Sum of 8 numbers = (Sum of 7 numbers) + x

⇒ 968 = 404 + x

⇒ x = 968 − 404

⇒ x = 564

∴ The correct answer is 564.

Additional Information

Arithmetic Mean

The average or arithmetic mean is calculated by adding all the values and dividing by the count of values: Average = Σx_i ÷ n.

Sum of Observations

Total sum can always be found if the average and total count are known: Sum = Average × Count.

Deviation Method

For larger numbers, the sum of deviations of all observations from the average is always zero.

Shortcut Trick

Multiply initial ratio parts by their respective percentage growth factors.

New Ratio = (4 × 190) : (18 × 190) : (21 × 140)

Simplify: (4 × 19) : (18 × 19) : (21 × 14) ⇒ 76 : 342 : 294

Divide by 2: 38 : 171 : 147.

∴ The correct answer is 38:171:147.

Alternate Method

Given: Initial ratio of salaries X : Y : Z = 4 : 18 : 21. Let salaries be 400, 1800, and 2100.

Formula: New Value = Original Value × (100 + Increment %)/100

⇒ New Salary of X = 400 + (90% of 400) = 400 + 360 = 760

⇒ New Salary of Y = 1800 + (90% of 1800) = 1800 + 1620 = 3420

⇒ New Salary of Z = 2100 + (40% of 2100) = 2100 + 840 = 2940

⇒ New Ratio = 760 : 3420 : 2940

⇒ New Ratio = 76 : 342 : 294 (Dividing by 10)

⇒ New Ratio = 38 : 171 : 147 (Dividing by 2)

∴ The correct answer is 38:171:147.

Additional Information

Compound Ratio

The ratio obtained by multiplying the corresponding terms of two or more ratios (a:b and c:d becomes ac:bd).

Percentage Increment

When a value increases by r%, the new value is (100 + r)/100 times the original value.

Ratio Simplification

A ratio remains unchanged if all its terms are multiplied or divided by the same non-zero constant.

Shortcut Trick

Assume a base mean of 50 and calculate deviations: (−15, +3, −12, +39, +21, +19, +36, −10, −9).

Net sum of deviations = 72.

Average deviation = 72 ÷ 9 = 8.

Actual Arithmetic Mean = 50 + 8 = 58.

∴ The correct answer is 58.

Alternate Method

Given: Observations are 35, 53, 38, 89, 71, 69, 86, 40, and 41.

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

Step 1: Calculate the Sum of Observations

⇒ Sum = 35 + 53 + 38 + 89 + 71 + 69 + 86 + 40 + 41

⇒ Sum = 522

Step 2: Identify Total Number of Observations

⇒ n = 9

Step 3: Calculate the Mean

⇒ Mean = 522 ÷ 9

⇒ Mean = 58

∴ The correct answer is 58.

Additional Information

Arithmetic Mean Property

The sum of the deviations of all observations from their arithmetic mean is always zero. If each observation is increased by a constant k, the mean also increases by k.

Sum of Observations Formula

Total Sum = Average × Total Number of Terms. This is used to find missing values in a data set when the mean is known.

Assumed Mean Method

Useful for large numbers: Actual Mean = A + (∑d ÷ n), where A is the assumed mean and d is the deviation (Value − A).

Shortcut Trick

The best scheme for the shopkeeper is the one that offers the minimum discount to the customer.

Compare total discounts: A (39+8−3.12=43.88%), B (4/12 × 100=33.33%), C (16%), D (38+34−12.92=59.08%).

Comparing the values 43.88%, 33.33%, 16%, and 59.08%, the minimum discount is 16%.

Therefore, Scheme C is the most profitable for the shopkeeper.

∴ The correct answer is C.

Alternate Method

Formulas: Successive Discount = a + b − (a × b) ÷ 100 | Buy x get y free Discount% = [y ÷ (x + y)] × 100

⇒ Scheme A: 39 + 8 − (39 × 8)/100 = 47 − 3.12 = 43.88% ⇒ Scheme B: (4 ÷ 12) × 100 = 1/3 × 100 = 33.33% ⇒ Scheme C: Single discount = 16% ⇒ Scheme D: 38 + 34 − (38 × 34)/100 = 72 − 12.92 = 59.08% ⇒ For a shopkeeper, the scheme with the lowest discount percentage results in the highest Selling Price (SP) and thus the most profit.

∴ The correct answer is C.

Additional Information

Customer vs Shopkeeper Preference

A customer always looks for the Maximum Discount (Scheme D), while the shopkeeper looks for the Minimum Discount (Scheme C).

Buy x Get y Free Concept

In 'Buy x Get y Free', the total items given to the customer is (x + y). The discount is calculated on the total items: Discount % = [Free Items ÷ Total Items] × 100.

Shortcut Trick

Let the pipes be A, B, and C (outlet).

Combined efficiency (A + B − C) = Total Work ÷ Time = 192 ÷ 48 = 4.

Individual efficiencies: A = 192 ÷ 96 = 2; B = 192 ÷ 64 = 3.

Outlet efficiency (C) = (A + B) − Combined efficiency = (2 + 3) − 4 = 1.

Time taken by pipe C = Total Work ÷ Efficiency = 192 ÷ 1 = 192 min.

∴ The correct answer is 192 min.

Alternate Method

Given: Filling time of Pipe 1 = 96 min, Pipe 2 = 64 min, and Combined time (with outlet) = 48 min.

Calculations:

⇒ Efficiency of Pipe A = 192 ÷ 96 = 2 units/min

⇒ Efficiency of Pipe B = 192 ÷ 64 = 3 units/min

⇒ Combined Efficiency (A + B − C) = 192 ÷ 48 = 4 units/min

⇒ Efficiency of Outlet Pipe C = (Efficiency of A + Efficiency of B) − Combined Efficiency

⇒ Efficiency of Pipe C = (2 + 3) − 4 = 1 unit/min

⇒ Time taken by Pipe C to empty the cistern = Total Work ÷ Efficiency of C

⇒ Time = 192 ÷ 1 = 192 min

∴ The correct answer is 192 min.

Additional Information

Inlet and Outlet Pipes

An inlet pipe fills a tank (positive work), while an outlet pipe empties it (negative work).

Net Efficiency Formula

Net efficiency = ∑(Efficiencies of Inlet pipes) − ∑(Efficiencies of Outlet pipes).

Work and Time Relationship

Time taken to complete a task is inversely proportional to efficiency: Time = Work ÷ Efficiency.

Shortcut Trick

In a quadrilateral, the angle formed by the bisectors of two adjacent angles is equal to half the sum of the other two angles.

∠TOS = (∠M + ∠N) ÷ 2

∠TOS = (63° + 99°) ÷ 2

∠TOS = 162° ÷ 2 = 81°

∴ The correct answer is 81°.

Alternate Method

Given: ∠M = 63°, ∠N = 99°

Formula Used: Sum of interior angles of a quadrilateral = 360°; Sum of angles in a triangle = 180°.

In quadrilateral MNST:

⇒ ∠M + ∠N + ∠S + ∠T = 360°

⇒ 63° + 99° + ∠S + ∠T = 360°

⇒ 162° + ∠S + ∠T = 360°

⇒ ∠S + ∠T = 360° − 162° = 198°

Since TO and SO are angle bisectors of ∠T and ∠S respectively:

⇒ ∠OTS = ∠T ÷ 2 and ∠OST = ∠S ÷ 2

In triangle ΔTOS:

⇒ ∠TOS + ∠OTS + ∠OST = 180°

⇒ ∠TOS + (∠T ÷ 2) + (∠S ÷ 2) = 180°

⇒ ∠TOS + (∠T + ∠S) ÷ 2 = 180°

⇒ ∠TOS + 198° ÷ 2 = 180°

⇒ ∠TOS + 99° = 180°

⇒ ∠TOS = 180° − 99° = 81°

∴ The correct answer is 81°.

Additional Information

Angle Sum Property of Polygons

The sum of all interior angles of an n-sided polygon is (n − 2) × 180°.

Angle Between Opposite Bisectors

The angle between the bisectors of two opposite angles of a quadrilateral is equal to half the difference of the other two angles.

Shortcut Trick

LCM of 10, 15, and 3 is 30.

Divide 539 by 30 to find the remainder: 539 ÷ 30 ⇒ Remainder = 29.

Number to be added = Divisor − Remainder = 30 − 29 = 1.

∴ The correct answer is 1.

Alternate Method

Given: Number = 539, Divisors = 10, 15, 3.

Formula: Least number to be added = LCM(divisors) − (Number % LCM(divisors)).

⇒ LCM of 10, 15, and 3:

⇒ 10 = 2 × 5

⇒ 15 = 3 × 5

⇒ 3 = 3

⇒ LCM = 2 × 3 × 5 = 30.

⇒ Divide 539 by 30:

⇒ 539 = 30 × 17 + 29.

⇒ Remainder = 29.

⇒ Required number to be added = 30 − 29 = 1.

∴ The correct answer is 1.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers. For divisibility problems involving multiple numbers, always find the LCM first.

Divisibility Rule of 10

A number is divisible by 10 if its last digit is 0. Since 539 ends in 9, the next number ending in 0 is 540, requiring 1 to be added.

Least Number to Subtract

To find the least number to subtract to make a number perfectly divisible, the answer is simply the Remainder obtained after division.

Shortcut Trick

Area of quadrilateral ABCD = Area(ΔBCD) + Area(ΔABD)

In right-angled triangle BCD, Area = 0.5 × 42 × 40 = 840 m²

Hypotenuse BD = √(42² + 40²) = 58 m

In isosceles triangle ABD (sides 36, 36, 58), height h = √(36² − 29²) = √455 ≈ 21.33 m

Area(ΔABD) = 0.5 × 58 × 21.33 ≈ 618.6 m²

Total Area = 840 + 618.6 = 1458.6 m²

∴ The correct answer is 1458.6 m².

Alternate Method

Given: AB = 36 m, BC = 42 m, CD = 40 m, AD = 36 m, and ∠C = 90°.

Formula Used: Area of right triangle = 0.5 × base × height; Pythagoras theorem: a² + b² = c²; Heron's formula: Area = √[s(s−a)(s−b)(s−c)].

A B C D 36 m 42 m 40 m

Shortcut Trick

Remaining distance = 1/3 of 168 = 56 km

Remaining time = (1 − 2/6) of 18 = 4/6 × 18 = 12 hours

Required Speed = 56 ÷ 12 = 4.66...

∴ The correct answer is 4.7 km/hr.

Alternate Method

Given:

Total distance = 168 km

Total time = 18 hours

Calculations:

⇒ Distance covered = 2/3 × 168 = 112 km

⇒ Remaining distance = 168 − 112 = 56 km

⇒ Time taken = 2/6 × 18 = 6 hours

⇒ Remaining time = 18 − 6 = 12 hours

⇒ Required speed = Remaining distance ÷ Remaining time

⇒ Speed = 56 ÷ 12

⇒ Speed = 4.666... ≈ 4.7 km/hr

∴ The correct answer is 4.7 km/hr.

Additional Information

Average Speed Formula

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

Relative Speed

Same direction: S₁ − S₂ | Opposite direction: S₁ + S₂.

Unit Conversion

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

Shortcut Trick

Required percentage = (Marks of B / Marks of A) × 100

Required percentage = (50 / 75) × 100 = (2 / 3) × 100

2/3 of 100 is approximately 66.67%.

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by B = 50

Marks obtained by A = 75

Maximum marks = 120 (Note: This is redundant for the specific ratio requested)

Formula Used:

Percentage of X relative to Y = (X / Y) × 100

Calculations:

⇒ Required % = (Marks of B / Marks of A) × 100

⇒ Required % = (50 / 75) × 100

⇒ Required % = (2 / 3) × 100

⇒ Required % = 200 / 3

⇒ Required % = 66.666...

⇒ Rounding off to two decimal places, we get 66.67%.

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

Percentage change = (Difference / Original Value) × 100. Used to find how much a value increased or decreased.

Base Value Identification

The value following the word 'than' or 'of' in a comparison is usually taken as the Base Value (denominator).

Fraction to Percentage Conversion

Common fractions: 1/2 = 50%, 1/3 = 33.33%, 2/3 = 66.67%, 1/4 = 25%, 3/4 = 75%.

Shortcut Trick

Ratio of Volumes = n = 8

Ratio of Radii (R/r) = √³(n) = √³(8) = 2

New radius (r) = 14 ÷ 2 = 7 cm

Surface Area = 4 × (22/7) × 7 × 7 = 616 cm²

∴ The correct answer is 616 cm².

Alternate Method

Given:

Radius of large sphere (R) = 14 cm

Number of solid balls (n) = 8

Formula Used:

Volume of Sphere = (4/3) × π × r³

Surface Area of Sphere = 4 × π × r²

Calculations:

⇒ Total Volume of large sphere = 8 × Volume of each small ball

⇒ (4/3) × π × (14)³ = 8 × (4/3) × π × r³

⇒ 14 × 14 × 14 = 8 × r³

⇒ 2744 = 8 × r³

⇒ r³ = 2744 ÷ 8

⇒ r³ = 343

⇒ r = √³(343) = 7 cm

⇒ Surface area of each ball = 4 × π × r²

⇒ Surface area = 4 × (22/7) × 7 × 7

⇒ Surface area = 4 × 22 × 7

⇒ Surface area = 88 × 7 = 616 cm²

∴ The correct answer is 616 cm².

Additional Information

Curved Surface Area of Hemisphere

The CSA of a hemisphere is exactly half of the sphere's surface area, given by 2 × π × r².

Total Surface Area of Hemisphere

The TSA of a solid hemisphere includes the base area, resulting in 3 × π × r².

Volume of Hemisphere

The Volume of a hemisphere is (2/3) × π × r³, which is half the volume of a full sphere.

Shortcut Trick

The Algebraic Identity used is: (Sum)² = (Sum of Squares) + 2 × (Sum of Products).

Substituting the given values: 15² = 83 + 2(ab + bc + ca)

225 = 83 + 2(ab + bc + ca)

2(ab + bc + ca) = 225 − 83 = 142

ab + bc + ca = 142 ÷ 2 = 71

∴ The correct answer is 71.

Alternate Method

Given:

a + b + c = 15

a² + b² + c² = 83

Formula Used:

The standard expansion of the square of a trinomial:

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Calculations:

⇒ (15)² = 83 + 2(ab + bc + ca)

⇒ 225 = 83 + 2(ab + bc + ca)

⇒ 225 − 83 = 2(ab + bc + ca)

⇒ 142 = 2(ab + bc + ca)

⇒ ab + bc + ca = 142 ÷ 2

⇒ ab + bc + ca = 71

∴ The correct answer is 71.

Additional Information

Trinomial Square Identity

The formula (a+b+c)² = a²+b²+c²+2(ab+bc+ca) relates the sum of three numbers, the sum of their squares, and the sum of their products taken in pairs.

Cubic Identity

A related identity is a³+b³+c³ − 3abc = (a+b+c)(a²+b²+c² − ab − bc − ca), which is frequently used in algebraic simplifications.

Value of ab+bc+ca

If a+b+c = 0, then a²+b²+c² = −2(ab+bc+ca), implying the sum of squares is twice the negative sum of the cyclic products.

Shortcut Trick

Profit = 15% ⇒ Fraction = 3/20.

Let Cost Price (CP) = 20 units and Profit = 3 units.

Then, Selling Price (SP) = 20 + 3 = 23 units.

Given: 23 units = ₹3,450 ⇒ 1 unit = 3,450 ÷ 23 = ₹150.

CP = 20 units = 20 × 150 = ₹3,000.

∴ The correct answer is ₹3,000.

Alternate Method

Given: Selling Price (SP) = ₹3,450, Profit Percentage = 15%. Formulas: Profit = SP − CP | SP = CP × (100 + P%) ÷ 100 | CP = (SP × 100) ÷ (100 + P%).

⇒ CP = (3,450 × 100) ÷ (100 + 15) ⇒ CP = (3,450 × 100) ÷ 115 ⇒ CP = 30 × 100 ⇒ CP = ₹3,000.

∴ The correct answer is ₹3,000.

Additional Information

Cost Price & Selling Price

Cost Price (CP) is the price at which an article is purchased. Selling Price (SP) is the price at which it is sold.

Profit and Loss Percentage

Profit % = (Profit ÷ CP) × 100. Loss % = (Loss ÷ CP) × 100. Both are always calculated on the Cost Price unless stated otherwise.

Given Series: RUS 36 PSQ 40 NQO 44 LOM 48 ?

The pattern followed here is:

∴ According to the pattern followed here JMK 52 will replace the (?) mark.

Hence, the correct answer is "Option 3".

Logic followed here is:

Each letter in the given cluster is shifted backward by 3 positions in the English alphabetical series to obtain the corresponding letter of the related cluster.

For

JFBC is related to GCYZ →

WSOP is related to TPLM →

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

Similarly, let's find the related cluster for IEAB:

Therefore, the related cluster for 'IEAB' is FBXY.

Hence, the correct answer is "Option 1".

Given:

• Only two people sit between G and S when counted from the right of G.
• Only one person sits between S and I when counted from the right of I.
• Only three people sit between G and R.
• H sits to the immediate left of Q.

Find: Who sits third to the left of R?

Starting from R and moving left:

R → I → G → Q

Therefore, the person sitting third to the left of R is Q.

Hence, the correct answer is Option 1 — Q.

Given:

Z ranked 21^st from the top and 39^th from the bottom.

Formula:

Total students = Rank from top + Rank from bottom − 1

Calculation:

= 21 + 39 − 1

= 60 − 1

= 59

Therefore, the total number of students in the class is 59.

Hence, the correct answer is Option 4 — 59.

Given Series:

(Left) # € % # 6 & 4 * 9 6 @ 5 % @ # © 8 # % * £ 2 (Right)

Required pattern: Symbol → Number → Symbol

Checking the series:

(Left) # € % **# 6 & 4 *** 9 6 @ 5 % @ # © 8 # % * £ 2 (Right)

Numbers satisfying the condition:

→ 6 in (# 6 &)

→ 4 in (& 4 *)

→ 5 in (@ 5 %)

→ 8 in (© 8 #)

Therefore, there are 4 such numbers.

Hence, the correct answer is Option 1 — 4.

Logic followed here is:

Each letter-cluster follows a pattern based on the addition and subtraction of positional values in alphabetical order.

The pattern is: 1st letter + 3 = 2nd letter; 2nd letter - 7 = 3rd letter.

For (1) LOH →

<br>

 

For (2) VYP →

For (3) CFY →

 

For (4) ZCV →

  

Therefore, the letter-cluster 'VYP' is the odd one out.

Hence, the correct answer is "Option 2".

Given Series:

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

Required pattern: Odd → Odd → Odd

Checking the series:

3 1 3 2 8 2 1 2 4 6 1 3 1 8 8 6 1 1 9 1 9 5 8 4 1 8 9 6 9

Odd numbers satisfying the condition:

→ 1 in (3 1 3)

→ 3 in (1 3 1)

→ 1 in (1 1 9)

→ 9 in (1 9 5)

Therefore, there are 4 such odd numbers.

Hence, the correct answer is Option 3 — 4.

Logic followed here is:

The letters in the first word are shifted by +5 and +4 alternatively to get the corresponding letters in the second word based on their positional values in the English alphabetical order.

For

ABHL is related to FFMP →

URBB is related to ZVGF →

 

The pattern observed is: (Letter Positional Value) + 5, + 4, + 5, + 4.

Similarly, let's find the related term for JDQN:

Therefore, 'JDQN' is related to 'OHVR'.

Hence, the correct answer is "Option 2".

Given:

• Only two people sit between A and B when counted from the left of A.
• G sits to the immediate left of A.
• Only three people sit between E and G when counted from the left of E.
• C sits to the immediate right of D.
• F is not an immediate neighbour of G.

Starting from H and moving right:

H → G → A

So, the person sitting second to the right of H is A.

Hence, the correct answer is Option 3 — A.

Logic followed here is:

The difference between the positional values of consecutive letters in each cluster follows a specific pattern (e.g., +6, +4, +4), except for the odd cluster.

1. SYDG:

2. UAEI:

3. HNRV:

  

4. XDHL:

 

 

Therefore, the letter-cluster 'SYDG' is the odd one out as it does not follow the +6, +4, +4 pattern.

Hence, the correct answer is "Option 1".

Logic followed here is.

Logic: The series is formed by adding consecutive odd numbers starting from 1 to each preceding term to obtain the next term.

The sequence of differences is 1, 3, 5, 7, which are consecutive odd numbers.

Therefore, the number that should come in place of the question mark is 148.

Hence, the correct answer is "Option 3".

Given:

Manish starts from Point Y.

Movement details:

37 km North

Left turn → 75 km West

Left turn → 69 km South

Left turn → 52 km East

Left turn → 32 km North

If Point Z is 23 km West of Point Y, then to reach Point Y again he must travel:

23 km towards East

Hence, the correct answer is "Option 3".