RRB Group D Official Paper (Held On: 27 Nov, 2025 Shift 3) (Previous Year Paper)

The correct answer is mgh.

Key Points

• Potential Energy (PE): Potential energy is the energy possessed by a body due to its position or configuration. For an object at height h above the Earth's surface, the potential energy is calculated as mgh, where:
• m: Mass of the object (in kilograms).
• g: Acceleration due to gravity (approximately 9.8 m/s² on Earth).
• h: Height above the Earth's surface (in meters).
• The formula PE = mgh is derived from the work done to raise the object against the force of gravity. Hence, this expression is universally accepted for calculating the gravitational potential energy.
• The other options are incorrect:
• mh/g: This formula incorrectly divides mass times height by gravity, which does not represent potential energy.
• m²gh: Squaring the mass is unnecessary and does not fit the physical definition of potential energy.
• hg/m: Dividing height times gravity by mass is mathematically and conceptually incorrect.
• Gravitational potential energy is a scalar quantity, meaning it has magnitude but no direction.
• Hence, the correct answer is mgh.

Additional Information

• Gravitational Potential Energy:
• It depends on the position of the object relative to a reference point, often taken as the Earth's surface.
• The formula PE = mgh applies near the Earth's surface where the gravitational field is uniform.
• For larger distances from Earth or in non-uniform gravitational fields, the formula becomes more complex and involves gravitational potential due to mass and distance.
• Examples of Potential Energy:
• An object placed on a shelf has potential energy due to its height above the ground.
• A stretched spring has elastic potential energy due to its deformation.
• Applications of Gravitational Potential Energy:
• Hydropower plants: Water stored at a height has potential energy, which is converted to kinetic energy and then to electrical energy.
• Roller coasters: Potential energy at the highest point is converted to kinetic energy during descent.
• Acceleration due to Gravity (g):
• The standard value of g on Earth's surface is 9.8 m/s².
• The value of g decreases with altitude and varies slightly depending on location (e.g., poles vs. equator).
• Important Points:
• Potential energy is always relative to a reference point. If the reference point is changed, the value of potential energy changes accordingly.
• In physics problems, the reference point is often assumed to be the ground or the Earth's surface unless specified otherwise.

The correct answer is m = v / u.

Key Points

• Magnification (m) produced by a lens is defined as the ratio of the height of the image to the height of the object.
• The formula for magnification is given as m = v / u, where:
• v is the image distance (distance of the image from the lens).
• u is the object distance (distance of the object from the lens).
• This relationship is derived from the lens formula: 1/f = 1/v - 1/u, where f is the focal length of the lens.
• The sign conventions for distances should be followed based on the Cartesian coordinate system. For instance:
• Positive values are used for distances measured in the direction of light propagation.
• Negative values are used for distances measured opposite to the direction of light propagation.
• As per the sign convention, the nature of the magnification determines whether the image is erect (positive m) or inverted (negative m).

Additional Information

• Lens Formula:
• The lens formula relates the focal length (f) of a lens to the object distance (u) and the image distance (v).
• The formula is expressed as: 1/f = 1/v - 1/u.
• This formula applies to both convex and concave lenses but follows sign conventions.
• Types of Magnification:
• Positive Magnification: Indicates an erect and virtual image (e.g., in magnifying glasses).
• Negative Magnification: Indicates an inverted and real image (e.g., in projection lenses).
• Sign Conventions:
• Distances are measured from the optical center of the lens.
• Object distance (u) is taken negative as it is measured against the direction of light.
• Image distance (v) is positive for real images and negative for virtual images.
• Applications:
• Microscopes: Use lenses to magnify small objects.
• Telescopes: Use lenses to magnify distant objects.
• Magnifying Glasses: Used for enlarging objects visually.
• Important Note: The formula m = v / u is derived for thin lenses and assumes the paraxial approximation where light rays make small angles with the principal axis.

The correct answer is No fixed shape and no fixed volume.

Key Points

• Characteristic of Gases: Gases have no fixed shape and no fixed volume. They take the shape and volume of their container, indicating their high level of fluidity and compressibility.
• Molecular Arrangement: Gas molecules are loosely packed and have a large amount of empty space between them, which allows them to move freely in all directions.
• Compressibility: Gases are highly compressible because their molecules are far apart and can be brought closer by applying pressure. This is not seen in solids and is limited in liquids.
• Fluid Nature: Gases are considered fluids because their molecules can flow freely. This property allows them to fill the entire volume of the container they are in.
• Diffusion: Gases exhibit high rates of diffusion, meaning they can mix evenly with other gases due to the constant, random motion of their particles.

Additional Information

• States of Matter: Matter exists in three primary states: solids, liquids, and gases. Gases are the most energetic state of matter, with particles that are widely spaced and move independently.
• Boyle's Law: This law states that the pressure of a gas is inversely proportional to its volume at constant temperature. It demonstrates the compressibility of gases.
• Charles's Law: This law describes how gases expand when heated. At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.
• Real-Life Examples: Examples of gases include oxygen (O₂), carbon dioxide (CO₂), and hydrogen (H₂). These gases are used in various industries, from respiration to manufacturing and energy production.
• Applications of Gases: Gases play a critical role in several applications, such as oxygen cylinders in hospitals, compressed natural gas (CNG) in vehicles, and carbon dioxide in the food and beverage industry for carbonation.

Important Points

• Incorrect Statements:
• No fixed shape but fixed volume: This is a property of liquids, not gases. Hence, this statement is incorrect.
• Fixed shape and volume: This is a property of solids, not gases. Hence, this statement is incorrect.
• Compressible but not fluid: While gases are compressible, they are also fluid, unlike solids. Hence, this statement is incorrect.
• Key Difference: The key difference between gases and other states of matter lies in their molecular arrangement and behavior under different conditions of pressure and temperature.

The correct answer is the recoil of the gun acts backward as reaction.

Key Points

• Newton's Third Law of Motion: When the gun is fired, it exerts a forward force on the bullet. In reaction, the bullet exerts an equal and opposite force on the gun. This is explained by Newton’s Third Law of Motion, which states, “For every action, there is an equal and opposite reaction.”
• Recoil of the gun: The backward motion of the gun after firing a bullet is called recoil. The gun moves backward as a reaction to the forward motion of the bullet.
• Conservation of Momentum: The firing process also illustrates the principle of conservation of momentum. Before firing, the total momentum of the system (gun and bullet) is zero. After firing, the backward momentum of the gun equals the forward momentum of the bullet, maintaining the total momentum of the system at zero.
• Air resistance and gravity: The forward motion of the bullet is not caused by air resistance or gravity. Air resistance acts as a retarding force, while gravity pulls the bullet downward. These forces do not contribute to the bullet's initial forward motion. Hence, statements 1 and 4 are incorrect.
• Force exerted by the barrel: The barrel guides the bullet's path but does not actively exert a forward force. The force comes from the combustion of gunpowder inside the chamber. Hence, statement 3 is incorrect.

Additional Information

• Newton's Laws of Motion:
• First Law: An object remains at rest or in uniform motion in a straight line unless acted upon by an external force.
• Second Law: The force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
• Third Law: For every action, there is an equal and opposite reaction.
• Conservation of Momentum:
• The total momentum of an isolated system remains constant if no external forces are acting on it.
• For example, in the case of firing a gun, the backward momentum of the gun is equal and opposite to the forward momentum of the bullet.
• This principle is fundamental in various fields, including rocketry and ballistics.
• Recoil Velocity:
• The backward velocity of the gun is called recoil velocity.
• It can be calculated using the formula derived from the conservation of momentum: m₁v₁ = -m₂v₂, where m₁ and v₁ are the mass and velocity of the bullet, and m₂ and v₂ are the mass and velocity of the gun.
• Applications of Conservation of Momentum:
• Rocket Propulsion: Rockets work on the same principle. The expulsion of gases in one direction results in the rocket moving in the opposite direction.
• Ballistics: The study of bullets and their motion also relies on the conservation of momentum.
• Water Sprinklers: Water exiting the nozzle of a sprinkler causes it to spin due to the reactionary force.
• Practical Implications:
• Recoil can be significant in larger firearms, and mechanisms like recoil pads or hydraulic systems are used to absorb the shock.
• In small firearms, the recoil is usually manageable but can still affect the shooter’s accuracy.

Important Points

• Gunpowder Combustion: The forward motion of the bullet is due to the rapid expansion of gases from the combustion of gunpowder. This generates a high-pressure force that propels the bullet.
• Air Resistance: While the bullet moves forward, air resistance acts as a drag force, slowing it down over time. It does not contribute to the bullet's initial motion.
• Gravity's Role: Gravity influences the trajectory of the bullet, causing it to follow a parabolic path. However, it does not provide any forward force.
• Ballistics: The study of the motion of projectiles like bullets is called ballistics. It involves understanding forces like gravity, air resistance, and the initial propulsion.
• Safety Considerations: Recoil can impact the shooter, especially in high-caliber firearms. Proper training and equipment are essential to manage the recoil effectively.

The correct answer is HIV can be transmitted through sexual acts.

Key Points

• Statement 1: HIV cannot be transmitted through aerosols such as sneezing, coughing, or air particles. It requires direct contact with certain body fluids like blood, semen, vaginal fluids, or breast milk. Hence, statement 1 is incorrect.
• Statement 2: HIV cannot be transmitted through mosquito bites. Mosquitoes do not inject blood from previous hosts into the next person they bite, making this mode of transmission biologically impossible. Hence, statement 2 is incorrect.
• Statement 3: HIV cannot be transmitted through casual contact like shaking hands, hugging, or sharing food. The virus does not survive well outside the human body and cannot be spread through touch. Hence, statement 3 is incorrect.
• Statement 4: HIV can be transmitted through sexual acts, specifically unprotected sex (vaginal, anal, or oral) with an infected person. This is a primary mode of transmission due to the exchange of bodily fluids like semen or vaginal fluids during intercourse. Hence, statement 4 is correct.
• Statement 5: No additional statement provided; the correct answer aligns with statement 4.

Additional Information

• HIV Transmission Modes:
• HIV is primarily transmitted through sexual contact, sharing of needles, contaminated blood transfusions, and from mother to child during childbirth or breastfeeding.
• It is not spread through saliva, sweat, tears, or casual contact like hugging, using the same toilet, or sharing utensils.
• Preventive Measures:
• Using condoms during sexual intercourse significantly reduces the risk of transmission.
• Individuals can opt for Pre-Exposure Prophylaxis (PrEP), a medication that helps prevent HIV infection.
• Avoid sharing needles or syringes and ensure blood transfusions are from certified and tested sources.
• Global Statistics:
• The UNAIDS Report 2022 states that approximately 38.4 million people globally were living with HIV.
• Most new infections occur in regions like Sub-Saharan Africa, where awareness and prevention measures need strengthening.
• Mother-to-Child Transmission:
• Without treatment, the transmission rate from mother to child can be as high as 25%-30%.
• With proper antiretroviral therapy during pregnancy and breastfeeding, transmission rates can be reduced to below 5%.

Important Points

• HIV Testing: Regular testing is essential for early diagnosis and treatment. The ELISA test and Western Blot are commonly used diagnostic methods.
• Antiretroviral Therapy (ART): ART helps manage HIV infection, allowing individuals to live longer and healthier lives by reducing the viral load to undetectable levels.
• Social Awareness: Educating communities about HIV transmission and prevention helps combat stigma and discrimination against affected individuals.
• Government Initiatives: In India, programs like the National AIDS Control Programme (NACP) aim to reduce HIV transmission rates and provide care for affected individuals.

The correct answer is CaCO₃.

Key Points

• When carbon dioxide (CO₂) is passed through lime water (Ca(OH)₂), calcium carbonate (CaCO₃) is formed as a precipitate. This is a well-known chemical reaction used to test for the presence of CO₂.
• The reaction occurs as follows:
• CO₂ + Ca(OH)₂ → CaCO₃ (precipitate) + H₂O.
• Calcium carbonate (CaCO₃) is insoluble in water, which is why it precipitates out of the solution as a white solid.
• If excess CO₂ is bubbled through the solution, the precipitate dissolves, forming calcium bicarbonate Ca(HCO₃)₂, which is soluble in water:
• CaCO₃ + CO₂ + H₂O → Ca(HCO₃)₂.
• This reaction is widely used in laboratories and is an important concept in the study of carbon dioxide's interaction with lime water.
• Key observation: The formation of a white precipitate upon passing CO₂ through lime water confirms the presence of CO₂.

Additional Information

• Calcium Hydroxide (Ca(OH)₂):
• Also known as lime water, it is a strong base used in various chemical reactions.
• It is slightly soluble in water, forming a clear solution that reacts with carbon dioxide to form calcium carbonate.
• Calcium Carbonate (CaCO₃):
• It is a white insoluble solid that is a common substance found in rocks as minerals like calcite and aragonite.
• It is widely used in construction (as limestone and marble), in the production of cement, and as a dietary calcium supplement.
• Calcium Bicarbonate (Ca(HCO₃)₂):
• This compound is soluble in water and is formed when excess carbon dioxide reacts with calcium carbonate in the presence of water.
• It is a key component in the natural carbon cycle and is often found in natural water systems, especially in hard water.
• Application of the Reaction:
• The reaction between carbon dioxide and lime water is a common laboratory test to detect the presence of CO₂.
• This reaction also occurs in nature, contributing to processes like the formation of stalactites and stalagmites in caves.
• 

Important Points

• The solubility of calcium carbonate decreases with an increase in temperature, which is why heating lime water causes the precipitate to settle faster.
• This reaction is a key process in the carbon cycle, playing a role in the sequestration of CO₂ in the form of carbonate minerals.
• The lime water test is a simple and effective way to confirm the presence of carbon dioxide in exhaled air or combustion products.

The correct answer is Gravitational acceleration on the moon is smaller..

Key Points

• Gravitational Acceleration on the Moon: The moon's gravitational acceleration is approximately 1/6^th that of Earth. This is because the moon has a much smaller mass and radius compared to Earth. Hence, the force of gravity acting on a body on the moon is significantly weaker. This explains why the weight of a body on the moon is less than on Earth.
• Weight vs. Mass: Weight is the force exerted by gravity on an object and is calculated as Weight = Mass × Gravitational Acceleration (g). Since the moon's gravitational acceleration is lower, the weight of the same object will also be smaller on the moon. However, the mass of the object remains constant, irrespective of its location.
• Atmosphere on the Moon: The moon has no significant atmosphere. However, the lack of an atmosphere does not directly affect the weight of an object. Weight is solely dependent on the gravitational force acting on it.
• Mass of the Moon vs. Earth: The moon's mass is approximately 7.35 × 10²² kg, which is much smaller than Earth's mass (5.97 × 10²⁴ kg). This reduced mass is a key reason for the weaker gravitational pull on the moon.
• Statement Analysis:
• Statement 1: "Gravitational acceleration on the moon is smaller." – This is correct. It explains the reduced weight of a body on the moon.
• Statement 2: "Mass of the body decreases on the moon." – This is incorrect. Mass is an intrinsic property and remains unchanged regardless of location.
• Statement 3: "Mass of the moon is equal to Earth's mass." – This is incorrect. The moon's mass is significantly smaller than Earth's mass.
• Statement 4: "The moon has no atmosphere." – While this is true, it does not affect the weight of a body directly. Therefore, it is irrelevant to the question.

Additional Information

• Gravitational Acceleration: Gravitational acceleration is the rate at which an object accelerates due to gravity. On Earth, it is approximately 9.8 m/s², while on the moon, it is only about 1.62 m/s².
• Weight on Different Planets: The weight of an object varies depending on the planet or celestial body it is on, as gravitational acceleration differs across celestial bodies. For example, the weight of an object on Jupiter would be much greater than on Earth because Jupiter has a stronger gravitational pull.
• Mass vs. Weight: Mass is a scalar quantity that represents the amount of matter in an object and is measured in kilograms (kg). Weight, on the other hand, is a vector quantity that depends on gravity and is measured in newtons (N).
• Lunar Surface: The moon's surface has a weak gravitational pull because of its smaller size and mass. This weak gravity is why astronauts can jump higher and carry heavier loads more easily on the moon than on Earth.
• Moon's Lack of Atmosphere: The absence of an atmosphere on the moon contributes to phenomena such as extreme temperature variations and the absence of weather or sound. However, it does not influence the weight of an object directly.

Important Points

• Formula for Weight: The weight of an object is calculated using the formula W = m × g, where W is weight, m is mass, and g is the gravitational acceleration.
• Moon's Gravity: The moon's gravitational acceleration is only about 16.5% of Earth's gravity. This means an object that weighs 60 kg on Earth will weigh only 10 kg on the moon.
• Scientific Exploration: The concept of gravity was first explained by Sir Isaac Newton in the 17^th century through his Law of Universal Gravitation. This law forms the foundation for understanding weight differences across celestial bodies.

The correct answer is Energy = Power × Time.

Key Points

• The relationship between electric energy, power, and time is given by the equation: Energy (E) = Power (P) × Time (T). This is the correct and universally accepted formula.
• Power is defined as the rate at which energy is transferred or converted. It is measured in watts (W), where 1 watt = 1 joule per second (J/s).
• Energy is the total amount of work done or the total power consumed over a period of time. It is measured in joules (J) in the International System of Units (SI).
• Time refers to the duration for which the power is being consumed or transferred. It is measured in seconds (s).
• For practical applications, energy is often measured in units like kilowatt-hours (kWh) when dealing with electrical energy consumption. For example, energy consumed by household appliances is calculated in kWh.

Additional Information

• Explanation of Units:
• Power (P): Measured in watts (W), it represents the rate of energy consumption or production.
• Energy (E): Measured in joules (J), it represents the total energy consumed or produced. For electricity billing, it is measured in kilowatt-hours (kWh), where 1 kWh = 3.6 × 10⁶ J.
• Time (T): Measured in seconds (s) or hours (h), it refers to the duration of power usage.
• Practical Applications:
• The formula E = P × T is used in calculating the energy consumption of electrical devices. For example, a 100 W bulb running for 10 hours consumes 1 kWh of energy (100 W × 10 h = 1000 Wh = 1 kWh).
• It is also used in industries to determine the energy efficiency of machines and devices.
• Common Misconceptions:
• The incorrect statement "Energy = Power + Time" violates the dimensional consistency of the formula, as power and time are measured in different units and cannot be added directly.
• Similarly, "Energy = Power / Time" is incorrect because dividing power by time does not yield energy.
• "Energy = Time - Power" is also incorrect, as subtraction of time and power is not mathematically or physically meaningful.
• Additional Insights:
• In mechanical systems, a similar relationship exists: Work = Force × Distance. This highlights the universality of the concept of energy transfer.
• Modern electricity meters calculate energy consumption in terms of kWh, which is a direct application of the formula E = P × T.

Important Points

• SI Units:
• Power: 1 watt (W) = 1 joule/second (J/s).
• Energy: 1 joule (J) = 1 watt-second (W·s).
• Time: Measured in seconds (s) for scientific calculations and hours (h) for practical purposes.
• Energy Billing: Energy consumed by households is typically billed in kilowatt-hours (kWh), a practical unit of electrical energy.
• Examples of Power and Energy:
• A 60 W lightbulb used for 5 hours consumes 0.3 kWh of energy.
• An electric heater rated at 2000 W running for 2 hours consumes 4 kWh of energy.

The correct answer is Lemon juice.

Key Points

• Lemon juice has the highest acidity among the given options due to its low pH value, which typically ranges between 2 to 3. This indicates a higher concentration of hydrogen ions compared to other substances.
• Acidity is measured by the pH scale, where lower values (below 7) denote higher acidity. Substances like lemon juice are highly acidic due to the presence of citric acid.
• Vinegar, while acidic, has a pH value ranging between 2.4 to 3.4, which is slightly higher than that of lemon juice. Its acidity is primarily due to the presence of acetic acid.
• Milk is slightly acidic with a pH value of approximately 6.5 to 6.7. Its acidity is much lower compared to lemon juice and vinegar.
• Water is neutral in nature with a pH value of approximately 7, indicating no acidity under normal conditions.

Additional Information

• Lemon Juice:
• Contains citric acid, which is responsible for its high acidity.
• Widely used in culinary applications, cleaning, and preservation due to its acidic nature.
• Rich in Vitamin C, making it beneficial for health and immunity.
• Lemon juice's acidity can be harmful to tooth enamel if consumed excessively.
• Vinegar:
• Contains acetic acid, which gives it its acidic properties.
• Used in cooking, cleaning, and as a preservative.
• Different types of vinegar, such as white vinegar, apple cider vinegar, and balsamic vinegar, vary in acidity levels.
• Milk:
• Slightly acidic due to the presence of lactic acid and other compounds.
• Rich in calcium and other nutrients, beneficial for bone health.
• Its pH can vary depending on freshness and storage conditions.
• Water:
• Considered neutral with a pH value of approximately 7.
• Pure water is not acidic or basic, but impurities can alter its pH.
• Essential for maintaining bodily functions and overall health.

Important Points

• pH Scale:
• The pH scale ranges from 0 to 14, where 0-6 indicates acidity, 7 is neutral, and 8-14 indicates alkalinity.
• Substances with lower pH values are more acidic, while higher pH values indicate basicity.
• Citric Acid:
• Found naturally in citrus fruits like lemons, limes, and oranges.
• Used in food and beverages as a flavoring agent and preservative.
• Acetic Acid:
• Key component of vinegar, contributing to its acidic properties.
• Commonly used in cleaning solutions and food preservation.

The correct answer is 75 J.

Key Points

• Work done in moving a charge is calculated using the formula: W = Q × V, where:
• W is the work done (in joules).
• Q is the charge (in coulombs).
• V is the potential difference (in volts).
• In the given question, the charge (Q) is 5 C, and the potential difference (V) is 15 V.
• Using the formula, W = 5 × 15 = 75 J.
• This means the work done in moving a charge of 5 C across a potential difference of 15 V is 75 joules.
• Option 3 is correct.

Additional Information

• Electric Potential Difference:
• The potential difference is the amount of work required to move a charge from one point to another in an electric field.
• It is measured in volts (V), where 1 volt equals 1 joule per coulomb.
• It plays a crucial role in understanding electric circuits and energy transfer.
• Charge (Q):
• Charge is a fundamental property of matter that causes it to experience a force in an electric field.
• It is measured in coulombs (C), where 1 coulomb equals the charge of approximately 6.24 × 10¹⁸ electrons.
• Work Done in Electricity:
• Work is the transfer of energy, and in the context of electricity, it happens when a charge moves under the influence of a potential difference.
• The relationship between work, charge, and voltage ensures the efficient functioning of various electrical systems.
• Applications:
• The concept of work done by charges is widely used in electric circuits, batteries, capacitors, and power generation systems.
• It is also essential in understanding energy consumption and transfer in electrical devices.

Important Points

• SI Units:
• The SI unit of work is joule (J), which is equivalent to one newton-meter.
• The SI unit of charge is coulomb (C).
• The SI unit of potential difference is volt (V), where 1 volt = 1 joule per coulomb.
• Key Formulas:
• Work done (W): W = Q × V.
• Energy: Energy = Work done, which is expressed in joules.
• Power: Power = Work done / Time, measured in watts (W).
• Practical Example:
• In a circuit, moving a charge of 2 C across a potential difference of 10 V would require 20 J of work.
• Such calculations are essential for designing efficient electrical systems and determining energy consumption.

The correct answer is Mitochondria.

Key Points

• Mitochondria are often referred to as the "powerhouses of the cell" because they are the primary site for ATP generation through oxidative phosphorylation.
• The inner membrane of mitochondria is highly folded into structures known as cristae, significantly increasing the surface area available for ATP-generating chemical reactions.
• ATP generation occurs in the electron transport chain, located within the inner mitochondrial membrane, which is essential for cellular energy production.
• Mitochondria have their own DNA and ribosomes, which allow them to produce some proteins necessary for their function, making them semi-autonomous organelles.
• Other cell organelles like lysosomes, vacuoles, and the cell wall do not participate in ATP generation as they serve different functions such as digestion, storage, and structural support, respectively.

Additional Information

• Mitochondria:
• Mitochondria are double-membrane-bound organelles found in nearly all eukaryotic cells.
• The outer membrane acts as a protective barrier, while the inner membrane contains the proteins and enzymes necessary for energy production.
• ATP synthesis involves processes like the Krebs cycle (occurring in the mitochondrial matrix) and oxidative phosphorylation (occurring in the inner membrane).
• They play roles in other metabolic tasks, including calcium homeostasis and apoptosis.
• Functions of Lysosomes:
• Lysosomes are membrane-bound organelles containing digestive enzymes. They break down waste materials, cellular debris, and foreign substances like pathogens.
• They are involved in autophagy, a process for recycling cellular components.
• Unlike mitochondria, lysosomes do not generate ATP.
• Functions of Vacuoles:
• Vacuoles are storage organelles found in plant and animal cells. They store nutrients, waste products, and water.
• Plant vacuoles also maintain turgor pressure, which is essential for structural integrity and growth.
• Vacuoles do not participate in energy production.
• Cell Wall:
• The cell wall is a rigid structure found in plant cells, fungi, and certain prokaryotes. It provides protection and structural support.
• It is composed of cellulose in plants, chitin in fungi, and peptidoglycan in bacteria.
• The cell wall does not facilitate ATP synthesis or any energy-related functions.

Important Points

• ATP Production: ATP, or adenosine triphosphate, is the primary energy currency of the cell, and mitochondria are the main site of its production.
• Electron Transport Chain: This is a series of complexes located on the inner mitochondrial membrane, where electrons are transferred to generate a proton gradient essential for ATP synthesis.
• Cristae: The folded structure of the inner membrane increases surface area, allowing more space for enzymes involved in ATP production.
• Semi-autonomy: Mitochondria have their own circular DNA, resembling bacterial DNA, which supports the theory that mitochondria evolved from ancient prokaryotic cells.

The correct answer is Gas in gas.

Key Points

• Diffusion refers to the process where particles move from a region of higher concentration to a region of lower concentration until equilibrium is reached.
• In the case of gases, the particles are widely spaced, have high kinetic energy, and move at high speeds. This enables them to diffuse faster compared to solids or liquids.
• Among the given options, diffusion of gas in gas occurs the fastest because gases have negligible intermolecular forces and maximum freedom of movement.
• Comparatively, solids have tightly packed particles with strong intermolecular forces, making diffusion extremely slow. Liquids are denser than gases, so diffusion in liquids is slower than in gases but faster than in solids.
• The molecular velocity of gases is governed by the Kinetic Molecular Theory, which states that lighter gas molecules (low molecular weight) diffuse more rapidly than heavier ones. For example, hydrogen diffuses faster than oxygen.

Additional Information

• Diffusion in Solids:
• Diffusion in solids is extremely slow because particles are closely packed and vibrate in fixed positions.
• This type of diffusion is usually observed in processes like alloy formation or interstitial diffusion in metals.
• Example: Diffusion of carbon atoms in iron to form steel.
• Diffusion in Liquids:
• Particles in liquids have more freedom of movement compared to solids but less than gases.
• Diffusion in liquids is slower than in gases due to higher density and intermolecular forces.
• Example: Diffusion of sugar in water.
• Diffusion in Gases:
• Gases exhibit the fastest diffusion due to high kinetic energy and negligible intermolecular forces.
• Example: The spread of perfume in a room.
• The rate of diffusion in gases can be mathematically represented by Graham’s Law of Diffusion, which states that the rate of diffusion is inversely proportional to the square root of the molar mass of the gas.
• Factors Affecting Diffusion Rate:
• Temperature: Higher temperatures increase the kinetic energy of particles, accelerating diffusion.
• Medium: Diffusion is faster in less dense mediums (e.g., gases) and slower in denser mediums (e.g., solids).
• Particle Size: Smaller particles diffuse faster due to lower mass.

Important Points

• Real-life Applications of Diffusion:
• Respiration: Oxygen diffuses into the blood from the alveoli in the lungs, while carbon dioxide diffuses out.
• Cooking: Aroma diffusion from spices or foods during cooking is an example of diffusion of gas in gas.
• Industrial Processes: Diffusion is crucial in processes like gas purification and separation techniques.
• Graham’s Law of Diffusion:
• This law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass.
• Mathematically, Rate₁/Rate₂ = √(M₂/M₁), where M₁ and M₂ are the molar masses of gases.
• Example: Hydrogen diffuses approximately 4 times faster than oxygen due to its lower molar mass.

The correct answer is Utilises nutrients efficiently and prevents spread of diseases.

Key Points

• Efficient Nutrient Utilisation: Intercropping allows multiple crops to grow together, enabling better utilisation of soil nutrients. Different crops have varying nutrient demands, reducing competition and enhancing soil fertility.
• Reduces Spread of Diseases: Growing different crops together can prevent the spread of diseases and pests. If one crop is affected, the others remain relatively safe, reducing overall crop loss.
• Improved Soil Health: Certain intercropping systems, such as planting legumes with cereals, can enhance nitrogen fixation in the soil, improving its health and reducing the need for chemical fertilisers.
• Higher Yield: Intercropping often results in greater productivity per unit area compared to monocropping, as crops complement each other in terms of growth and resource use.
• Enhanced Biodiversity: Intercropping promotes biodiversity, which can improve ecological balance, attract beneficial insects, and reduce reliance on chemical pesticides.

Additional Information

• Statement 1: Increases irrigation needs: Intercropping does not necessarily increase irrigation needs. In fact, it can reduce water wastage as the different root systems of crops help in better water absorption. Hence, this statement is incorrect.
• Statement 2: Reduces labour cost: Intercropping can sometimes increase labour requirements for tasks such as planting, managing, and harvesting due to the complexity of managing multiple crops. Hence, this statement is incorrect.
• Statement 3: Requires less land: Intercropping doesn’t reduce the total land required but optimises land use by growing more than one crop in the same area. Hence, this statement is incorrect.
• Statement 4: Utilises nutrients efficiently and prevents spread of diseases: As explained earlier, intercropping improves nutrient use efficiency and reduces the spread of diseases, making it highly beneficial. Hence, this statement is correct.

Important Points

• Intercropping Types:
• Mixed Intercropping: Two or more crops are grown together without a specific arrangement.
• Row Intercropping: Crops are grown in distinct rows to minimise competition and optimise sunlight exposure.
• Strip Intercropping: Crops are grown in alternate strips to improve pest control and soil conservation.
• Relay Intercropping: A second crop is planted before the first crop is harvested, maximising productivity.
• Examples of Intercropping: Common intercropping systems include maize and beans, wheat and mustard, and sugarcane with pulses.
• Environmental Benefits: Intercropping reduces reliance on chemical inputs, minimises soil erosion, and enhances carbon sequestration in the soil.
• Economic Benefits: Farmers can diversify their income sources by growing different crops simultaneously, reducing economic risks associated with market fluctuations.
• Global Relevance: Intercropping is widely practised in countries like India, China, Brazil, and African nations for sustainable agriculture and food security.

The correct answer is Galvanisation.

Key Points

• Galvanisation refers to the process of coating steel or iron with a layer of zinc to protect it from corrosion and rusting.
• The zinc layer acts as a barrier that prevents moisture and oxygen from coming into contact with the underlying metal.
• Even if the zinc coating is slightly damaged, it continues to protect the metal by acting as a sacrificial anode. Zinc oxidizes faster than steel or iron, thus preventing rust formation.
• Galvanisation is widely used in industries such as construction, automobile manufacturing, and infrastructure development due to its ability to prolong the life of metal products.
• Common examples include galvanized pipes, roofing sheets, and wire mesh used in fencing.

Additional Information

• Cathodic Protection:
• Cathodic protection is a technique used to control the corrosion of a metal surface by making it the cathode of an electrochemical cell.
• This method is commonly applied in pipelines, ships, and storage tanks.
• It does not involve coating the metal but rather uses methods such as attaching sacrificial anodes or impressed current systems.
• Passivation:
• Passivation involves treating or coating metal to reduce the chemical reactivity of its surface and improve corrosion resistance.
• It is commonly used for stainless steel to enhance its natural resistance to rust.
• The process includes the formation of a thin protective oxide layer, often through chemical treatments.
• Electroplating:
• Electroplating is the process of depositing a layer of metal (such as nickel, chromium, or gold) onto a surface using an electrochemical process.
• Unlike galvanisation, electroplating is used more for decorative purposes or to improve electrical conductivity.
• It is widely applied in jewelry manufacturing, electrical components, and automotive parts.
• Galvanisation:
• Galvanisation is a cost-effective method for protecting metals, particularly in environments exposed to moisture or harsh weather conditions.
• The process involves dipping the metal into molten zinc or using electrochemical methods.
• It is essential for industries that require durable materials for outdoor applications.

Important Points

• Features of Zinc Coating:
• Zinc provides dual protection—a physical barrier and sacrificial protection.
• Galvanised steel lasts significantly longer than uncoated steel, making it ideal for applications like bridges, railings, and highways.
• Galvanisation Methods:
• Hot-Dip Galvanisation: The metal is dipped in molten zinc, forming a thick protective layer.
• Electro-Galvanisation: Zinc is deposited through an electrochemical process, resulting in a thinner coating.
• Applications:
• Used in construction materials like roofing sheets and steel beams.
• Essential in the automotive sector for producing rust-resistant car parts.
• Widely applied in household items such as wire mesh and storage containers.

The correct answer is amplitude of vibration is small.

Key Points

• Amplitude of Vibration: The loudness of sound depends on the amplitude of vibration. A smaller amplitude results in a softer sound, while a larger amplitude leads to a louder sound. When the table is hit lightly, the vibrations produced have a small amplitude, resulting in a soft sound. Hence, this statement is correct.
• Frequency of Vibration: Frequency refers to the number of vibrations or oscillations per second and determines the pitch of the sound. It is not directly related to the loudness of the sound in this context. Hence, this statement is incorrect.
• Speed of Sound: The speed of sound depends on the medium through which it travels and does not change based on how lightly or strongly an object is hit. Hence, this statement is incorrect.
• Wavelength: Wavelength is inversely related to frequency and does not directly affect the loudness of the sound. In this scenario, the wavelength remains unaffected by the force of the hit. Hence, this statement is incorrect.

Additional Information

• Amplitude:
• It is the maximum displacement of particles in a medium from their mean position when a wave propagates.
• The amplitude of a sound wave determines its loudness. Higher amplitude results in louder sound and vice versa.
• For example, when a drum is struck hard, the amplitude of the vibrations increases, producing a louder sound.
• Frequency:
• Frequency is measured in hertz (Hz) and determines the pitch of a sound.
• A higher frequency corresponds to a higher-pitched sound, whereas a lower frequency produces a lower pitch.
• For example, the sound of a flute has a higher frequency compared to the sound of a bass drum.
• Speed of Sound:
• The speed of sound is determined by the medium through which it travels. For example, sound travels faster in solids compared to liquids and gases.
• In air at room temperature (approximately 20°C), the speed of sound is about 343 m/s.
• The speed remains constant in a given medium and is unaffected by the force of the sound-producing action.
• Wavelength:
• It is the distance between two consecutive points in phase on a wave, such as crest to crest or trough to trough.
• The wavelength is inversely proportional to the frequency of the wave: λ = v/f, where λ is the wavelength, v is the speed of sound, and f is the frequency.
• In this context, the wavelength of the sound produced does not directly affect its loudness.

Important Points

• Loudness and Amplitude: Loudness is a subjective property of sound that depends on the amplitude of the sound wave. When the amplitude increases, more energy is transmitted, resulting in a louder sound.
• Pitch and Frequency: Pitch is directly related to frequency. A high-frequency sound (e.g., a whistle) has a high pitch, while a low-frequency sound (e.g., thunder) has a low pitch.
• Practical Examples:
• In musical instruments, striking or plucking with more force increases the amplitude of vibrations, creating louder sounds.
• Conversely, soft strikes reduce the amplitude, resulting in softer sounds.

The correct answer is Tertiary consumers.

Key Points

• Tertiary consumers occupy the highest position in a food chain. These organisms are generally carnivores and feed on secondary consumers. Examples include apex predators like lions, tigers, and hawks.
• The population at the tertiary consumer level is the smallest due to the 10% energy transfer rule, which states that only about 10% of energy is passed on from one trophic level to the next, leading to limited energy availability for higher trophic levels.
• The energy pyramid follows a pyramid shape, with producers forming the base and tertiary consumers forming the apex, indicating a decrease in both energy and population as one moves up the levels.
• Compared to producers and primary consumers, tertiary consumers are fewer because they depend on multiple levels of the food chain for sustenance, and their energy requirements are higher.
• Ecological stability is maintained by the balance between the populations of producers, primary consumers, secondary consumers, and tertiary consumers.

Additional Information

• Producers:
• Producers are organisms, such as plants, algae, and cyanobacteria, that can perform photosynthesis to synthesize their own food from sunlight.
• They form the base of the food chain and have the largest population among all trophic levels.
• Producers are the primary source of energy for all other organisms in the ecosystem.
• Primary Consumers:
• These are herbivores that feed on producers. Examples include deer, rabbits, and caterpillars.
• They occupy the second trophic level in the food chain.
• They play a crucial role in transferring energy from producers to higher trophic levels.
• Secondary Consumers:
• These are carnivores or omnivores that feed on primary consumers. Examples include frogs, small fish, and snakes.
• Secondary consumers occupy the third trophic level and are fewer in number than primary consumers due to energy loss at each level.
• Energy Transfer in Food Chain:
• Energy transfer in an ecosystem is governed by the 10% law, which states that only 10% of the energy is transferred to the next level, while the remaining 90% is lost as heat, respiration, or waste.
• This explains why the population and biomass decrease significantly at higher trophic levels.
• Importance of Tertiary Consumers:
• They help regulate the populations of secondary and primary consumers, ensuring ecological balance.
• Tertiary consumers are often keystone species, meaning their presence or absence can significantly impact an ecosystem.

Important Points

• Food Chain Dynamics:
• Each step in a food chain is called a trophic level. The movement of energy and nutrients occurs through these levels.
• Producers are at the first trophic level, followed by primary consumers, secondary consumers, and tertiary consumers.
• Ecological Pyramids:
• Ecological pyramids represent the relative energy or biomass at each trophic level, with the base being the widest (producers) and the apex being the narrowest (tertiary consumers).
• Impact of Energy Loss:
• Due to energy loss at each trophic level, ecosystems can support fewer organisms as one moves higher in the food chain.
• This makes tertiary consumers the least abundant group in any ecosystem.

The correct answer is The cell membrane is permeable to selective substances..

Key Points

• The cell membrane is semi-permeable: The cell membrane, also known as the plasma membrane, is a selectively permeable structure. This means it allows certain substances, such as small molecules like oxygen and carbon dioxide, to pass through while restricting larger molecules like starch or proteins.
• Size and polarity determine permeability: Small, non-polar molecules like oxygen (O₂) can easily diffuse through the lipid bilayer of the membrane due to their size and lack of charge. Larger molecules like starch and proteins are too big to pass through without assistance.
• Role of transport proteins: For larger or charged molecules, the cell membrane uses specific transport proteins, such as carrier or channel proteins, to facilitate their movement across the membrane.
• Breaking down of larger molecules: Larger molecules like starch need to be broken down into smaller components (e.g., glucose) by enzymes before they can pass through the cell membrane.
• Selective permeability is vital for survival: The selective nature of the cell membrane helps maintain the cell's internal environment, regulating the entry and exit of substances to ensure proper functioning and homeostasis.

Additional Information

• Composition of the cell membrane:
• The cell membrane is primarily made up of a phospholipid bilayer, with hydrophobic (water-repelling) tails and hydrophilic (water-attracting) heads. This unique structure makes the membrane selectively permeable.
• It also contains proteins, cholesterol, and carbohydrates that contribute to its functionality and fluidity.
• Simple diffusion: Small and non-polar molecules, such as oxygen and carbon dioxide, move across the cell membrane through simple diffusion without requiring energy.
• Facilitated diffusion: Larger or polar molecules, such as glucose and ions, require the assistance of specific transport proteins to cross the membrane. This process does not require energy.
• Active transport: Some molecules are transported against their concentration gradient using energy in the form of ATP. This process is called active transport and involves protein pumps.
• Endocytosis and exocytosis: For very large molecules or particles, the cell uses processes like endocytosis (engulfing substances into the cell) and exocytosis (expelling substances out of the cell) to move materials across the membrane.

Important Points

• Oxygen and carbon dioxide: These gases are crucial for cellular respiration and can easily diffuse across the membrane due to their small size and non-polar nature.
• Starch and proteins: These are large, complex molecules that require enzymatic breakdown (e.g., starch into glucose) or specific transport mechanisms to pass through the cell membrane.
• Role of enzymes: Enzymes like amylase break down starch into simpler sugars, and proteases break down proteins into amino acids, which can then be absorbed by cells.
• Homeostasis: The selective permeability of the cell membrane is essential for maintaining homeostasis, ensuring that cells function efficiently by regulating the internal environment.

The correct answer is  Make some of their own proteins.

Key Points

• Mitochondria are known as the powerhouses of the cell because they are responsible for producing energy in the form of ATP (adenosine triphosphate) through cellular respiration.
• Unlike other organelles, mitochondria possess their own DNA (mitochondrial DNA or mtDNA) and ribosomes, which are similar to those found in prokaryotic cells.
• Due to the presence of their own DNA and ribosomes, mitochondria can make some of their own proteins essential for their functions, although most mitochondrial proteins are encoded by nuclear DNA and imported into the mitochondria.
• The ability to synthesize proteins independently is a feature inherited from their evolutionary origin, as mitochondria are believed to have evolved from free-living bacteria through endosymbiosis.
• Other functions of mitochondria include regulating calcium levels, apoptosis (programmed cell death), and the production of reactive oxygen species as signaling molecules.

Additional Information

• Option 1: Digest materials like lysosomes do
• Lysosomes are specialized organelles responsible for digestion and recycling of cellular waste through hydrolytic enzymes. Mitochondria do not perform this function. Hence, this statement is incorrect.
• Option 2: Move around the cell on their own
• Mitochondria exhibit dynamic behavior, including movement within the cytoplasm along microtubules, but this movement is facilitated by cytoskeletal elements and motor proteins rather than autonomous movement. Hence, this statement is incorrect.
• Option 3: Make food using sunlight like plants do
• Plants use chloroplasts for photosynthesis, a process that converts sunlight, water, and carbon dioxide into glucose. Mitochondria do not perform photosynthesis. Hence, this statement is incorrect.
• Mitochondrial DNA:
• Mitochondrial DNA is circular and contains genes that code for essential proteins involved in oxidative phosphorylation. This DNA is maternally inherited and plays a role in tracing ancestry.
• Endosymbiotic Theory:
• The endosymbiotic theory explains the origin of mitochondria and chloroplasts as descendants of free-living prokaryotes that were engulfed by ancestral eukaryotic cells.
• This theory is supported by the presence of double membranes, their own DNA, and ribosomes in both organelles.
• Protein Synthesis in Mitochondria:
• Mitochondrial ribosomes (mitoribosomes) are responsible for translating mitochondrial mRNA to produce proteins needed for energy generation and other mitochondrial functions.

Important Points

• Comparison with Nuclear DNA:
• Mitochondrial DNA is much smaller than nuclear DNA and primarily encodes proteins involved in energy production.
• Most mitochondrial proteins are encoded by nuclear DNA, synthesized in the cytoplasm, and imported into mitochondria.
• Diseases Related to Mitochondrial Dysfunction:
• Mutations in mitochondrial DNA can lead to disorders such as Leber's hereditary optic neuropathy, mitochondrial myopathy, and MELAS syndrome.
• Mitochondrial dysfunction is also implicated in aging and neurodegenerative diseases like Parkinson’s and Alzheimer’s.

The correct answer is Water takes the shape of its container but retains its volume.

Key Points

• Liquids have a definite volume: Liquids maintain a consistent volume regardless of the container they are poured into. This property differentiates liquids from gases, which expand to occupy the entire volume of the container.
• No fixed shape: Liquids adapt to the shape of the container they are poured into. For instance, water poured into a cylindrical container takes on a cylindrical shape, while in a square container, it assumes a square shape.
• Correct observation: The statement "Water takes the shape of its container but retains its volume" accurately highlights the defining characteristics of liquids. This observation demonstrates the ability of liquids to change shape while maintaining their volume.
• Statement analysis:
• Option 1: "Water takes the shape of its container but retains its volume" – Correct. This accurately describes the behavior of liquids.
• Option 2: "Air expands to fill the entire container" – Incorrect. This describes the behavior of gases, not liquids.
• Option 3: "A sponge changes its shape when squeezed" – Incorrect. This refers to solids with elastic properties, not liquids.
• Option 4: "A solid iron block retains its shape in any container" – Incorrect. This describes the fixed shape characteristic of solids, not liquids.
• Physical properties of liquids: Liquids are incompressible to a certain extent and have fluidity, allowing them to flow and take the shape of their container while retaining their mass and volume.

Additional Information

• States of matter: Matter exists in three primary states: solid, liquid, and gas. Each state is defined by its physical properties such as shape, volume, and compressibility.
• Solids: Have a fixed shape and volume. Particles are closely packed and have minimal movement.
• Liquids: Have a fixed volume but no fixed shape. Particles are less tightly packed compared to solids and can move freely.
• Gases: Have neither a fixed shape nor a fixed volume. Particles are widely spaced and move randomly.
• Compressibility: Liquids are relatively incompressible compared to gases because their particles are closely packed. However, they can exhibit slight compressibility under extreme pressure.
• Fluidity: Liquids and gases are classified as fluids due to their ability to flow. This property is due to the mobility of their particles.
• Applications of liquids: Liquids play a crucial role in various processes and industries:
• Transport: Liquids like water and oil are transported in containers suited to their properties.
• Cooling systems: Liquids like water and coolants are used in engines and machinery to absorb and dissipate heat.
• Hydraulic systems: Liquids like oil are used in hydraulic machinery to transmit force.
• Scientific principle: The unique properties of liquids are explained by intermolecular forces that are weaker than those in solids but stronger than those in gases.

Important Points

• Intermolecular forces: Liquids exhibit moderate intermolecular forces, allowing particles to stay close together while still enabling mobility.
• Surface tension: Liquids have surface tension due to cohesive forces between molecules. This explains phenomena like the formation of droplets.
• Viscosity: The resistance of a liquid to flow is termed viscosity. Higher viscosity indicates slower flow, e.g., honey compared to water.
• Evaporation: Liquids can transition to a gaseous state at their surface when molecules gain sufficient energy to overcome intermolecular forces.
• Boiling point: The temperature at which a liquid turns into a gas throughout its volume is its boiling point, which varies based on pressure and intermolecular forces.

The correct answer is + 20 cm.

Key Points

• The distance between the pole (P) and the center of curvature (C) of a convex mirror is defined as the radius of curvature (R), which is given as + 40 cm.
• The focal length (f) of a mirror is related to its radius of curvature (R) by the formula:

f = R / 2

• Substituting the value of R = + 40 cm into the formula:

f = 40 / 2 = + 20 cm

• The focal length of a convex mirror is always positive because the focus lies on the same side as the reflective surface, as per the sign conventions in optics.
• Hence, the distance between the pole and the focus of the convex mirror is + 20 cm.

Additional Information

• Convex Mirror:
• A convex mirror is a spherical mirror with the reflective surface bulging outward.
• It is also known as a diverging mirror because it spreads out light rays after reflection.
• Convex mirrors are commonly used in rear-view mirrors for vehicles and in surveillance systems.
• Sign Conventions in Optics:
• Distances measured in the direction of the incident light (towards the reflective surface) are considered positive.
• Distances measured against the direction of the incident light are considered negative.
• For convex mirrors, the focal length (f) is always positive.
• Radius of Curvature (R) and Focal Length (f):
• The radius of curvature is the distance from the pole to the center of curvature of the mirror.
• The focal length is half the radius of curvature: f = R / 2.
• This relationship applies to both convex and concave mirrors.
• Applications of Convex Mirrors:
• Convex mirrors provide a wider field of view, making them suitable for vehicle rear-view mirrors.
• They are used in security mirrors in stores to monitor large areas.
• In automated teller machines (ATMs), convex mirrors are installed for surveillance purposes.

Important Points

• Mirror Formula:
• The mirror formula is:

1/f = 1/v + 1/u

where:

• f = focal length
• v = image distance
• u = object distance
• This formula is applicable for both convex and concave mirrors.
• Significance of Positive Focal Length:
• A positive focal length indicates that the focus lies on the same side as the reflective surface of the convex mirror.
• It also signifies that the mirror forms virtual, upright, and diminished images for all object positions.

The correct answer is Acquired Immuno Deficiency Syndrome.

Key Points

• Acquired Immuno Deficiency Syndrome (AIDS) is caused by the Human Immunodeficiency Virus (HIV), which is primarily transmitted through unprotected sexual contact, contaminated blood, and from mother to child during childbirth or breastfeeding.
• AIDS weakens the immune system, making individuals susceptible to opportunistic infections and certain cancers. The virus attacks CD4 cells (a type of T-cell), which play a crucial role in the immune response.
• Unprotected sexual intercourse with an infected partner is one of the most common modes of HIV transmission globally. Hence, statement 2 is correct.
• Diseases like Protein Energy Malnutrition (Statement 1), Diabetes (Statement 3), and Cancer (Statement 4) are not sexually transmitted. These are either non-communicable diseases or caused by other factors. Hence, statements 1, 3, and 4 are incorrect.
• AIDS is preventable through safe sexual practices, regular testing, and the use of antiretroviral therapy for those diagnosed with HIV.

Additional Information

• Characteristics of HIV/AIDS:
• HIV belongs to a group of viruses called retroviruses. It has two main types: HIV-1 and HIV-2, with HIV-1 being more prevalent globally.
• Symptoms of HIV infection include fever, fatigue, weight loss, recurrent infections, and skin rashes. In advanced stages, it leads to AIDS.
• There is no cure for HIV/AIDS; however, Antiretroviral Therapy (ART) helps manage the disease and prolong the lives of infected individuals.
• Protein Energy Malnutrition (PEM):
• PEM is a form of malnutrition caused by a lack of calories and protein in the diet.
• It is common in children in developing countries and leads to conditions such as Kwashiorkor and Marasmus.
• PEM is not a communicable disease and is not transmitted through sexual contact.
• Diabetes:
• Diabetes is a chronic metabolic disorder characterized by high blood sugar levels due to insufficient insulin production or ineffective use of insulin by the body.
• It is categorized into two types: Type 1 Diabetes (autoimmune) and Type 2 Diabetes (lifestyle-related).
• Diabetes is a non-communicable disease and cannot be transmitted through sexual contact.
• Cancer:
• Cancer is the uncontrolled growth of abnormal cells in the body. It can occur in various organs and tissues.
• While certain types of cancers, such as cervical cancer, are associated with sexually transmitted infections like HPV (Human Papillomavirus), cancer itself is not a communicable disease.

The correct answer is 10 m/s.

Key Points

• Uniform acceleration: The car starts from rest and accelerates uniformly, meaning the rate of change in velocity is constant (2 m/s²). This is a crucial factor in using the equations of motion.
• Initial velocity (u): Since the car starts from rest, the initial velocity is 0 m/s.
• Time (t): The car accelerates for a duration of 5 seconds, as mentioned in the question.
• Equation of motion: To calculate the final velocity, the equation v = u + at is used, where:
• v = Final velocity
• u = Initial velocity
• a = Acceleration
• t = Time
• Calculation: Substituting the given values into the equation:
• v = u + at
• v = 0 + (2 × 5)
• v = 10 m/s

Thus, the final velocity of the car is 10 m/s.

• Correct option: The correct answer is 10 m/s.

Additional Information

• Equations of motion:
• These equations are fundamental in kinematics, used to solve problems involving linear motion under uniform acceleration.
• Key equations include:
• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• They are applicable only when acceleration is constant.
• Uniform acceleration:
• Uniform acceleration refers to a scenario where the rate of change of velocity remains constant.
• Examples include free-fall motion under gravity and linear acceleration in vehicles.
• SI units:
• In physics, the SI unit for velocity is meters per second (m/s), and for acceleration, it is meters per second squared (m/s²).
• Using consistent units is critical to obtaining correct numerical results.
• Applications:
• Understanding these concepts is crucial in fields such as mechanical engineering, automobile design, and physics-based simulations.
• They are also essential for calculating speeds and distances in real-world transportation scenarios.

Important Points

• Constant acceleration: The assumption of constant acceleration simplifies calculations and is often used in introductory physics problems.
• Initial velocity: In many problems, starting from rest (u = 0) is explicitly stated, making calculations straightforward.
• Real-world relevance: While uniform acceleration is idealized, it provides a foundational understanding of motion, which can be adapted to more complex, real-world scenarios.
• Critical thinking: Always verify the units and ensure that the given values align with the conditions of uniform acceleration before applying equations of motion.

The correct answer is Bromopropane.

Key Points

• Saturated hydrocarbons are compounds where all the carbon-carbon bonds are single bonds. These are also known as alkanes or their derivatives.
• Bromopropane is a derivative of propane, which is an alkane. In bromopropane, one hydrogen atom in propane is replaced by a bromine atom, but the structure remains saturated as it contains only single bonds between carbon atoms.
• Propyne contains a triple bond between two carbon atoms, making it an unsaturated compound. Hence, it is not saturated.
• 2-Bromopropene contains a double bond between two carbon atoms, which also makes it unsaturated.
• Propene contains a double bond between two carbon atoms, classifying it as an unsaturated compound.
• Therefore, among the given options, Bromopropane is the only saturated compound because it contains only single bonds between carbon atoms.

Additional Information

• Saturated Hydrocarbons:
• Saturated hydrocarbons are also known as alkanes. The general formula for alkanes is C_nH_2n+2.
• They are characterized by single covalent bonds between carbon atoms, which makes them relatively less reactive compared to unsaturated hydrocarbons.
• Examples include methane (CH₄), ethane (C₂H₆), and propane (C₃H₈).
• Saturated hydrocarbons undergo substitution reactions rather than addition reactions due to the absence of multiple bonds.
• Unsaturated Hydrocarbons:
• Unsaturated hydrocarbons include alkenes and alkynes. Alkenes have at least one double bond (general formula: C_nH_2n), while alkynes have at least one triple bond (general formula: C_nH_2n-2).
• They are more reactive due to the presence of double or triple bonds, which can participate in addition reactions.
• Examples include ethene (C₂H₄), propene (C₃H₆), and ethyne (C₂H₂).
• Functional Groups and Saturation:
• The presence of functional groups (like bromine in bromopropane) does not affect the saturation of a hydrocarbon as long as the carbon-carbon bonds remain single.
• For example, bromopropane is saturated despite the presence of a bromine atom because its carbon backbone contains only single bonds.
• Applications of Saturated Compounds:
• Saturated hydrocarbons like alkanes are widely used as fuels, such as propane and butane in LPG (Liquefied Petroleum Gas).
• They are also used as solvents and in the production of various chemicals in the petrochemical industry.

Important Points

• Bromopropane:
• It is a saturated organic compound with the formula C₃H₇Br.
• It is used as a solvent, especially in degreasing and cleaning applications.
• Bromopropane is also employed in the synthesis of pharmaceuticals and agrochemicals.
• Propyne:
• Propyne (C₃H₄) is an alkyne with a triple bond and is used as a fuel and in organic synthesis.
• 2-Bromopropene:
• It contains a double bond and is classified as an alkene derivative. It is used in organic synthesis.
• Propene:
• Propene (C₃H₆) is an unsaturated hydrocarbon (alkene) used in the production of polypropylene and other chemicals.

The correct answer is Butane.

Key Points

• Butane is a straight-chain hydrocarbon belonging to the alkane family. Its chemical formula is C4H10, and it consists of four carbon atoms arranged in a continuous chain.
• 3-Methylpentane, 2-Methylpropane, and Isobutane are branched-chain hydrocarbons. They contain substituent groups attached to the main chain, making them different from straight-chain hydrocarbons. Hence, these options are incorrect.
• In straight-chain hydrocarbons, all the carbon atoms are connected in a linear sequence without any branching. This structure is characteristic of Butane.
• Isomerism: While butane has isomers such as Isobutane, the straight-chain structure specifically refers to the unbranched form of butane.
• Hence, the correct answer is Butane.

Additional Information

• Hydrocarbons:
• Hydrocarbons are organic compounds composed entirely of carbon and hydrogen atoms. They are classified as alkanes, alkenes, and alkynes based on the type of bonding between carbon atoms.
• Alkanes are saturated hydrocarbons with single bonds between carbon atoms, while alkenes and alkynes have double and triple bonds, respectively.
• Straight-chain hydrocarbons:
• Straight-chain hydrocarbons have no branching in their structure. Examples include methane, ethane, propane, and butane.
• They are usually named using the IUPAC nomenclature system, where the prefix indicates the number of carbon atoms in the chain.
• Branched-chain hydrocarbons:
• In branched-chain hydrocarbons, substituent groups (such as methyl or ethyl groups) are attached to the main chain.
• Examples include 3-Methylpentane, 2-Methylpropane, and Isobutane.
• Applications of Butane:
• Butane is commonly used as a fuel in lighters and portable stoves.
• It is also used in the production of liquefied petroleum gas (LPG) and as a raw material in the petrochemical industry.
• Isomerism in Butane:
• Butane has two isomers: n-Butane (straight-chain) and Isobutane (branched-chain).
• Both isomers exhibit different physical properties, such as boiling points and melting points, due to differences in molecular structure.

Important Points

• Alkane Naming Conventions: Alkanes are named based on the number of carbon atoms in the longest continuous chain. Prefixes like "meth-", "eth-", "prop-", and "but-" indicate one, two, three, and four carbon atoms, respectively.
• Physical Properties: Straight-chain hydrocarbons generally have higher boiling points than their branched-chain counterparts due to stronger intermolecular forces.
• Environmental Impact: Combustion of hydrocarbons such as butane releases carbon dioxide (CO2), a greenhouse gas contributing to global warming.
• Industrial Usage: Hydrocarbons like butane are widely used in the manufacturing of plastics, synthetic rubber, and other chemical products.

The correct answer is Bryophyllum.

Key Points

• Bryophyllum is a plant known for its unique ability to reproduce asexually through budding along the margins of its leaves.
• In Bryophyllum, small plantlets, often called buds, develop at the notches of the leaf margins. These plantlets have their own roots and can grow into a new plant when they fall off and come into contact with soil.
• This method of reproduction is an example of vegetative propagation, a type of asexual reproduction where new plants grow from vegetative parts like leaves, stems, or roots.
• The ability of Bryophyllum to reproduce in this manner ensures its survival and rapid propagation, especially in conditions where seeds may not germinate effectively.
• Other plants, like Hydra and Spirogyra, reproduce asexually too, but their methods (budding in Hydra and fragmentation in Spirogyra) differ significantly from Bryophyllum's vegetative propagation through leaf margins.

Additional Information

• Bryophyllum Characteristics:
• Belongs to the family Crassulaceae and is native to Madagascar but is widely cultivated in tropical and subtropical regions.
• Commonly known as the "Mother of Thousands" or "Miracle Leaf" because of its unique reproduction method.
• Its leaves are fleshy and succulent, designed to store water, which makes it drought-resistant.
• Asexual Reproduction in Plants:
• Asexual reproduction is a mode of reproduction where offspring arise from a single parent without the involvement of gamete fusion.
• Methods include vegetative propagation, budding, fragmentation, and spore formation.
• Plants like Bryophyllum, onion, and potato reproduce through vegetative propagation.
• Comparison with Other Options:
• Hydra: Reproduces asexually through budding, where a new organism grows from the body of the parent (incorrect for this question).
• Spirogyra: Reproduces asexually through fragmentation, where filaments break into pieces, and each piece grows into a new organism (incorrect for this question).
• Mango: Reproduces sexually through seeds and does not show asexual reproduction through budding (incorrect for this question).
• Importance of Vegetative Propagation:
• Ensures rapid reproduction and survival of plants in unfavorable conditions.
• Used in agriculture and horticulture to produce identical plants with desirable traits.
• Allows propagation of plants that do not produce viable seeds.

Important Points

• Key Terminology:
• Asexual Reproduction: Reproduction without the involvement of gametes.
• Vegetative Propagation: A method of asexual reproduction where new plants grow from vegetative parts like stems, roots, or leaves.
• Budding: A process where a new organism develops from an outgrowth of a parent, common in organisms like Hydra.
• Applications of Bryophyllum:
• Used in traditional medicine for its anti-inflammatory and wound-healing properties.
• Often cultivated as an ornamental plant due to its unique appearance and propagation method.

Shortcut Trick

Ratio of Volume of Sphere to TSA of Hemisphere = (4/3) π r³ : 3 π r² = 4r : 9

Given 4r / 9 = 28 / 3 ⇒ r = (28 × 9) / (3 × 4) = 7 × 3 = 21 cm

Volume = (4/3) × (22/7) × 21 × 21 × 21 = 88 × 441 = 38,808 cm³

Check Divisibility by 11: 38,808 ⇒ (3 + 8 + 8) − (8 + 0) = 19 − 8 = 11 (Divisible).

∴ The correct answer is 38,808 cm³.

Alternate Method

Given:

Radius of Sphere = Radius of Hemisphere = r

Ratio of Volume of Sphere (V_s) to Total Surface Area of Hemisphere (A_h) = 28 : 3

Formula Used:

V_s = (4/3) π r³

A_h = 3 π r²

Calculations:

⇒ V_s / A_h = [(4/3) × π × r³] / [3 × π × r²] = 28 / 3

⇒ [(4/3) × r] / 3 = 28 / 3

⇒ 4r / 9 = 28 / 3

⇒ r = (28 × 9) / (3 × 4)

⇒ r = 7 × 3 = 21 cm

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

⇒ Volume = (4/3) × (22/7) × 9261

⇒ Volume = 4 × 22 × (9261 / 21)

⇒ Volume = 88 × 441

⇒ Volume = 38,808 cm³

∴ The correct answer is 38,808 cm³.

Additional Information

Surface Area of Sphere

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

Hemisphere Formulas

The curved surface area (CSA) is 2πr² and the volume is (2/3)πr³.

Ratio of Volumes

For a sphere and hemisphere with the same radius, the ratio of their volumes is always 2 : 1.

Given:

Simplify: $\dfrac{[72 ÷ {6 × (3 + 2)} + 5^2 - 8] × 5}{97}$

Formula used:

Follow BODMAS: Brackets → Orders → Division → Multiplication → Addition → Subtraction

Calculation:

$\dfrac{[72 ÷ {6 × (3 + 2)} + 5^2 - 8] × 5}{97}$

⇒ $\dfrac{[72 ÷ {6 × 5} + 25 - 8] × 5}{97}$

⇒ $\dfrac{[72 ÷ 30 + 25 - 8] × 5}{97}$

⇒ $\dfrac{[2.4 + 25 - 8] × 5}{97}$

⇒ $\dfrac{[19.4] × 5}{97}$

⇒ $\dfrac{97}{97}$

⇒ 1

∴ The correct answer is option (2).

Shortcut Trick

Profit Ratio of Anmol to Ayushi = 2700 : (7500 − 2700) = 2700 : 4800 = 9 : 16.

Since the investment time is the same, Profit Ratio = Investment Ratio.

Sum of ratio parts = 9 + 16 = 25 units.

25 units = ₹46,600 ⇒ 1 unit = ₹1,864.

Anmol's Investment = 9 × 1864 = ₹16,776.

∴ The correct answer is ₹16,776.

Alternate Method

Given: Total Investment = ₹46,600 | Total Profit = ₹7,500 | Anmol's Profit = ₹2,700

Formula Used: Profit Ratio = Investment Ratio × Time Ratio

⇒ Ayushi's Profit = Total Profit − Anmol's Profit

⇒ Ayushi's Profit = 7,500 − 2,700 = ₹4,800

⇒ Ratio of Profits (Anmol : Ayushi) = 2,700 : 4,800 = 9 : 16

⇒ Since time is constant, Ratio of Investment = 9 : 16

⇒ Anmol's Investment = (9 / (9 + 16)) × 46,600

⇒ Anmol's Investment = (9 / 25) × 46,600

⇒ Anmol's Investment = 9 × 1,864 = ₹16,776

∴ The correct answer is ₹16,776.

Additional Information

Simple Partnership

In a simple partnership, all partners invest their capital for the same time period. The profit is distributed simply in the ratio of their investments.

Compound Partnership

If partners invest for different time periods, the profit is divided in the ratio of the product of capital and time: (C₁ × T₁) : (C₂ × T₂).

Ratio of Capitals

If the profit ratio and time ratio are known, the ratio of capitals can be found as (P₁/T₁) : (P₂/T₂).

Shortcut Trick

Direct formula for Marked Price (MP) to Cost Price (CP) ratio:

MP / CP = (100 + Profit%) / (100 − Discount%)

MP / CP = (100 + 4) / (100 − 29) = 104 / 71

Mark-up% = [(104 − 71) / 71] × 100 = (33 / 71) × 100 ≈ 46.478...

∴ The correct answer is 46.48%.

Alternate Method

Given:

Discount% = 29%

Profit% = 4%

Let CP be 100 units.

⇒ Selling Price (SP) = 100 + 4% of 100 = 104 units.

⇒ SP is obtained after 29% Discount on MP.

⇒ 0.71 × MP = 104

⇒ MP = 104 / 0.71 ≈ 146.4788 units.

⇒ Mark-up = MP − CP = 146.4788 − 100 = 46.4788 units.

⇒ Mark-up% = (46.4788 / 100) × 100 = 46.48%

∴ The correct answer is 46.48%.

Additional Information

Mark-up Percentage

It is the percentage by which the Marked Price is higher than the Cost Price. Formula: [(MP − CP) / CP] × 100.

Relation between CP, MP, Profit, and Discount

The bridge between CP and MP is SP. SP = CP × (1 + Profit/100) and SP = MP × (1 − Discount/100).

Successive Percentage Change

If mark-up is x% and discount is y%, the net Profit% is given by: x − y − (xy/100).

Shortcut Trick

Let the cost of a TV be 100 units. Then, the cost of a washing machine is 60 units (40% less).

Total initial cost = (7 × 60) + (6 × 100) = 420 + 600 = 1020 units.

Total final cost = (420 × 1.80) + (600 × 0.78) = 756 + 468 = 1224 units.

Percentage change = [(1224 − 1020) ÷ 1020] × 100 = (204 ÷ 1020) × 100 = 20%.

∴ The correct answer is Increase by 20%.

Alternate Method

Given: Cost of washing machine (W) is 40% less than TV (T). Number of items: 7W and 6T.

⇒ Let Cost of TV (T) = 100 and Cost of Washing Machine (W) = 100 − 40 = 60.

⇒ Total Initial Cost (7W + 6T) = (7 × 60) + (6 × 100) = 420 + 600 = 1020.

⇒ New Cost of W = 60 + 80% of 60 = 60 + 48 = 108.

⇒ New Cost of T = 100 − 22% of 100 = 78.

⇒ Total Final Cost = (7 × 108) + (6 × 78) = 756 + 468 = 1224.

⇒ Net Increase = 1224 − 1020 = 204.

⇒ Percentage Change = (204 ÷ 1020) × 100 = 20%.

∴ The correct answer is Increase by 20%.

Additional Information

Percentage Change Formula

Percentage change is calculated as [(Final Value − Initial Value) ÷ Initial Value] × 100.

Weighted Average Concept

When components of a total change by different percentages, the total percentage change depends on the weight (initial value) of each component.

Successive Percentage Change

If a value changes by a% and then by b%, the net change is [a + b + (ab/100)]%.

Shortcut Trick

Cost Price (CP) = ₹460

Selling Price (SP) = ₹483

Profit % = (SP − CP) ÷ CP × 100

Profit % = (23 ÷ 460) × 100 = 1/20 × 100 = 5%

∴ The correct answer is 5%.

Alternate Method

Given:

Cost Price (CP) of the watch = ₹460

Selling Price (SP) of the watch = ₹483

Formula Used:

Profit = Selling Price − Cost Price

Profit Percentage = (Profit ÷ Cost Price) × 100

Calculations:

⇒ Profit = 483 − 460

⇒ Profit = ₹23

⇒ Profit % = (23 ÷ 460) × 100

⇒ Profit % = (1 ÷ 20) × 100

⇒ Profit % = 5%

∴ The correct answer is 5%.

Additional Information

Loss Percentage

When Cost Price is greater than Selling Price, Loss % = (CP − SP) ÷ CP × 100.

Marked Price & Discount

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

Effective Profit/Loss

For successive changes, the effective percentage is calculated using the formula a + b + (ab/100).

Shortcut Trick

Effective Compound Interest rate for 2 years at 20% = 20 + 20 + (20 × 20)/100 = 44%.

Total Amount percentage = 100% (Principal) + 44% (Interest) = 144%.

Amount = 144% of ₹6,500 = 1.44 × 6500.

Amount = ₹9,360.

∴ The correct answer is ₹9,360.

Alternate Method

Given:

Principal (P) = ₹6,500

Rate (R) = 20% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 6500 × (1 + 20/100)²

⇒ A = 6500 × (120/100) × (120/100)

⇒ A = 6500 × 1.2 × 1.2

⇒ A = 6500 × 1.44

⇒ A = 9360

∴ The correct answer is ₹9,360.

Additional Information

Effective Rate for 2 Years

For a rate R%, the effective compound interest rate for 2 years is given by 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 becomes R/2 and the time becomes 2n.

Shortcut Trick

When the teacher joins, the average age of all 52 people increases by 1 year (from 24 to 25).

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

Teacher's Age = 25 + (51 × 1) = 25 + 51 = 76 years.

∴ The correct answer is 76 years.

Alternate Method

Given:

Number of students = 51

Initial Average age = 24 years

Total number of people (including teacher) = 51 + 1 = 52

New Average age = 25 years

Formula: Total Age = Average Age × Number of Persons

Calculations:

⇒ Total age of 51 students = 51 × 24 = 1224 years

⇒ Total age of 52 people (including teacher) = 52 × 25 = 1300 years

⇒ Age of the teacher = Total age of group − Total age of students

⇒ Age of the teacher = 1300 − 1224

⇒ Age of the teacher = 76 years

∴ The correct answer is 76 years.

Additional Information

Definition of Average

Average is the sum of all observations divided by the total number of observations. Average = Σx ÷ n.

Inclusion Property

If a new member joins and the average increases, the new member's value is higher than the original average. If the average decreases, the value is lower.

Age/Observation Change Formula

New Entry Value = (New Average × New Strength) − (Old Average × Old Strength).

Shortcut Trick

Relative speed (same direction) = 84 − 54 = 30 kmph.

Total distance (P + Q) = Speed × Time ⇒ 30 × (5/18) × 270 = 2250 m.

Length of train P = 2250 − 447 = 1803 m.

New relative speed (opposite direction) = 84 + 87 = 171 kmph.

Time to cross R = (1803 + 287) ÷ (171 × 5/18) = 2090 ÷ 47.5 = 44 seconds.

∴ The correct answer is 44 seconds.

Alternate Method

Given: Speed of P = 84 kmph, Speed of Q = 54 kmph, Length Q = 447 m, Time = 4.5 min = 270 sec.

Formula Used: Distance = Relative Speed × Time; Relative Speed (Same) = S₁ − S₂; Relative Speed (Opposite) = S₁ + S₂.

⇒ Relative Speed (P and Q) = (84 − 54) × 5/18 = 30 × 5/18 = 25/3 m/s.

⇒ Total distance = (Length P + Length Q) = (25/3) × 270 = 2250 m.

⇒ Length P = 2250 − 447 = 1803 m.

⇒ Relative Speed (P and R) = (84 + 87) × 5/18 = 171 × 5/18 = 47.5 m/s.

⇒ Total length to cross R = 1803 + 287 = 2090 m.

⇒ Time = 2090 ÷ 47.5 = 44 seconds.

∴ The correct answer is 44 seconds.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed is S₁ − S₂. When they move in opposite directions, relative speed is S₁ + S₂.

Unit Conversion

To convert speed from km/hr to m/s, multiply by 5/18. To convert minutes to seconds, multiply by 60.

Train Crossing Length

When a train crosses another train, the total distance covered is always the sum of the lengths of both trains (L₁ + L₂).

Shortcut Trick

Apply the Chain Rule formula: (M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

Here, depth is the work (W). Since depth is doubled (59 → 118) and time is significantly reduced, the number of men must increase.

(11 × 19) ÷ 59 = (M₂ × 2) ÷ 118

M₂ = (11 × 19 × 118) ÷ (59 × 2) = 11 × 19 = 209

∴ The correct answer is 209 men.

Alternate Method

Given:

Initial men (M₁) = 11, Initial days (D₁) = 19, Initial depth (W₁) = 59 feet

Final days (D₂) = 2, Final depth (W₂) = 118 feet

Formula Used:

M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂

Calculations:

⇒ (11 × 19) ÷ 59 = (M₂ × 2) ÷ 118

⇒ 11 × 19 × 118 = M₂ × 2 × 59

⇒ 209 × 118 = M₂ × 118

⇒ M₂ = 209

∴ The correct answer is 209 men.

Additional Information

Chain Rule proportionality

Men (M) is inversely proportional to Days (D) and directly proportional to Work (W).

Work & Wages

Wages are always distributed in the ratio of the work done, which is usually Efficiency × Time.

Efficiency of Men

If efficiency changes, the formula becomes (M₁D₁E₁)/W₁ = (M₂D₂E₂)/W₂.

Given:

$\dfrac{1}{20}$ is what percentage of $\dfrac{2}{60}$?

Formula used:

Percentage = $\dfrac{\text{Part}}{\text{Whole}} \times 100$

Calculation:

Part = $\dfrac{1}{20}$, Whole = $\dfrac{2}{60}$

⇒ Percentage = $\dfrac{\dfrac{1}{20}}{\dfrac{2}{60}} \times 100$

⇒ Percentage = $\dfrac{1}{20} \times \dfrac{60}{2} \times 100$

⇒ Percentage = $\dfrac{60}{40} \times 100$

⇒ Percentage = $\dfrac{3}{2} \times 100$

⇒ Percentage = 150%

∴ The correct answer is option (2).

Given:

Simplify: 4 ÷ 24 + 3.6 × (1)/(9) - { (1)/(15) + (1 - 1.5) }

Formula used:

Use BODMAS (Brackets, Order, Division/Multiplication, Addition/Subtraction).

Calculation:

Expression: 4 ÷ 24 + 3.6 × (1)/(9) - { (1)/(15) + (1 - 1.5) }

⇒ 4 ÷ 24 = 1/6

⇒ 3.6 × (1)/(9) = 3.6 ÷ 9 = 0.4

⇒ (1)/(15) = 0.0667

⇒ (1 - 1.5) = -0.5

⇒ { (1)/(15) + (1 - 1.5) } = 0.0667 - 0.5 = -0.4333

⇒ 1/6 + 0.4 - (-0.4333)

⇒ 0.1667 + 0.4 + 0.4333 = 1

∴ The correct answer is option (4).

Shortcut Trick

Assume all 118 coins are ₹10 coins.

Total value = 118 × 10 = ₹1180.

Difference from actual value = 1180 − 855 = ₹325.

Difference per coin replaced = 10 − 5 = ₹5.

Number of ₹5 coins = 325 ÷ 5 = 65.

∴ The correct answer is 65.

Alternate Method

Given:

Total coins = 118

Total value = ₹855

Let:

Number of ₹5 coins = x

Number of ₹10 coins = 118 − x

Calculation:

⇒ 5x + 10(118 − x) = 855

⇒ 5x + 1180 − 10x = 855

⇒ −5x = 855 − 1180

⇒ −5x = −325

⇒ x = 325 ÷ 5

⇒ x = 65

∴ The correct answer is 65.

Additional Information

Alligation Method

Find average value per coin (855/118) and use the alligation cross with ₹5 and ₹10 to find the ratio of quantities.

Substitution Method

Express one variable in terms of another from the total count equation and substitute it into the total value equation.

Shortcut Trick

For third proportional x of two numbers a and b, use the formula x = b² ÷ a.

Here, a = 32 and b = 24.

⇒ x = (24 × 24) ÷ 32

⇒ x = 576 ÷ 32 = 18.

∴ The correct answer is 18.

Alternate Method

Given: First number (a) = 32, Second number (b) = 24, Third proportional = x

Formula Used: If a, b, and x are in continued proportion, then a : b = b : x.

⇒ 32 ÷ 24 = 24 ÷ x

⇒ 32 × x = 24 × 24

⇒ 32x = 576

⇒ x = 576 ÷ 32

⇒ x = 18

∴ The correct answer is 18.

Additional Information

Mean Proportional

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

Fourth Proportional

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

Compound Ratio

The compound ratio of a : b and c : d is expressed as (a × c) : (b × d).

Shortcut Trick

Total Distance = Sum of lengths = 267 + 153 = 420 m

Relative Speed (Same direction) = 162 − 36 = 126 km/hr

Convert speed to m/s: 126 × (5/18) = 7 × 5 = 35 m/s

Time = Distance ÷ Speed = 420 ÷ 35 = 12 sec

∴ The correct answer is 12 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 267 m, Speed (S₁) = 162 km/hr

Length of Train 2 (L₂) = 153 m, Speed (S₂) = 36 km/hr

Formula Used:

Time = (L₁ + L₂) ÷ (S₁ − S₂)

Calculations:

⇒ Total Distance = 267 + 153 = 420 m

⇒ Relative Speed = 162 − 36 = 126 km/hr

⇒ Speed in m/s = 126 × (5/18) m/s

⇒ Speed = 35 m/s

⇒ Time taken = 420 ÷ 35

⇒ Time taken = 12 sec

∴ The correct answer is 12 sec.

Additional Information

Relative Speed Concept

When two objects move in the same direction, relative speed = S₁ − S₂. When they move in opposite directions, relative speed = 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.

Train Overtaking/Crossing

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

Shortcut Trick

Savings = 60%, so Expenditure = 100% − 60% = 40%.

Given, 40% of Income = ₹440.

100% of Income = (440 ÷ 40) × 100 = 11 × 100 = ₹1,100.

∴ The correct answer is ₹1,100.

Alternate Method

Given:

Savings Percentage = 60%

Expenditure = ₹440

Calculations:

⇒ Total Income = Expenditure + Savings

⇒ 100% = Expenditure % + 60%

⇒ Expenditure % = 40%

⇒ 40% of Income = ₹440

⇒ (40 ÷ 100) × Income = 440

⇒ Income = (440 × 100) ÷ 40

⇒ Income = 11 × 100 = ₹1,100

∴ The correct answer is ₹1,100.

Additional Information

Income Equation

Income is always the sum of Expenditure and Savings. Income = Expenditure + Savings.

Percentage Base Change

If savings are s%, then expenditure is (100 − s)% of the total income.

Ratio Method

Savings of 60% means Savings : Income = 3 : 5. Thus, Expenditure : Income = 2 : 5.

Shortcut Trick

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

Given value of n = 50.

⇒ Average = 50 + 1 = 51.

∴ The correct answer is 51.

Alternate Method

Given: Total count of numbers (n) = 50.

The first 50 even natural numbers are 2, 4, 6, ..., up to the 50^th term.

Formula Used:

Average of an Arithmetic Progression (AP) = (First term + Last term) ÷ 2

Calculations:

⇒ First term (a) = 2

⇒ 50^th term (l) = 2 × 50 = 100

⇒ Average = (2 + 100) ÷ 2

⇒ Average = 102 ÷ 2

⇒ Average = 51

∴ The correct answer is 51.

Additional Information

Average of first n natural numbers

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

Average of first n odd natural numbers

The average of the first n odd natural numbers (1, 3, 5...up to n terms) is simply equal to n.

Sum of first n even natural numbers

The total sum of the first n even natural numbers is given by the formula n × (n + 1).

Shortcut Trick

Discount = 485 − 293 = 192

Discount % = (192 ÷ 485) × 100

Discount % ≈ 39.587% ≈ 39.59%

∴ The correct answer is 39.59%.

Alternate Method

Given:

Marked Price (MP) = ₹485

Selling Price (SP) = ₹293

Formula Used:

Discount = Marked Price − Selling Price

Discount Percentage = (Discount ÷ Marked Price) × 100

Calculations:

⇒ Discount = 485 − 293 = 192

⇒ Discount % = (192 ÷ 485) × 100

⇒ Discount % = 19200 ÷ 485

⇒ Discount % = 39.5876...

⇒ Rounding off to two decimal places, we get 39.59%.

∴ The correct answer is 39.59%.

Additional Information

Marked Price (MP)

It is the price displayed on the label of an item. Discount is always calculated on the Marked Price.

Successive Discounts

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

Profit and Loss on CP

Unlike discounts, Profit and Loss percentages are always calculated on the Cost Price (CP) unless stated otherwise.

Shortcut Trick

The angle formed by the bisectors of two exterior angles of a triangle is given by the direct formula:

∠MON = 90° − ∠P ÷ 2

Given, ∠P = 76°

∠MON = 90° − 76° ÷ 2 = 90° − 38° = 52°

∴ The correct answer is 52°.

Alternate Method

Given: In ΔPMN, ∠P = 76°. The bisectors of the exterior angles at M and N intersect at point O.

Formula Used: Sum of angles in a triangle = 180°, Linear pair angle sum = 180°

In ΔPMN:

⇒ ∠P + ∠PMN + ∠PNM = 180°

⇒ 76° + ∠PMN + ∠PNM = 180°

⇒ ∠PMN + ∠PNM = 180° − 76° = 104°

Let the exterior angles at M and N be ∠Ext_M and ∠Ext_N:

⇒ ∠Ext_M = 180° − ∠PMN

⇒ ∠Ext_N = 180° − ∠PNM

Since MO and NO are the bisectors of these exterior angles:

⇒ ∠OMN = ∠Ext_M ÷ 2 = (180° − ∠PMN) ÷ 2 = 90° − ∠PMN ÷ 2

⇒ ∠ONM = ∠Ext_N ÷ 2 = (180° − ∠PNM) ÷ 2 = 90° − ∠PNM ÷ 2

In ΔMON:

⇒ ∠MON + ∠OMN + ∠ONM = 180°

⇒ ∠MON + (90° − ∠PMN ÷ 2) + (90° − ∠PNM ÷ 2) = 180°

⇒ ∠MON + 180° − (∠PMN + ∠PNM) ÷ 2 = 180°

⇒ ∠MON = (∠PMN + ∠PNM) ÷ 2

⇒ ∠MON = 104° ÷ 2 = 52°

∴ The correct answer is 52°.

Additional Information

Incenter Angle Formula

The angle formed by the interior angle bisectors of ∠B and ∠C at the incenter I is given by ∠BIC = 90° + ∠A ÷ 2.

Excenter Angle Formula

The angle formed by the exterior angle bisectors of ∠B and ∠C at the excenter O is given by ∠BOC = 90° − ∠A ÷ 2.

Relation between Incenter and Excenter

The sum of the angles formed at the incenter and the opposite excenter is always 180° (i.e., ∠BIC + ∠BOC = 180°).

Shortcut Trick

The equation x² + 1 = 2x is a standard perfect square identity.

Rearranging gives x² − 2x + 1 = 0, which is (x − 1)² = 0.

This implies the value of x = 1.

Substitute x = 1 into the expression x⁴ − 1:

(1)⁴ − 1 = 1 − 1 = 0.

∴ The correct answer is 0.

Alternate Method

Given: x² + 1 = 2x

To Find: x⁴ − 1

⇒ x² − 2x + 1 = 0

⇒ x² − x − x + 1 = 0

⇒ x(x − 1) − 1(x − 1) = 0

⇒ (x − 1)(x − 1) = 0

⇒ (x − 1)² = 0

⇒ x − 1 = 0

⇒ x = 1

Now, substituting the value of x in the required algebraic expression:

⇒ x⁴ − 1 = (1)⁴ − 1

⇒ 1 − 1 = 0

∴ The correct answer is 0.

Additional Information

Perfect Square Identity

The formula (a − b)² = a² − 2ab + b² is used to identify expressions that can be simplified into a single squared term.

Difference of Squares

The expression x⁴ − 1 is a difference of squares and can be factored as (x² − 1)(x² + 1) or (x − 1)(x + 1)(x² + 1).

Nature of Roots

For a quadratic equation ax² + bx + c = 0, if the discriminant (D = b² − 4ac) is zero, the roots are real and equal.

Shortcut Trick

Combined efficiency of A, B, and C = (1/3 + 1/8 + 1/24) = (8 + 3 + 1)/24 = 12/24 = 1/2 unit/day.

Days to complete the original work = 1 ÷ (1/2) = 2 days.

Days to complete thrice the same work = 2 × 3 = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given: Time taken by A = 3 days, B = 8 days, C = 24 days.

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

⇒ Total Efficiency of (A + B + C) = 8 + 3 + 1 = 12 units/day.

⇒ New Work to be done = 3 × 24 = 72 units.

⇒ Required Time = New Work ÷ Total Efficiency

⇒ Required Time = 72 ÷ 12 = 6 days.

∴ The correct answer is 6 days.

Additional Information

Work and Efficiency Relationship

Work is directly proportional to Efficiency and Time. If work is tripled, the time required at constant efficiency also triples.

Combined Work Formula

For three people, combined time T = (D₁×D₂×D₃) ÷ (D₁D₂ + D₂D₃ + D₃D₁). Multiply this result by 'n' for 'n' times the work.

Shortcut Trick

Given cos θ = 5/13. In a Pythagorean triplet (5, 12, 13), sin θ = 12/13.

Expression √[(1 − sin θ)/(1 + sin θ)] = √[(1 − sin θ)² / (1 − sin² θ)] = (1 − sin θ) / cos θ.

Substituting values: (1 − 12/13) / (5/13) = (1/13) / (5/13) = 1/5.

∴ The correct answer is 1/5.

Alternate Method

Given: cos θ = Base / Hypotenuse = 5 / 13

Formula: Perpendicular (p) = √(h² − b²) and sin θ = p / h

⇒ p = √(13² − 5²) = √(169 − 25) = √144 = 12

⇒ sin θ = 12 / 13

⇒ Value = √[(1 − 12/13) / (1 + 12/13)]

⇒ Value = √[(1/13) / (25/13)] = √(1 / 25)

⇒ Value = 1 / 5

∴ The correct answer is 1/5.

Additional Information

Trigonometric Identities

Fundamental identity: sin² θ + cos² θ = 1. Also, 1 − sin² θ = cos² θ.

Rationalization Shortcut

√[(1 − sin θ)/(1 + sin θ)] = sec θ − tan θ and √[(1 + sin θ)/(1 − sin θ)] = sec θ + tan θ.

Shortcut Trick

Recognize Pythagorean Triplets to split the base 42 cm into two parts: 12 cm and 30 cm.

For side 20 cm, triplet (3, 4, 5) × 4 → (12, 16, 20).

For side 34 cm, triplet (8, 15, 17) × 2 → (16, 30, 34).

Check: 12 + 30 = 42 cm (the longest side). The common height is 16 cm.

∴ The correct answer is 16 cm.

Alternate Method

Given: Sides of the triangle are a = 42 cm, b = 34 cm, and c = 20 cm.

Formula Used: Heron's Formula for Area = √[s(s−a)(s−b)(s−c)] and Area = 1/2 × base × height.

⇒ Semi-perimeter (s) = (42 + 34 + 20) ÷ 2

⇒ s = 96 ÷ 2 = 48 cm

⇒ Area = √[48 × (48 − 42) × (48 − 34) × (48 − 20)]

⇒ Area = √[48 × 6 × 14 × 28]

⇒ Area = √[(16 × 3) × (3 × 2) × (14) × (14 × 2)]

⇒ Area = √[16 × 3² × 2² × 14²] = 4 × 3 × 2 × 14 = 336 cm²

⇒ For longest side (base = 42 cm), Area = 1/2 × 42 × height

⇒ 336 = 21 × height

⇒ height = 336 ÷ 21 = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Heron's Formula

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

Altitude vs. Side Length

In any triangle, the longest side always has the shortest altitude corresponding to it, as Area = 1/2 × b × h remains constant.

Pythagorean Triplets

Common sets of integers (a, b, c) such that a² + b² = c², helping solve triangle problems quickly without complex calculations.

Shortcut Trick

Sum difference = (Average₂ − Average₁) × Number of persons

(I + J + L + M) − (I + J + K + L) = (12 − 9) × 4

M − K = 3 × 4 = 12 years

M = K + 12 = 5 + 12 = 17 years

∴ The correct answer is 17 years.

Alternate Method

Given:

Average age of I, J, K, L = 9 years

Average age of I, J, L, M = 12 years

Age of K = 5 years

Formula Used:

Sum of observations = Average × Number of observations

Calculations:

⇒ Total age of I, J, K, L = 9 × 4 = 36 years

⇒ Total age of I, J, L, M = 12 × 4 = 48 years

⇒ (I + J + L + M) − (I + J + K + L) = 48 − 36

⇒ M − K = 12

⇒ M − 5 = 12

⇒ M = 12 + 5 = 17 years

∴ The correct answer is 17 years.

Additional Information

Net Change in Average

When one person is replaced by another, the change in sum equals the difference between the new and old person's values: Change = (New − Old).

Weighted Average

Combined average of two groups is given by (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Shortcut Trick

Apply the Chain Rule formula: M₁ × D₁ = M₂ × D₂

3 × 20 = M₂ × 2

60 = M₂ × 2

M₂ = 30 men

∴ The correct answer is 30 men.

Alternate Method

Given:

Number of men in first case (M₁) = 3

Number of days in first case (D₁) = 20

Number of days in second case (D₂) = 2

Formula Used:

Total Work = Men × Days

Calculations:

⇒ Total work = 3 × 20 = 60 units

⇒ Let the required number of men be M₂.

⇒ M₂ × 2 = 60

⇒ M₂ = 60 ÷ 2

⇒ M₂ = 30 men

∴ The correct answer is 30 men.

Additional Information

General Chain Rule Formula

For different work or hours: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂, where H is hours and W is work.

Inverse Proportion

Men and Days are inversely proportional. If more men are employed, fewer days are required to complete the same work.

Work and Wages

Wages are generally distributed in the ratio of the work done by individuals (Efficiency × Time).

Given: Seven boxes B, C, D, I, J, K and L are kept one over the other but not necessarily in the same order.

Only I is kept above C.

Only L is kept between C and J.

Only K is kept below D.

 

Now, placing remaining box B in the remaining position:

Now, The boxes are J, L, and C.

Hence, the correct answer is "Option 4".

The logic followed here is:

thus, "137" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

The logic followed here is:

‘nail post fish’ is coded as ‘pl mg qn’,

‘fish urn knife’ is coded as ‘ix pl ko’.

Here,

Here, the code for ‘fish’ is pl.

Hence, the correct answer is "Option 4".

Given Statements:

Some paddy are rice.

Some rice are wheat.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some paddy are wheat – False (Because, no definite relation is given between paddy and wheat, so it can be possible only).

II. All wheat are rice – False (Because, only some rice are wheat is given, so all wheat being rice is not certain).

So, Neither conclusion I nor II follow.

Hence, the correct answer is "Option 3".

The logic followed here is:

thus, "858" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

The logic followed here is:

thus, "144" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Family Chart:

Decoding the Symbols:

Given: ‘O + P @ T - G # V’

It means: O is the Sister of P, P is the Brother of T, T is the Wife of G, G is the Father of V.

 

So, O is the Mother's sister of V.

Hence, the correct answer is "Option 4".

BODMAS Table: 

Given equation: 21 B 16 C 20 D 26 A 2 = ?

Now, if ‘A’ stands for ‘÷’, ‘B’ stands for ‘×’, ‘C’ stands for ‘+’ and ‘D’ stands for ‘−’, then:

⇒ 21 × 16 + 20 − 26 ÷ 2 = ?

⇒ 336 + 20 − 13 = ?

⇒ 356 − 13 = ?

⇒ 343

Hence, the correct answer is "Option 3".

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

Now,

Even numbers which are immediately followed by odd numbers.

So, Needed pattern is: Even number − Odd number.

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

There are 2 even numbers which are immediately followed by odd numbers.

Hence, the correct answer is 'Option 3'.

The logic followed here is:

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

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Rearrange letters in a specific pattern:

First word → second word (letters rearranged in a fixed order)

Second word → third word (letters rearranged in another fixed order)

Given,

MIST → TIMS → ISMT

HOPE → EOHP → OPHE

Now, checking the options:

Option 1) PUTS → UPTS → SUTP

Option 2) BUSY → YUBS → USBY

Option 3) WIDE → EIWD → DIWE

Option 4) NUMB → NMUB → BMUN

Hence, the correct option is "Option 2".