RRB Group D Official Paper (Held On: 01 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is Larger Amplitude.

Key Points

• Amplitude is directly related to the loudness of a sound. A louder sound corresponds to a sound wave with a larger amplitude. Hence, the correct answer is Larger Amplitude.
• The frequency of the sound wave determines its pitch and does not change in this scenario, as the question specifies that both sounds have the same frequency.
• The wavelength of a sound wave is inversely proportional to its frequency, but since the frequency is constant in this case, the wavelength remains unchanged.
• The speed of sound depends on the medium through which the sound travels and is not affected by the loudness or amplitude of the sound.
• Thus, a louder sound wave is characterized by a larger amplitude, which results in higher energy being transmitted through the medium.

Additional Information

• Amplitude:
• The amplitude of a sound wave refers to the maximum displacement of particles in the medium from their equilibrium position.
• A higher amplitude corresponds to a louder sound, while a lower amplitude corresponds to a softer sound.
• Amplitude is measured in terms of decibels (dB) when describing the loudness of a sound.
• Frequency and Pitch:
• Frequency is the number of oscillations or cycles of a wave per second and is measured in Hertz (Hz).
• The pitch of a sound is determined by its frequency: higher frequency results in a higher pitch, while lower frequency results in a lower pitch.
• In this question, frequency is constant, so the pitch does not change.
• Wavelength:
• The wavelength of a sound wave is the distance between two consecutive crests or troughs.
• It is inversely related to the frequency: higher frequency results in a shorter wavelength, and lower frequency results in a longer wavelength.
• Since the frequency remains the same, the wavelength also remains constant in this case.
• Speed of Sound:
• The speed of sound depends on the properties of the medium, such as its density and elasticity.
• In air, the speed of sound at room temperature is approximately 343 meters per second (m/s).
• The speed of sound is independent of the amplitude or loudness of the sound wave.

Important Points

• Loudness and Energy: The loudness of a sound is related to the energy carried by the sound wave. A wave with a higher amplitude carries more energy, which makes it louder.
• Sound Intensity: The intensity of a sound wave is proportional to the square of its amplitude. This means that even a small increase in amplitude can result in a significant increase in loudness.
• Perception of Loudness: The human ear perceives loudness on a logarithmic scale. A 10 dB increase in sound level is perceived as approximately twice as loud.
• Applications: Understanding amplitude and loudness is crucial in fields such as audio engineering, acoustics, and noise control.

The correct answer is Different colours of light travel at different speeds in glass..

Key Points

• Dispersion of white light occurs when white light passes through a glass prism. It splits into its constituent colors, forming a spectrum (VIBGYOR: Violet, Indigo, Blue, Green, Yellow, Orange, Red).
• This phenomenon is due to the fact that different colors of light travel at different speeds in glass. Each color has a different wavelength, and the refractive index of glass varies with the wavelength of light.
• Violet light bends the most, while red light bends the least. This is because violet has the shortest wavelength, and red has the longest wavelength among the visible spectrum.
• The separation of white light into its constituent colors is referred to as dispersion.
• The prism's geometry and material properties enhance this effect by refracting light at different angles, ensuring the colors become distinct and visible to the observer.

Additional Information

• Refraction:
• Refraction is the bending of light as it passes from one medium to another with a different refractive index, such as air to glass or water.
• In the case of a prism, the change in speed of light in glass compared to air causes the light to bend and split into different colors.
• Refractive Index:
• The refractive index of a material determines how much light bends when entering the material. It is wavelength-dependent, which is why light splits into colors.
• The refractive index is higher for shorter wavelengths (violet) and lower for longer wavelengths (red).
• Visible Spectrum:
• The visible spectrum consists of colors ranging from violet (shortest wavelength, ~400 nm) to red (longest wavelength, ~700 nm).
• The range of wavelengths determines the color of light perceived by the human eye.
• Applications of Dispersion:
• Dispersion is used in spectroscopy to analyze the composition of light sources.
• It is also the principle behind the formation of rainbows, where water droplets act as prisms dispersing sunlight.
• Important Experiments:
• Isaac Newton demonstrated the dispersion of light through a prism in the 17th century, proving that white light is a mixture of different colors.

Important Points

• Statement 1: "The prism absorbs some colors." This is incorrect because the prism only bends light but does not absorb colors.
• Statement 2: "Different colors of light travel at different speeds in glass." This is correct. The varying speed of light in glass causes dispersion.
• Statement 3: "White light contains only one color." This is incorrect. White light is a combination of all visible spectrum colors.
• Statement 4: "Different colors of light travel at the same speed in glass." This is incorrect. Different colors have different speeds due to varying refractive indices.

The correct answer is Thumb.

Key Points

• Fleming’s Left-Hand Rule is a visual tool used in electromagnetism to determine the directions of force, magnetic field, and current in devices such as electric motors.
• It states that if you stretch out your thumb, index finger, and middle finger perpendicular to each other:
• Thumb indicates the direction of the force or motion (correct answer).
• Index finger represents the direction of the magnetic field.
• Middle finger shows the direction of the current.
• The rule is particularly useful for understanding the working of an electric motor where electromagnetic forces interact to create motion.
• It is applicable in scenarios where a conductor carrying current is placed in a magnetic field, resulting in motion due to the Lorentz force.
• Fleming’s Left-Hand Rule is different from the Right-Hand Rule, which is used for generators.

Additional Information

• Electric Motors:
• Electric motors convert electrical energy into mechanical energy through electromagnetic interactions.
• The key components include a rotor, stator, commutator, and brushes, which work together to produce motion.
• Fleming’s Left-Hand Rule helps engineers design motors by predicting the direction of force on the rotor.
• Magnetic Field:
• The magnetic field is a region around a magnet or current-carrying conductor where magnetic forces can be felt.
• It is represented by magnetic field lines, which indicate the direction and strength of the field.
• In the Left-Hand Rule, the index finger indicates the direction of these magnetic field lines.
• Current:
• Electric current is the flow of electrons through a conductor.
• It is represented by the middle finger in Fleming's Left-Hand Rule.
• In electric motors, current interacts with the magnetic field to produce force and motion.
• Applications of Fleming’s Left-Hand Rule:
• Used in designing and understanding electric motors, such as DC motors and induction motors.
• Applicable in railguns, loudspeakers, and other devices where electromagnetic forces are involved.
• Helps in visualizing the interaction of current and magnetic fields in practical scenarios.

Important Points

• Lorentz Force: The force acting on a current-carrying conductor in a magnetic field is called the Lorentz force. It is the basis for Fleming’s Left-Hand Rule.
• Difference Between Left and Right-Hand Rule: The Left-Hand Rule is used for motors, while the Right-Hand Rule is applicable to generators.
• Historical Context: The rule was named after John Ambrose Fleming, who introduced it in the early 20th century.
• Visualization Tip: Always ensure the thumb, index finger, and middle finger are perpendicular to each other while applying the rule.

The correct answer is four times the original value.

Key Points

• Kinetic Energy (KE) is given by the formula: KE = (1/2)mv², where m is the mass of the body and v is its velocity.
• When the velocity v is doubled, the new velocity becomes 2v.
• Substituting the new velocity into the formula, the new kinetic energy becomes: KE' = (1/2)m(2v)² = (1/2)m(4v²) = 4 × (1/2)mv².
• This shows that the kinetic energy increases by a factor of 4 when the velocity is doubled.
• Hence, the kinetic energy becomes four times the original value when the velocity is doubled.

Additional Information

• Kinetic Energy:
• Kinetic Energy is a type of mechanical energy associated with a body in motion. It depends on the mass and velocity of the body.
• It is a scalar quantity and is always positive.
• The SI unit of Kinetic Energy is the joule (J).
• Example: A moving car, a flying airplane, and a rolling ball all have kinetic energy.
• Effect of Velocity on Kinetic Energy:
• Kinetic Energy is directly proportional to the square of the velocity of the body. This means that a small change in velocity has a significant impact on kinetic energy.
• For instance, if velocity is tripled, the kinetic energy increases by a factor of 9 (3²).
• Similarly, reducing the velocity to half will decrease the kinetic energy to one-fourth of its original value.
• Relation with Mass:
• Kinetic Energy is directly proportional to the mass of the body. If the mass is doubled, the kinetic energy also doubles, assuming velocity remains constant.
• This implies that heavier objects moving at the same speed have more kinetic energy than lighter ones.
• Applications:
• Transport: Kinetic Energy is critical in understanding vehicle motion, fuel efficiency, and safety measures like braking systems.
• Energy Generation: Hydropower plants use the kinetic energy of flowing water to generate electricity.
• Sports: In games like cricket or football, the kinetic energy of the ball determines its speed and impact.
• Important Facts:
• The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy.
• Kinetic Energy is a component of mechanical energy, which also includes potential energy.
• In a closed system with no external forces, the total mechanical energy (kinetic + potential) remains conserved.

The correct answer is the solute particles do not settle down when left undisturbed.

Key Points

• A solution is considered stable when the solute particles are uniformly distributed throughout the solvent and do not settle at the bottom when left undisturbed. This ensures the solution remains homogeneous over time.
• In a stable solution, the solute particles are dissolved at the molecular or ionic level in the solvent, forming a single-phase system (e.g., salt dissolved in water).
• Option 1 is incorrect as solute particles floating on the surface indicate incomplete dissolution, leading to an unstable solution.
• Option 2 is incorrect because solute particles settling down when left undisturbed indicates it is a suspension, not a solution.
• Option 3 is incorrect as the formation of a separate layer signifies immiscibility, often seen in mixtures like oil and water.
• Stable solutions are typical of true solutions such as sugar in water or alcohol in water. These solutions are characterized by transparency and uniform composition.

Additional Information

• Types of Solutions:
• True Solution: Homogeneous mixture where solute particles are less than 1 nm in size (e.g., sugar in water).
• Colloidal Solution: Heterogeneous mixture where solute particles range between 1-100 nm and do not settle down (e.g., milk).
• Suspension: Heterogeneous mixture where solute particles are larger than 100 nm and settle down over time (e.g., sand in water).
• Factors Affecting Stability of a Solution:
• Temperature: Increasing temperature often increases solubility, enhancing stability.
• Nature of Solvent and Solute: Polar solvents dissolve polar solutes, while nonpolar solvents dissolve nonpolar solutes.
• Concentration: Over-saturation can lead to precipitation, reducing stability.
• Examples of Stable Solutions:
• Salt dissolved in water.
• Alcohol mixed with water.
• Vinegar (acetic acid in water).
• Applications:
• Pharmaceutical Industry: Preparation of syrups and injectables.
• Food Industry: Beverages and flavoring solutions.
• Chemical Industry: Solutions for reactions and processes.

Important Points

• True Solution Characteristics:
• Particle size is less than 1 nm.
• Does not scatter light (Tyndall effect absent).
• Cannot be separated by filtration.
• Stable and homogeneous in nature.
• Colloidal vs. True Solutions:
• Colloids exhibit the Tyndall effect, while true solutions do not.
• Colloidal particles are larger than those in true solutions.
• Unstable Mixtures: Suspensions and immiscible liquids like oil and water are examples of unstable mixtures, where solute and solvent do not form a homogeneous system.

The correct answer is carbon dioxide and water.

Key Points

• When a hydrocarbon is burnt in the presence of an excess of air (oxygen), it undergoes a complete combustion reaction.
• The products of complete combustion of a hydrocarbon are always carbon dioxide (CO₂) and water (H₂O). This is because all carbon atoms in the hydrocarbon are oxidized to form CO₂, and hydrogen atoms form H₂O.
• The general chemical equation for the combustion of a hydrocarbon (e.g., C_xH_y) is: C_xH_y + O₂ → xCO₂ + (y/2)H₂O.
• This process is exothermic, meaning it releases a significant amount of energy in the form of heat and light.
• Examples of hydrocarbons include methane (CH₄), ethane (C₂H₆), propane (C₃H₈), and butane (C₄H₁₀), all of which produce CO₂ and H₂O upon complete combustion.
• Hence, the correct answer is carbon dioxide and water.

Additional Information

• Incomplete Combustion:
• If the hydrocarbon burns in a limited supply of oxygen, the reaction results in incomplete combustion.
• Incomplete combustion produces carbon monoxide (CO) or even elemental carbon (soot) along with water.
• This occurs because there is not enough oxygen to fully oxidize the carbon atoms to CO₂.
• Example: CH₄ + O₂ → CO + 2H₂O (incomplete combustion of methane).
• Environmental Impact:
• Complete combustion of hydrocarbons produces CO₂, a significant greenhouse gas that contributes to global warming and climate change.
• Incomplete combustion, on the other hand, releases CO, which is a toxic gas that can cause severe health issues.
• Examples of Hydrocarbons and Their Combustion:
• Methane (CH₄): CH₄ + 2O₂ → CO₂ + 2H₂O.
• Propane (C₃H₈): C₃H₈ + 5O₂ → 3CO₂ + 4H₂O.
• Butane (C₄H₁₀): 2C₄H₁₀ + 13O₂ → 8CO₂ + 10H₂O.
• Applications of Combustion:
• Combustion of hydrocarbons is widely used as a source of energy in domestic (e.g., cooking gas), industrial (e.g., power plants), and transportation (e.g., fuels like petrol and diesel) sectors.
• Fuels like natural gas, LPG, gasoline, and kerosene are commonly used for their ability to produce significant heat upon combustion.
• Types of Hydrocarbons:
• Alkanes: Saturated hydrocarbons (e.g., methane, ethane).
• Alkenes: Unsaturated hydrocarbons with one or more double bonds (e.g., ethene, propene).
• Alkynes: Unsaturated hydrocarbons with one or more triple bonds (e.g., ethyne).

The correct answer is 2, 7, 4, 6.

Key Points

• The reaction is: C₂H₆ + O₂ → CO₂ + H₂O.
• To balance a chemical equation, the number of atoms of each element must be the same on both sides of the equation.
• Start by balancing the carbon (C) atoms. Ethane (C₂H₆) has 2 carbon atoms, so we place a coefficient of 2 for CO₂ on the product side:
• C₂H₆ + O₂ → 2CO₂ + H₂O
• Next, balance the hydrogen (H) atoms. Ethane has 6 hydrogen atoms, and water (H₂O) has 2 hydrogen atoms per molecule. Therefore, place a coefficient of 3 for H₂O:
• C₂H₆ + O₂ → 2CO₂ + 3H₂O
• Finally, balance the oxygen (O) atoms. There are 4 oxygen atoms in 2CO₂ and 3 oxygen atoms in 3H₂O, totaling 7 oxygen atoms on the product side. To balance the oxygen, place a coefficient of 7/2 for O₂ on the reactant side:
• C₂H₆ + (7/2)O₂ → 2CO₂ + 3H₂O
• Multiply through by 2 to eliminate the fraction:
• 2C₂H₆ + 7O₂ → 4CO₂ + 6H₂O
• Thus, the coefficients are 2, 7, 4, and 6.

Additional Information

• Law of Conservation of Mass:
• In a chemical reaction, mass is neither created nor destroyed.
• The total mass of reactants must be equal to the total mass of products.
• Balancing Chemical Equations:
• A balanced equation ensures the same number of atoms of each element on both sides of the equation.
• It is essential for correctly representing chemical reactions and obeying the Law of Conservation of Mass.
• Combustion Reactions:
• Combustion involves the reaction of a substance with oxygen to produce energy in the form of heat and light.
• Hydrocarbon combustion typically produces carbon dioxide (CO₂) and water (H₂O).
• Steps to Balance Equations:
• Identify the reactants and products.
• Count the number of atoms for each element on both sides of the equation.
• Adjust coefficients to balance each element.
• Ensure the coefficients are the smallest possible integers.

Important Points

• Hydrocarbons:
• Hydrocarbons consist of hydrogen and carbon atoms.
• Examples include methane (CH₄), ethane (C₂H₆), and propane (C₃H₈).
• They are classified as alkanes, alkenes, and alkynes based on the type of bonds between carbon atoms.
• Applications of Combustion Reactions:
• Combustion reactions are used in energy production, such as in internal combustion engines and power plants.
• They are also fundamental to heating systems and industrial processes.
• Oxidation Process:
• Combustion is a type of oxidation reaction where oxygen reacts with a substance.
• Oxidation plays a significant role in many chemical and biological processes.

The correct answer is Newton's First Law of Motion.

Key Points

• Newton's First Law of Motion, also known as the Law of Inertia, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
• When the bus suddenly stops, the body of the person inside the bus tends to continue moving forward due to inertia, as no external force initially acts directly on their body to stop this motion.
• The phenomenon occurs because the lower part of the person’s body comes to rest due to contact with the bus floor, while the upper part continues moving forward until balanced by external forces like friction or a physical obstruction.
• This tendency of the body to resist changes to its state of motion directly illustrates Newton's First Law. Hence, this law is responsible for the phenomenon described in the question.
• Other laws of motion like Newton's Second and Third Laws are not directly applicable here as they deal with acceleration due to force and action-reaction pairs, respectively.

Additional Information

• Newton's First Law of Motion:
• It is also referred to as the Law of Inertia.
• The concept of inertia was first introduced by Galileo Galilei and later formulated as a law by Isaac Newton.
• Inertia is the property of an object to resist changes in its state of motion or rest.
• Examples: A book on a table remains stationary unless pushed, and a ball rolling on a smooth surface continues moving unless stopped by friction or another force.
• Newton's Second Law of Motion:
• It states that the rate of change of momentum of an object is directly proportional to the external force applied to it and occurs in the direction of the force.
• Mathematical Expression: F = ma, where F is force, m is mass, and a is acceleration.
• This law explains how force and acceleration are related but does not directly apply to the scenario in the question.
• Newton's Third Law of Motion:
• It states that for every action, there is an equal and opposite reaction.
• Example: When you push against a wall, the wall pushes back with an equal force in the opposite direction.
• This law focuses on force pairs rather than the inertia described in the question.
• Law of Conservation of Momentum:
• This principle states that the total momentum of a closed system remains conserved unless acted upon by external forces.
• It is mathematically expressed as: m₁v₁ + m₂v₂ = constant.
• While this law is applicable in collision or explosion scenarios, it is not relevant to the phenomenon described in the question.

Important Points

• Real-life Applications of Newton's First Law:
• Seat belts in vehicles are designed to prevent passengers from moving forward abruptly during sudden stops, counteracting inertia.
• Objects placed on a moving vehicle tend to slide forward when the vehicle halts suddenly due to inertia.
• A ball rolling on the ground stops eventually because friction acts as an external force to counteract its motion.
• Historical Context:
• Newton's Laws of Motion were published in 1687 in his work Philosophiæ Naturalis Principia Mathematica.
• These laws form the foundation of classical mechanics and are widely applied in engineering, physics, and everyday phenomena.

The correct answer is Sweet Potato.

Key Points

• The roots of the Sweet Potato have the ability to give rise to new plants. This process occurs through a phenomenon known as vegetative propagation.
• Sweet Potato roots are modified adventitious roots that store food and can develop shoots, which eventually grow into a new plant under suitable conditions.
• Unlike other plants like Maize, Potato, and Gram, which propagate through seeds, tubers, or other parts, the vegetative propagation of sweet potatoes is specifically through roots.
• In vegetative propagation, a part of the plant other than seeds is used to grow new plants. This is commonly seen in crops like Sweet Potato, Dahlia, and Turmeric.
• This method is widely used in agriculture and gardening as it ensures the propagation of plants with identical genetic properties.

Additional Information

• Sweet Potato:
• It is a dicotyledonous plant belonging to the family Convolvulaceae.
• Sweet potatoes are rich in carbohydrates, vitamins (especially Vitamin A), and dietary fiber, making them an important food crop.
• They are primarily grown in tropical and subtropical regions, with India, China, and the United States being major producers.
• They are propagated through vegetative means such as root slips or cuttings.
• Vegetative Propagation:
• This is a form of asexual reproduction where new plants are produced from the roots, stems, or leaves of the parent plant.
• Examples include Sweet Potato (roots), Potato (stem tubers), Ginger (rhizomes), and Bryophyllum (leaves).
• This method is advantageous as it is faster, bypasses the seed stage, and retains the parental plant's genetic characteristics.
• Comparison of Options:
• Maize: Propagates through seeds and cannot regenerate from roots.
• Potato: Propagates through stem tubers, not roots.
• Gram: Propagates through seeds only.
• Sweet Potato: Propagates through roots, which can generate new plants, making it the correct answer.
• Agricultural Importance:
• Vegetative propagation ensures uniform crop quality, which is vital in commercial farming.
• It also allows propagation of plants that produce few or no viable seeds, such as Sweet Potato.
• Additionally, it enables cultivation in regions where seed germination is difficult due to climatic conditions.

The correct answer is It increases the chance of survival under changing environmental conditions..

Key Points

• Variation within a population is essential because it provides the genetic diversity necessary for adaptation to changing environmental conditions, ensuring survival.
• Genetic variation allows organisms to develop traits that can help them cope with challenges such as climate change, new predators, or diseases.
• Populations with greater variation are less likely to face extinction, as some individuals are more likely to survive adverse conditions and pass their genes to the next generation.
• Variation is the basis of natural selection, a process described by Charles Darwin, where advantageous traits become more prevalent over generations.
• Environmental changes, such as drought, temperature shifts, or habitat loss, act as selective pressures. Populations with higher genetic variation have a better chance to persist and evolve in response to these pressures.

Additional Information

• Importance of Variation:
• Variation arises due to mutations, genetic recombination during reproduction, and migration of individuals between populations.
• It is also influenced by environmental factors such as temperature, food availability, and geographical barriers.
• Examples include the development of antibiotic resistance in bacteria and the variation in beak sizes of Darwin’s finches.
• Option-Specific Details:
• Option 2: "It ensures that all individuals are identical"—incorrect because identical individuals lack genetic diversity, making the population vulnerable to environmental changes.
• Option 3: "It always gives instant results within the same generation"—incorrect as evolutionary changes due to variation occur over multiple generations, not instantly.
• Option 4: "It helps organisms reproduce faster"—incorrect because variation does not directly affect reproduction speed but influences adaptability and survival.
• Natural Selection:
• Natural selection operates on variation within populations, favoring traits that enhance survival and reproduction.
• Example: Peppered moths in industrial regions evolved darker coloration to blend with soot-covered trees, demonstrating survival advantages due to variation.
• Relevance to Human Evolution:
• Genetic variation in humans has led to adaptations such as sickle-cell anemia resistance to malaria and lactose tolerance in populations that domesticated cattle.
• Understanding variation aids in medical research, such as developing treatments for genetic disorders and improving crop resilience.

Important Points

• Role of Mutation: Mutations are random changes in DNA that introduce new genetic variations in populations, providing raw material for evolution.
• Genetic Drift: Random changes in allele frequencies in small populations can lead to significant evolutionary changes over time.
• Examples in Conservation: Conservation efforts focus on maintaining genetic diversity in endangered species to enhance their adaptability and survival.
• Artificial Selection: Humans use variation for selective breeding in agriculture and livestock to enhance desirable traits such as higher yield or disease resistance.

The correct answer is Inter-cropping.

Key Points

• Inter-cropping is the agricultural practice of growing two or more crops simultaneously on the same piece of land in a defined spatial arrangement.
• This technique ensures the efficient utilization of resources such as light, water, and nutrients, improving productivity and sustainability.
• Crops are planted in a systematic pattern, such as alternate rows or blocks, to avoid competition and maximize benefits.
• Common examples of inter-cropping include growing legumes with cereals (e.g., maize and beans) or combining shallow-rooted crops with deep-rooted crops.
• It helps reduce the risk of complete crop failure due to pests or diseases, as multiple crops are grown together, offering diversity.
• Farmers adopt this method to enhance soil fertility, particularly by incorporating nitrogen-fixing crops like legumes into the cropping system.

Additional Information

• Aeroponics:
• A method of growing plants without soil, where plant roots are suspended in the air and sprayed with a nutrient-rich solution.
• It is used primarily in controlled environments like greenhouses for crops such as lettuce, herbs, and strawberries.
• Aeroponics minimizes water use and enhances nutrient absorption but is not related to inter-cropping.
• Hydroponics:
• A soil-less cultivation technique where plants grow in water containing dissolved nutrients.
• Hydroponics is ideal for urban farming and indoor agriculture but does not involve planting multiple crops together in a systematic pattern.
• Examples include growing spinach, tomatoes, and cucumbers in hydroponic systems.
• Monocropping:
• Refers to cultivating a single crop on a piece of land year after year, leading to soil degradation and nutrient depletion.
• Unlike inter-cropping, monocropping lacks diversity and is more susceptible to pests and diseases.
• Crops like wheat, rice, and corn are commonly grown using this technique.
• Benefits of Inter-cropping:
• Improves yield stability by diversifying crop production.
• Enhances soil fertility through nitrogen fixation by legumes.
• Minimizes the spread of pests and diseases due to crop diversity.
• Promotes sustainable agriculture by efficient use of natural resources.

Important Points

• Inter-cropping is widely practiced in developing countries like India due to its economic and ecological benefits.
• It is a key component of integrated farming systems, which aim to maximize productivity while conserving resources.
• Examples of inter-cropping systems include the three sisters method (corn, beans, and squash) in Native American farming practices.
• Scientific studies have shown that inter-cropping can increase productivity by up to 20-30% compared to monocropping.
• Government programs like National Mission on Sustainable Agriculture promote inter-cropping as a climate-resilient farming practice.

The correct answer is Water pushes the swimmer forward.

Key Points

• When a swimmer pushes water backward, it is an application of Newton's Third Law of Motion, which states: "For every action, there is an equal and opposite reaction."
• The backward force exerted by the swimmer on the water is met with an equal and opposite reaction force exerted by the water on the swimmer, propelling the swimmer forward.
• The reaction force acts in the forward direction, enabling the swimmer to move through the water effectively.
• This principle is fundamental to swimming techniques, where the swimmer uses their arms and legs to push against the water, creating propulsion.
• Other forces, such as drag and buoyancy, also influence swimming, but the forward propulsion is solely due to the reaction force from the water.

Additional Information

• Newton's Laws of Motion:
• First Law (Law of Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
• Second Law (F = ma): The force acting on an object is equal to the mass of the object multiplied by its acceleration.
• Third Law: For every action, there is an equal and opposite reaction. This law is directly applicable to swimming.
• Buoyancy in Swimming:
• Buoyancy is the upward force exerted by water, counteracting the swimmer's weight and allowing them to float.
• This upward force is governed by Archimedes' Principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
• Drag Force in Swimming:
• As a swimmer moves through water, they experience resistance or drag force.
• Drag force depends on the swimmer's speed, body position, and surface area exposed to the water.
• Streamlined body positions can minimize drag, improving swimming efficiency.
• Applications of Newton's Third Law:
• Propulsion in swimming.
• Rocket propulsion, where gases are expelled backward to propel the rocket forward.
• Recoil of firearms, where the gun exerts a backward force when the bullet is fired forward.
• Examples in Daily Life:
• Walking: When you push the ground backward with your foot, the ground exerts an equal and opposite force propelling you forward.
• Cycling: The wheels push the ground backward, and the reaction force moves the cycle forward.

Important Points

• Action-Reaction Pair: The force exerted by the swimmer on the water (action) and the force exerted by the water on the swimmer (reaction) form an action-reaction pair.
• Role of Muscles: Efficient swimming requires strong muscles to generate sufficient backward force against the water.
• Streamlining in Swimming: To reduce drag, swimmers adopt streamlined body positions, such as keeping the body straight and minimizing unnecessary movements.
• Swimming Strokes: Different strokes, such as freestyle, breaststroke, and butterfly, utilize Newton's Third Law to generate forward motion through specific arm and leg movements.

The correct answer is Ribosome.

Key Points

• Rough Endoplasmic Reticulum (RER) appears rough under a microscope due to the presence of ribosomes attached to its surface.
• Ribosomes are responsible for protein synthesis, a critical cellular function essential for growth, repair, and maintaining cellular processes.
• The RER is primarily involved in the synthesis of proteins that are either secreted by the cell, incorporated into the cell membrane, or sent to lysosomes.
• These ribosomes are bound to the cytosolic surface of the RER and help in translating messenger RNA (mRNA) into polypeptide chains.
• The proteins produced by the RER are often modified and folded within its lumen before being transported to their final destinations.

Additional Information

• Ribosomes:
• Ribosomes are small organelles composed of ribosomal RNA (rRNA) and proteins.
• They exist in two forms: free ribosomes (floating in the cytoplasm) and bound ribosomes (attached to the RER).
• Free ribosomes synthesize proteins that function within the cytoplasm, while bound ribosomes produce proteins destined for secretion or for use in the cell membrane.
• Functions of Rough Endoplasmic Reticulum:
• It plays a critical role in the synthesis and modification of proteins.
• The RER is involved in the addition of carbohydrate chains to proteins, forming glycoproteins.
• It provides a pathway for the transport of synthesized proteins to the Golgi apparatus for further processing and packaging.
• Comparison with Smooth Endoplasmic Reticulum (SER):
• SER lacks ribosomes and is involved in lipid synthesis, detoxification, and calcium storage.
• Unlike the RER, the SER does not contribute directly to protein synthesis.
• Other Organelles:
• Centrosome: Involved in organizing microtubules and plays a vital role during cell division. Not related to protein synthesis.
• Lysosome: Contains digestive enzymes to break down cellular waste. Not related to the RER's rough appearance.
• Peroxisome: Responsible for breaking down fatty acids and detoxifying harmful substances. Not linked to the RER's function or appearance.
• Important Points:
• The RER is prominent in cells specialized for protein secretion, such as pancreatic cells (produce digestive enzymes) and plasma cells (produce antibodies).
• The discovery of ribosomes and their role in protein synthesis was a landmark finding by George Palade, who received the Nobel Prize in Physiology or Medicine in 1974.
• The ribosomes attached to the RER are 70S in prokaryotes and 80S in eukaryotes.

The correct answer is Heart.

Key Points

• The structural organization of a cell consists of essential components such as the cytoplasm, mitochondria, cell membrane, nucleus, and cell wall (in plants and some microorganisms). These parts work together to maintain cellular function.
• The cytoplasm is the gel-like substance inside the cell where most cellular activities occur, including metabolic pathways like glycolysis. It is a key structural component of the cell.
• Mitochondria, known as the "powerhouse of the cell," are organelles responsible for energy production through the process of cellular respiration. They are integral to the cell's structure and function.
• The cell wall is present in plant cells, fungi, and some bacteria, providing structural support and protection. Animal cells lack a cell wall but have a flexible cell membrane.
• The heart, however, is an organ of the circulatory system and not a structural part of a cell. It pumps blood throughout the body and is unrelated to the cellular organization. Thus, the heart is not a component of cell structure.
• Hence, the correct answer is Heart.

Additional Information

• Components of a cell:
• Nucleus: The nucleus contains genetic material (DNA) and regulates cellular activities like growth, metabolism, and reproduction.
• Cell membrane: It is the outer boundary of the cell that controls the entry and exit of substances, maintaining the internal environment of the cell.
• Endoplasmic reticulum (ER): The ER is involved in protein and lipid synthesis. It is divided into rough ER (with ribosomes) and smooth ER (without ribosomes).
• Golgi apparatus: It is responsible for modifying, sorting, and packaging proteins and lipids for secretion or internal use.
• Ribosomes: These are the sites of protein synthesis within the cell. Ribosomes can be free-floating in the cytoplasm or attached to the rough ER.
• Types of cells:
• Prokaryotic cells: These cells lack a nucleus and membrane-bound organelles (e.g., bacteria).
• Eukaryotic cells: These cells have a nucleus and membrane-bound organelles (e.g., plant and animal cells).
• Importance of cellular components:
• Cellular components work together to maintain homeostasis, facilitate energy production, and support overall cellular functions.
• They contribute to processes such as growth, repair, reproduction, and environmental response.
• The heart as an organ:
• The heart is a muscular organ responsible for pumping oxygenated and deoxygenated blood throughout the body.
• It is part of the circulatory system and plays no role in the structural organization or functional components of a cell.
• The heart is composed of tissues and cells, including cardiomyocytes (heart muscle cells), but it is a system-level structure.

Important Points

• Structural differences between plant and animal cells:
• Plant cells have a cell wall, chloroplasts, and large central vacuoles, which are absent in animal cells.
• Animal cells have centrioles and smaller vacuoles compared to plant cells.
• Cell specialization:
• Cells specialize to perform specific functions in multicellular organisms (e.g., nerve cells, muscle cells, and skin cells).
• This specialization ensures efficient functioning and division of labor within the body.
• Cellular organelles and diseases:
• Dysfunction of cellular organelles can lead to diseases such as mitochondrial disorders or lysosomal storage diseases.
• Understanding cell biology helps in developing treatments for such conditions.

The correct answer is A strip of impure metal to be refined.

Key Points

• Electrolytic refining is a process used to purify metals using the principles of electrolysis. It is commonly used for refining metals such as copper, zinc, silver, gold, and nickel.
• In this process, the impure metal is used as the anode, while a thin sheet of the pure metal is used as the cathode. Hence, the statement “A strip of impure metal to be refined is used as the anode” is correct.
• The electrolyte used is usually a solution of a soluble salt of the same metal. For example, during the refining of copper, the electrolyte is typically acidified copper sulfate solution (CuSO₄).
• When electric current is passed through the electrolyte, the impure metal at the anode dissolves into the solution as positively charged metal ions.
• These metal ions move through the electrolyte and get deposited on the pure metal cathode, forming a thicker layer of highly pure metal.
• The impurities present in the impure metal either remain dissolved in the electrolyte or settle down at the bottom of the cell as anode mud.
• Therefore, the anode must be made of the impure metal so that it can dissolve during electrolysis and allow the purification process to occur.
• Hence, the correct answer is A strip of impure metal to be refined.

Additional Information

• Working Principle of Electrolytic Refining:
• The process is based on the principle that metal ions migrate in an electric field and get deposited on the electrode where reduction occurs.
• At the anode, the impure metal undergoes oxidation, releasing metal ions into the solution.
• At the cathode, these ions gain electrons and are deposited as pure metal.
• Example: Refining of Copper
• In the electrolytic refining of copper, the impure copper block acts as the anode and a thin sheet of pure copper acts as the cathode.
• The electrolyte used is acidified copper sulphate solution.
• When current passes, Cu → Cu²⁺ + 2e⁻ occurs at the anode.
• At the cathode, the reaction Cu²⁺ + 2e⁻ → Cu results in the deposition of pure copper.
• Nature of Impurities:
• Less reactive impurities such as gold (Au), silver (Ag), and platinum (Pt) settle down as anode mud.
• More reactive impurities like iron (Fe) and zinc (Zn) dissolve in the electrolyte.
• These by-products are sometimes recovered because precious metals like gold and silver have significant commercial value.

Important Points

• Industrial Applications:
• Electrolytic refining is widely used in industries to obtain metals with purity levels above 99.9%.
• This method is especially important in the electronics industry where highly pure copper is required for manufacturing electrical wires and circuits.
• Advantages of Electrolytic Refining:
• Produces very high purity metals.
• Allows recovery of valuable metals such as gold, silver, and platinum from anode mud.
• It is an efficient and widely adopted technique in modern metallurgical industries.

The correct answer is Only a and b.

Key Points

• Law of Constant Proportions: This law, also known as Proust's Law, states that in a chemical compound, the elements are always present in definite proportions by mass, irrespective of the source or preparation method.
• Statement a: "In water, the ratio of the mass of hydrogen to the mass of oxygen is always 1:8." This is correct since water (H₂O) always contains hydrogen and oxygen in the ratio of 2:16 by mass, which simplifies to 1:8. Hence, this statement adheres to the law of constant proportions.
• Statement b: "The ratio of carbon and oxygen in carbon dioxide is always 3:8 by mass." This is correct because in CO₂, carbon (12 g/mol) and oxygen (32 g/mol) have a fixed mass ratio of 12:32, which simplifies to 3:8. This confirms the law of constant proportions.
• Statement c: "In ammonia, nitrogen and hydrogen are always present in the ratio 14:1 by mass." This is incorrect as ammonia (NH₃) consists of nitrogen (14 g/mol) and hydrogen (3 g/mol), giving a mass ratio of 14:3, not 14:1.
• Hence, only statements a and b are correct.

Additional Information

• Concept of Law of Constant Proportions:
• Proposed by Joseph Proust in 1799, this law is a fundamental principle in chemistry.
• It applies to pure chemical compounds and states that the composition of a compound is independent of its source.
• The law is significant in understanding the stoichiometry of compounds and chemical reactions.
• Exceptions to the Law:
• Non-stoichiometric compounds: Some compounds, like ferrous oxide (FeO), do not follow this law as their composition varies due to defects in their crystal structure.
• Isotopic variations: When isotopes of an element are present, the mass ratio may slightly differ due to their different atomic masses.
• Applications of the Law:
• Used in determining the chemical formula of compounds.
• Helps in verifying the purity of a compound by analyzing its elemental composition.
• Plays an essential role in the development of the modern atomic theory.
• Key Examples:
• Water (H₂O): Always contains hydrogen and oxygen in the mass ratio 1:8.
• Carbon dioxide (CO₂): Always contains carbon and oxygen in the mass ratio 3:8.
• Methane (CH₄): Always contains carbon and hydrogen in the mass ratio 3:1.

Important Points

• Law of Multiple Proportions: Distinct from the law of constant proportions, this law states that when two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element are in a simple whole-number ratio.
• Modern Atomic Theory: The law of constant proportions laid the groundwork for John Dalton's atomic theory, which explains the fixed composition of compounds based on the arrangement and combination of atoms.
• Analytical Techniques: Modern analytical tools like mass spectrometry and spectroscopy validate the law of constant proportions by accurately determining the elemental composition of compounds.

The correct answer is In the abdomen, on either side of the back bone.

Key Points

• The kidneys are located in the human body in the abdomen, specifically on either side of the vertebral column (backbone). This placement ensures their proximity to important blood vessels, such as the aorta and the inferior vena cava.
• Each kidney is situated in the retroperitoneal space, meaning they are positioned behind the peritoneum (a membrane lining the abdominal cavity). This unique location protects the kidneys from external injuries.
• The right kidney is slightly lower than the left kidney due to the presence of the liver on the right side of the body.
• The kidneys are bean-shaped organs, each measuring approximately 11-14 cm in length and weighing around 120-150 grams. These vital organs help in filtering blood and excreting waste products.
• The kidneys play a critical role in regulating the body's fluid balance, maintaining electrolyte levels, and ensuring proper blood pressure control through their hormone production.

Additional Information

• Functions of the Kidneys:
• The kidneys filter around 50 gallons of blood daily to remove waste products and excess fluids, which are then excreted as urine.
• They help maintain the body's acid-base balance, ensuring the pH levels of the blood remain stable.
• Kidneys produce hormones such as erythropoietin, which stimulates red blood cell production, and renin, which plays a role in blood pressure regulation.
• They also activate Vitamin D, which is essential for maintaining healthy bones and calcium levels in the body.
• Structure of the Kidneys:
• Each kidney is composed of millions of tiny filtering units called nephrons. Nephrons are responsible for filtering blood and forming urine.
• The kidneys have three main regions: the renal cortex (outer layer), the renal medulla (middle layer), and the renal pelvis (inner region where urine collects).
• Diseases Related to Kidneys:
• Chronic Kidney Disease (CKD): A gradual loss of kidney function over time, often caused by diabetes or high blood pressure.
• Kidney Stones: Hard mineral deposits formed in the kidneys, often leading to pain and difficulty in urination.
• Urinary Tract Infections (UTIs): Infections that can affect any part of the urinary system, including the kidneys.
• Interesting Facts:
• Humans can survive with just one kidney, as a single kidney is capable of performing all essential functions.
• The kidneys receive 20-25% of the total cardiac output, which highlights their importance in maintaining overall health.

Important Points

• The kidneys are part of the excretory system, which includes other organs such as the ureters, urinary bladder, and urethra.
• The term renal is often used to describe kidney-related conditions, functions, or anatomical structures (e.g., renal artery, renal failure).
• The kidneys are protected by layers of fat and connective tissue, which cushion them against mechanical shocks.
• Dialysis is a medical procedure used to artificially replicate kidney functions in individuals with renal failure.

The correct answer is Mass is constant and Weight is variable.

Key Points

• Mass is a measure of the amount of matter in an object. It is a scalar quantity and remains constant regardless of the object's location. This is because mass depends on the object's intrinsic properties and not on external factors like gravity. Hence, this statement is correct.
• Weight is the force exerted on an object due to gravity. It is a vector quantity and depends on the gravitational acceleration of the location. Weight can vary depending on the gravitational field strength. For example, an object’s weight on the Moon is less than its weight on Earth due to the Moon's weaker gravity. Hence, this statement is correct.
• The relationship between weight and mass is given by the formula: Weight = Mass × Gravitational acceleration (W = m × g). Since "g" (gravitational acceleration) varies with location, weight is not constant.
• Mass remains constant across the universe, while weight changes depending on the gravitational field strength. For instance, astronauts experience weightlessness in space due to the absence of significant gravity, but their mass remains unchanged.
• Thus, the statement "Mass is constant and Weight is variable" is the correct answer.

Additional Information

• Mass:
• Mass is measured in kilograms (kg) in the International System of Units (SI).
• It remains the same on Earth, the Moon, or any other celestial body.
• Mass is a fundamental property of matter and does not change with changes in gravitational force or location.
• Weight:
• Weight is measured in newtons (N) in the SI system, as it is a force.
• It depends on the local gravitational acceleration (e.g., Earth's gravitational acceleration is approximately 9.8 m/s²).
• On the Moon, the gravitational acceleration is about 1/6th of Earth's, so an object’s weight on the Moon is only about 1/6th of its weight on Earth.
• Examples of Mass vs. Weight:
• A person with a mass of 50 kg will have the same mass on Earth and the Moon.
• Their weight on Earth would be approximately 490 N (50 × 9.8), while on the Moon, it would be approximately 81.6 N (50 × 1.63).
• Gravitational Variations:
• Gravitational acceleration can vary even on Earth, depending on altitude and latitude. For example, "g" is slightly less at the equator compared to the poles.
• In space, where there is no significant gravitational pull, objects are in a state of weightlessness despite retaining their mass.

Important Points

• SI Units:
• Mass is measured in kilograms (kg).
• Weight is measured in newtons (N).
• Real-Life Applications:
• Weight is a crucial factor in fields like engineering, construction, and space exploration.
• Mass is essential in scientific research, chemistry, and physics as it determines an object's resistance to acceleration.
• Key Formula: Weight (W) = Mass (m) × Gravitational acceleration (g).

The correct answer is 1.45 × 10⁶ J.

Key Points

• The heat produced in an electric heater is calculated using Joule's Law of Heating: H = (V²/R) × t
• Given:
• Voltage, V = 220 V
• Resistance, R = 20 Ω
• Time, t = 10 minutes = 600 s
• Substituting the values:
  

H = (220²/20) × 600

= (48400/20) × 600

= 2420 × 600

= 1,452,000 J

= 1.452 × 10⁶ J

• Hence, the heat produced is 1.45 × 10⁶ J.

Additional Information

• Joule's Law of Heating:
• H = I²Rt
• H = VIt
• H = (V²/R)t
• All three expressions are equivalent.
• SI Unit of Heat Energy:
• The SI unit is the joule (J).
• Electric Power:
• P = VI = I²R = V²/R.
• For this heater, P = 2420 W.
• Applications of Joule Heating:
• Electric heaters
• Electric irons
• Toasters
• Electric kettles

The correct answer is Sum of atomic masses of all atoms in the formula unit of an ionic compound.

Key Points

• Formula Unit: A formula unit in chemistry refers to the simplest ratio of ions represented in an ionic compound. It highlights the composition of a compound without detailing its molecular structure.
• Definition: The formula unit mass is the sum of the atomic masses of all atoms in the formula unit of an ionic compound. This is the correct definition that matches the question's context.
• Examples: For sodium chloride (NaCl), the formula unit mass is calculated by adding the atomic mass of sodium (23) and chlorine (35.5), which equals 58.5.
• Difference from Molecular Mass: Unlike molecular compounds that have distinct molecules, ionic compounds like NaCl exist as crystal lattices. Hence, the term "formula unit" is used instead of "molecule."
• Relevance: Understanding formula unit mass is critical in stoichiometric calculations, especially when dealing with ionic compounds in chemical reactions.

Additional Information

• Atomic Mass: The mass of a single atom, typically measured in atomic mass units (amu). For example, the atomic mass of sodium (Na) is approximately 23 amu.
• Ionic Compounds: Ionic compounds are formed through the electrostatic attraction between positively charged cations and negatively charged anions. Examples include NaCl (sodium chloride), MgO (magnesium oxide), etc.
• Molecular Mass vs. Formula Unit Mass: Molecular mass is the sum of atomic masses in a molecule, while formula unit mass is used for ionic compounds and is calculated based on the empirical formula.
• Applications: Formula unit mass is essential in quantitative chemical analysis, such as calculating the amount of reactants or products in a chemical reaction involving ionic compounds.
• Key Difference: Ionic compounds do not form discrete molecules; instead, they form a continuous network or lattice structure. Hence, the term "formula unit" is used to simplify representation.

Important Points

• Option 1: "Mass of an atom" refers to the atomic mass of a single atom and is not equivalent to the formula unit mass. This option is incorrect.
• Option 2: "Mass of one molecule" is applicable to covalent compounds, not ionic compounds. Formula unit mass is used for ionic compounds. This option is incorrect.
• Option 3: "Sum of atomic masses of all atoms in the formula unit of an ionic compound" is the accurate definition of formula unit mass. This option is correct.
• Option 4: "Mass of one mole of a compound" refers to the molar mass, which is different from formula unit mass. This option is incorrect.

The correct answer is Parthenium.

Key Points

• Parthenium, commonly known as Congress Grass, is a highly invasive weed that causes significant harm to agricultural productivity and biodiversity.
• It competes with crops for essential nutrients, water, sunlight, and space, drastically reducing yields. Hence, its removal benefits farmers.
• Parthenium contains toxic compounds like parthenin, which are harmful to humans, animals, and other plants. It causes dermatitis and respiratory problems in humans and is toxic to livestock.
• This weed poses a threat to biodiversity by outcompeting native vegetation and reducing the availability of resources for pollinators and other beneficial organisms.
• Farmers and agricultural scientists recommend physical removal, biological control (using insects like Zygogramma bicolorata), and chemical herbicides to manage its spread.

Additional Information

• Characteristics of Parthenium:
• Belongs to the family Asteraceae.
• It is a fast-growing annual herbaceous weed that thrives in disturbed soils, roadsides, and agricultural fields.
• Originated in North and South America and was accidentally introduced to India in the 1950s through imported food grains.
• Produces a large number of seeds (up to 15,000 per plant), facilitating its rapid spread.
• Impact on Agriculture:
• Parthenium affects the growth and yield of major crops like wheat, rice, and maize.
• It also disrupts soil properties by altering the microbial composition, which impacts crop health.
• Control Measures:
• Biological Control: The beetle Zygogramma bicolorata is used to control Parthenium populations by feeding on its leaves and flowers.
• Chemical Control: Herbicides like glyphosate and 2,4-D are effective in managing Parthenium.
• Physical Removal: Manual uprooting of Parthenium before it flowers and sets seeds is recommended to prevent its spread.
• Comparison with Other Options:
• Wheat: A staple crop and a major food source; beneficial for farmers.
• Sunflower: A commercial crop grown for edible oil and seeds; highly profitable for farmers.
• Soybean: A significant source of protein and oil; essential for agricultural economy.
• Parthenium: A harmful weed with no economic or agricultural benefits; its removal is critical.

Important Points

• Health Hazards of Parthenium:
• Causes allergic reactions, including asthma, hay fever, and dermatitis, in humans.
• Livestock that graze on Parthenium suffer from reduced milk production and other health issues due to its toxicity.
• Economic Loss:
• Parthenium infestation leads to significant economic losses in agriculture by reducing crop productivity and increasing weed management costs.
• In India, it has been reported to cause losses worth several crores annually.
• Government Initiatives:
• Several states in India have undertaken awareness campaigns and weed eradication programs to control Parthenium.
• Research organizations like the Indian Council of Agricultural Research (ICAR) are actively working on sustainable management strategies for this weed.

The correct answer is Spectrum.

Key Points

• Dispersion occurs when a ray of white light passes through a prism, splitting into its constituent colors. This phenomenon is due to the variation in the refractive index of the material for different wavelengths of light.
• The band of seven colors produced on a screen as a result of dispersion is called a spectrum. These colors include violet, indigo, blue, green, yellow, orange, and red (VIBGYOR).
• Each color in the spectrum corresponds to a specific wavelength of visible light, with violet having the shortest wavelength and red having the longest.
• The phenomenon of dispersion and the formation of a spectrum were scientifically explained by Isaac Newton, who demonstrated that white light is composed of multiple colors.
• This property of light is crucial in understanding optical phenomena, such as rainbows, which are natural examples of spectrum formation due to dispersion of light through water droplets in the atmosphere.

Additional Information

• Definition of Dispersion: Dispersion refers to the splitting of white light into its constituent colors when it passes through a medium like a prism. It occurs because different colors of light travel at different speeds when passing through a medium.
• Applications of Spectrum:
• Spectroscopy: Used in scientific fields to analyze the composition of substances based on their light spectrum.
• Optical Instruments: Instruments like spectrometers and spectrophotometers rely on dispersion and spectrum analysis.
• Astronomy: Astronomers use spectra to study the composition and movement of celestial bodies.
• Refractive Index: The refractive index of a material determines the degree of bending of light and varies with the wavelength of light, leading to dispersion.
• Real-World Examples:
• Rainbows: Formed by the dispersion and total internal reflection of sunlight in water droplets.
• CDs and DVDs: The grooves on the surface act as a diffraction grating, creating a spectrum.
• Other Phenomena Related to Light:
• Reflection: Bouncing back of light when it hits a surface.
• Refraction: Bending of light as it passes from one medium to another.
• Diffraction: The bending of light around obstacles or openings.

Important Points

• Isaac Newton's Experiment: Newton demonstrated dispersion using a glass prism and white light, proving that sunlight consists of different colors.
• Color Wavelengths: The visible spectrum ranges from approximately 400 nm (violet) to 700 nm (red).
• Prism Material: The degree of dispersion depends on the material of the prism; for example, glass disperses light more than air.
• Difference Between Spectrum and Image: A spectrum is a band of colors due to dispersion, whereas an image is a reproduction of an object through reflection or refraction.

The correct answer is Prostate gland.

Key Points

• The prostate gland is a part of the male reproductive system in humans. It is located just below the bladder and surrounds the urethra.
• This gland plays a crucial role in the production of seminal fluid, which nourishes and transports sperm during ejaculation.
• The prostate gland produces a fluid that constitutes approximately 30% of the semen volume. This fluid contains enzymes, zinc, and citric acid, which contribute to sperm viability and motility.
• It also contains the enzyme prostate-specific antigen (PSA), which helps in liquefying the semen after ejaculation for better sperm mobility.
• The prostate gland is regulated by hormones such as testosterone, which is produced by the testes, emphasizing its connection to the male reproductive system.

Additional Information

• Vagina:
• The vagina is part of the female reproductive system. It is the muscular tube that connects the cervix to the external genitalia.
• It plays a role in childbirth, menstruation, and sexual intercourse. It is not part of the male reproductive system.
• Uterus:
• The uterus, also known as the womb, is another component of the female reproductive system.
• It is where the fertilized egg implants and develops into a fetus during pregnancy. It is not found in males.
• Fallopian Tube:
• The fallopian tubes are part of the female reproductive system, responsible for transporting the egg from the ovary to the uterus.
• Fertilization of the egg by sperm typically occurs in the fallopian tubes.
• Prostate Gland:
• The prostate gland is a vital part of the male reproductive system, contributing to the production of semen.
• It is prone to certain health conditions such as prostatitis (inflammation), benign prostatic hyperplasia (BPH), and prostate cancer, especially in older men.

Important Points

• The male reproductive system consists of organs such as the testes, vas deferens, prostate gland, seminal vesicles, and penis.
• The testes produce sperm and hormones like testosterone, which regulate secondary sexual characteristics and reproductive functions.
• The prostate gland and seminal vesicles contribute to the production of seminal fluid, which ensures the nourishment and transport of sperm.
• Health conditions related to the prostate gland, such as prostate cancer, are common in older men. Regular screening for PSA levels is recommended for early detection.

The correct answer is Acid + Base → Salt + Water.

Key Points

• A neutralisation reaction occurs when an acid reacts with a base to form a salt and water. This reaction is a fundamental chemical process widely observed in various fields of chemistry.
• The general equation of a neutralisation reaction is: Acid + Base → Salt + Water.
• For example, when hydrochloric acid (HCl) reacts with sodium hydroxide (NaOH), the products are sodium chloride (NaCl) (a salt) and water (H₂O):
• HCl + NaOH → NaCl + H₂O
• This reaction is an exothermic process, meaning it releases heat energy.
• The pH of the resulting solution after a complete neutralisation is typically around 7, indicating a neutral solution.
• This concept is widely applied in various industries like pharmaceuticals, wastewater treatment, and agriculture to neutralise acidic or basic substances.

Additional Information

• Explanation of Incorrect Options:
• Option 1 (Acid + Salt → Water): This is incorrect because acids and salts do not typically react to form water unless there are special conditions, such as the presence of a base or specific reaction mechanisms.
• Option 3 (Base + Oxide → Salt + Water): This is partially incorrect. While certain bases react with acidic oxides to form salts and water, this is not the definition of a neutralisation reaction.
• Option 4 (Acid + Metal → Salt + Hydrogen): This represents a reaction where an acid reacts with a metal to produce a salt and hydrogen gas, not water. For example: Zn + HCl → ZnCl₂ + H₂.
• Characteristics of Neutralisation Reactions:
• The reaction involves the combination of H⁺ ions from the acid and OH⁻ ions from the base to form water (H₂O).
• Neutralisation reactions are used in antacid formulations to reduce excess stomach acidity by neutralising hydrochloric acid (HCl) in the stomach.
• They are also used in the treatment of industrial effluents to neutralise acidic or basic waste before disposal into water bodies.
• Examples of Neutralisation Reactions:
• NaOH + HCl → NaCl + H₂O: Sodium hydroxide reacts with hydrochloric acid to form sodium chloride and water.
• Ca(OH)₂ + H₂SO₄ → CaSO₄ + 2H₂O: Calcium hydroxide reacts with sulfuric acid to produce calcium sulfate and water.
• Importance in Daily Life:
• Used in agriculture to neutralise acidic soils with lime (Ca(OH)₂).
• Helps in the formulation of antacids to relieve acidity in the human stomach.
• Essential in water treatment plants to neutralise acidic water before distribution.

Important Points

• Real-world Applications of Neutralisation Reactions:
• Medicinal Use: Antacids like magnesium hydroxide (milk of magnesia) neutralise excess stomach acid to relieve heartburn and indigestion.
• Environmental Use: Neutralising acidic rainwater using basic substances like lime to prevent damage to soil and aquatic ecosystems.
• Industrial Use: In chemical manufacturing, neutralisation reactions are used to control pH levels of products and waste materials.
• Indicators of Neutralisation: Common indicators like phenolphthalein and methyl orange are used to signify completion of the reaction by showing a colour change.

The correct answer is protons and neutrons.

Key Points

• Nucleons are the subatomic particles that constitute the nucleus of an atom.
• The two types of nucleons are protons and neutrons, which are bound together by the strong nuclear force.
• Protons carry a positive charge (+1) and have a mass approximately equal to 1 atomic mass unit (amu).
• Neutrons are electrically neutral particles with a mass similar to that of protons (approximately 1 amu).
• Electrons are not considered nucleons because they are located outside the nucleus and have negligible mass compared to protons and neutrons.
• The nucleus is responsible for most of the atom's mass, as protons and neutrons are significantly heavier than electrons.

Additional Information

• Protons:
• Protons are positively charged particles found in the nucleus of an atom.
• The number of protons in an atom defines its atomic number, which determines the element's identity.
• Protons are composed of three quarks (two up quarks and one down quark) held together by the strong nuclear force.
• Neutrons:
• Neutrons are neutral particles that also reside in the atomic nucleus.
• They play a crucial role in stabilizing the nucleus by reducing the repulsive forces between positively charged protons.
• Neutrons consist of three quarks (one up quark and two down quarks) bound by the strong nuclear force.
• Electrons:
• Electrons are negatively charged particles that orbit the nucleus in specific energy levels or shells.
• Their mass is approximately 1/1836 of a proton, making their contribution to the atom's total mass negligible.
• Electrons are not nucleons because they are not part of the nucleus.
• Strong Nuclear Force:
• The strong nuclear force is the fundamental interaction responsible for binding protons and neutrons together in the nucleus.
• It operates over very short distances (approximately 1 femtometer) and is the strongest force in nature.
• This force overcomes the electrostatic repulsion between positively charged protons in the nucleus.
• Applications:
• Understanding nucleons is critical in fields like nuclear physics, radiology, and energy production.
• Nuclear reactions, such as fission and fusion, involve changes in the arrangement of protons and neutrons in the nucleus.
• Isotopes of an element differ in their number of neutrons and have applications in medicine, dating techniques, and scientific research.

Shortcut Trick

Calculate semi-perimeter (s): (75 + 34 + 61) ÷ 2 = 85 cm.

Find Area using Heron's: √[85 × (85−75) × (85−34) × (85−61)] = √[85 × 10 × 51 × 24] = 1020 cm².

Altitude (h) for base 34 cm: (2 × Area) ÷ Base = (2 × 1020) ÷ 34.

h = 2040 ÷ 34 = 60 cm.

∴ The correct answer is 60 cm.

Alternate Method

Given: Sides of triangle a = 75 cm, b = 34 cm, and c = 61 cm.

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

Calculations:

⇒ Semi-perimeter (s) = (a + b + c) ÷ 2

⇒ s = (75 + 34 + 61) ÷ 2 = 170 ÷ 2 = 85 cm

⇒ Area = √[85 × (85 − 75) × (85 − 34) × (85 − 61)]

⇒ Area = √[85 × 10 × 51 × 24]

⇒ Area = √[(5 × 17) × (5 × 2) × (3 × 17) × (2³ × 3)]

⇒ Area = √[5² × 17² × 3² × 2⁴]

⇒ Area = 5 × 17 × 3 × 2² = 1020 cm²

⇒ Now, Area = 1/2 × 34 × Altitude

⇒ 1020 = 17 × Altitude

⇒ Altitude = 1020 ÷ 17 = 60 cm

∴ The correct answer is 60 cm.

Additional Information

Heron's Formula

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

Altitude of a Triangle

The altitude is the perpendicular line segment from a vertex to the opposite side. Every triangle has three altitudes.

Relation between Area and Altitude

For any triangle, Altitude = (2 × Area) ÷ Corresponding Base. The shortest side of a scalene triangle corresponds to the longest altitude.

Shortcut Trick

Total Marked Price = 250 × 1,014 = ₹2,53,500

Net Selling Price = 2,53,500 × 0.70 × 0.85 × 0.80

Using effective multiplier: 2,53,500 × 0.476 = ₹1,20,666

∴ The correct answer is ₹1,20,666.

Alternate Method

Given:

Quantity = 250 mixers

Price per mixer = ₹1,014

Discounts = 30%, 15% and 20% (card payment)

Calculations:

⇒ Total MP = 250 × 1,014 = ₹2,53,500

⇒ Price after 30% discount = 2,53,500 × (70/100) = ₹1,77,450

⇒ Price after 15% discount = 1,77,450 × (85/100) = ₹1,50,832.5

⇒ Price after 20% card discount = 1,50,832.5 × (80/100)

⇒ Final Amount = 1,50,832.5 × 0.8 = ₹1,20,666

∴ The correct answer is ₹1,20,666.

Additional Information

Successive Discount Formula

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

Net Selling Price

SP = MP × (100−d₁)/100 × (100−d₂)/100 × (100−d_n)/100. This is the standard chain rule for discounts.

Marked Price (MP)

It is the price printed on the product, also known as the list price, upon which discounts are calculated.

Shortcut Trick

Combined efficiency of A, B, and C using LCM of 3, 2, 6 = 6 units.

Efficiencies: A = 2, B = 3, C = 1. Total Efficiency = 6.

Time for original work = 6 ÷ 6 = 1 day.

Time for thrice the work = 1 × 3 = 3 days.

∴ The correct answer is 3 days.

Alternate Method

Given: Time taken by A = 3 days, B = 2 days, C = 6 days.

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

⇒ Total Efficiency of (A + B + C) = 2 + 3 + 1 = 6 units/day

⇒ New Work = 3 × Original Work = 3 × 6 = 18 units

⇒ Time required = New Work ÷ Total Efficiency

⇒ Time required = 18 ÷ 6 = 3 days

∴ The correct answer is 3 days.

Additional Information

Combined Work Formula

For three people, the time taken together is T = (a×b×c) ÷ (ab + bc + ca). Multiply the result by 'n' for n-times the work.

Work and Efficiency Relationship

Efficiency is inversely proportional to the time taken. If work increases by a factor of 3, time also increases by a factor of 3 for constant efficiency.

Shortcut Trick List the frequencies of existing numbers: 8 (occurs 3 times), 7 (occurs 3 times), 9 (2 times), 10 (2 times), 2 (1 time), 5 (1 time). The Mode is defined as the value with the highest frequency. Currently, both 7 and 8 have the same maximum frequency of 3. For the Mode of the data to be 7, the frequency of 7 must be greater than that of any other number. Therefore, the additional variable x must be 7 to make its frequency 4. ∴ The correct answer is 7.

Alternate Method Given Data: 8, 7, 8, 7, 9, 10, 2, 8, 9, 7, 5, 10 and x Given Mode = 7

Analyze frequencies in the set (excluding x): ⇒ Count of 8 = 3 ⇒ Count of 7 = 3 ⇒ Count of 9 = 2 ⇒ Count of 10 = 2 ⇒ Count of 2 = 1 ⇒ Count of 5 = 1 ⇒ Since Mode = 7, the total frequency of 7 must be greater than the frequency of 8. ⇒ Frequency(7) including x > Frequency(8) ⇒ 3 + 1 > 3 ⇒ 4 > 3 ⇒ This condition is satisfied only when x = 7. ∴ The correct answer is 7.

Additional Information Mode The value that appears most frequently in a given data set. A set can be unimodal (one mode), bimodal (two modes), or multimodal. Empirical Relationship For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean. Median The middle value of a data set when it is arranged in ascending or descending order. If the number of terms n is even, it is the average of the two middle terms. Mean The average of all observations calculated as: Mean = (Sum of all observations) ÷ (Total number of observations).

Shortcut Trick Let Son = x. Then Mother = 2x, Father = 2x + 3, and Daughter = x + 3. Sum: x + (2x) + (2x + 3) + (x + 3) = 144 ⇒ 6x + 6 = 144. 6x = 138 ⇒ x = 23 years. Daughter = 23 + 3 = 26 years. ∴ The correct answer is 26 years. Alternate Method Given: Total age of Father, Mother, Son, and Daughter = 144 years. Age order: Father > Mother > Daughter > Son.

Calculations: ⇒ Let Son's age = S ⇒ Mother's age = 2 × S = 2S ⇒ Father's age = Mother + 3 = 2S + 3 ⇒ Daughter's age = Son + 3 = S + 3 ⇒ (2S + 3) + 2S + (S + 3) + S = 144 ⇒ 6S + 6 = 144 ⇒ 6S = 138 ⇒ S = 23 years ⇒ Daughter's age = 23 + 3 = 26 years ∴ The correct answer is 26 years. Additional Information Linear Equations in Age Problems Always express all individual ages in terms of a single variable (usually the youngest person) to simplify the equation. Age Difference Property The age difference between two people remains constant regardless of the number of years that pass. Averages in Age Average age = (Sum of ages of all members) ÷ (Total number of members).

Shortcut Trick

Loss = Cost Price (CP) − Selling Price (SP)

Loss Percentage = (Loss ÷ CP) × 100

Calculation: (10720 − 9862.40) ÷ 10720 × 100 = 857.6 ÷ 10720 × 100 = 8%

∴ The correct answer is 8%.

Alternate Method

Given:

Cost Price (CP) = ₹10,720

Selling Price (SP) = ₹9,862.40

Formula Used:

Loss = CP − SP

Loss % = (Loss / CP) × 100

Calculations:

⇒ Loss = 10720 − 9862.40

⇒ Loss = 857.60

⇒ Loss % = (857.60 ÷ 10720) × 100

⇒ Loss % = 0.08 × 100

⇒ Loss % = 8%

∴ The correct answer is 8%.

Additional Information

Profit Percentage

Calculated when SP > CP. Profit % = [(SP − CP) ÷ CP] × 100.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Discount % = (Discount ÷ MP) × 100.

Successive Percentage Change

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

Shortcut Trick

Initial Income = 100%, Savings = 27%, Expenditure = 73%.

Net change in Savings (S) × 27 = Total Income change × 100 − Expenditure change × 73.

Change in Savings % = (100 × 20 − 73 × 15) ÷ 27 = (2000 − 1095) ÷ 27 = 905 ÷ 27 ≈ 33.518%.

Absolute increase in Savings = Initial Savings × (905/27)% = (12000 × 27/100) × (905/2700) = 3240 × 905/2700 = ₹1,086.

∴ The correct answer is increase by ₹1,086.

Alternate Method

Given: Initial Income = ₹12,000, Savings % = 27%, Income Increase = 20%, Expenditure Increase = 15%.

Formula Used: Income = Expenditure + Savings

⇒ Initial Savings = 27% of 12,000 = 0.27 × 12,000 = ₹3,240

⇒ Initial Expenditure = 12,000 − 3,240 = ₹8,760

⇒ New Income = 12,000 + 20% of 12,000 = 12,000 + 2,400 = ₹14,400

⇒ New Expenditure = 8,760 + 15% of 8,760 = 8,760 + 1,314 = ₹10,074

⇒ New Savings = New Income − New Expenditure = 14,400 − 10,074 = ₹4,326

⇒ Change in Savings = 4,326 − 3,240 = ₹1,086 (Increase)

∴ The correct answer is increase by ₹1,086.

Additional Information

Percentage Change in Components

If components A and B sum up to C, the percentage change in C is the weighted average of the percentage changes in A and B based on their initial values.

Alligation Method for Income-Expenditure

Ratio of Expenditure to Savings = (% Change in Savings − % Change in Income) : (% Change in Income − % Change in Expenditure).

Base Value Stability

When calculating final changes, always use the initial total Income or Expenditure as the base unless specified otherwise.

Shortcut Trick

Change in average age = 14 − 13 = 1 year.

Total persons now = 35 students + 1 teacher = 36.

Teacher age = New Average + (Original Number × Increase in Average)

Teacher age = 14 + (35 × 1) = 14 + 35 = 49 years.

∴ The correct answer is 49 years.

Alternate Method

Given:

Number of students = 35

Initial average age = 13 years

New average age (including teacher) = 14 years

Calculations:

⇒ Total age of 35 students = 35 × 13

⇒ Total age of students = 455 years

⇒ Total number of people (students + teacher) = 36

⇒ Total age of 36 people = 36 × 14

⇒ Total age of 36 people = 504 years

⇒ Teacher age = Total age of 36 people − Total age of 35 students

⇒ Teacher age = 504 − 455 = 49 years

∴ The correct answer is 49 years.

Additional Information

Average Formula

The average of a group is calculated as: Average = (Sum of all observations) ÷ (Total number of observations).

Inclusion/Exclusion Concept

When a new person joins a group, the Age of the new person = (New Average) + (Number of initial people × Increase in Average).

Combined Average

If two groups with n₁ and n₂ members have averages A₁ and A₂, the combined average is (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

In melting and recasting problems, if a large sphere of radius R is converted into smaller spheres of radius r, the ratio of Total Surface Area of all small spheres to the Surface Area of the original sphere is always R : r.

Here, R = 10.5 cm and r = 3.5 cm.

Ratio = 10.5 : 3.5

Ratio = 3 : 1.

∴ The correct answer is 3 : 1.

Alternate Method

Given:

Radius of the original sphere (R) = 10.5 cm

Radius of each smaller sphere (r) = 3.5 cm

Formula Used:

Volume of a sphere = (4/3) × π × radius³

Surface Area of a sphere = 4 × π × radius²

Calculations:

Since the volume remains constant during melting:

⇒ Volume of Original Sphere = n × Volume of one small sphere

⇒ (4/3) × π × (10.5)³ = n × (4/3) × π × (3.5)³

⇒ n = (10.5 / 3.5)³

⇒ n = 3³ = 27

Total Surface Area of 27 smaller spheres = 27 × 4 × π × (3.5)²

Surface Area of original sphere = 4 × π × (10.5)²

⇒ Required Ratio = [27 × 4 × π × (3.5)²] / [4 × π × (10.5)²]

⇒ Ratio = 27 × (3.5 / 10.5)²

⇒ Ratio = 27 × (1 / 3)²

⇒ Ratio = 27 × (1 / 9) = 3

∴ The correct answer is 3 : 1.

Additional Information

Volume and Area Relation

When a 3D object is melted and recast, the total volume remains invariant, but the total surface area usually increases as the number of pieces increases.

Scaling Factor

If the linear dimensions (radius) are scaled by a factor k, the Area scales by k² and the Volume scales by k³.

Hemisphere Properties

For a hemisphere of radius r, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Shortcut Trick

Sheets per folder = 8 + 6 = 14

Booklet pages per folder = (24 ÷ 6) + 4 = 4 + 4 = 8

Total items per folder = 14 + 8 = 22

Total for 15 folders = 15 × 22 = 330

∴ The correct answer is 330 sheets and pages.

Alternate Method

Given:

Number of exam folders = 15

Question sheets per folder = 8

Answer sheets per folder = 6

Booklet pages calculation = $[\frac{27-3}{6}+4]$

Formula Used:

Total = Folders × (Question Sheets + Answer Sheets + Booklet Pages)

Calculations:

⇒ Booklet pages = [(24 ÷ 6) + 4]

⇒ Booklet pages = [4 + 4] = 8

⇒ Total items in 1 folder = 8 + 6 + 8 = 22

⇒ Total items for 15 folders = 15 × 22

⇒ Total items = 330

∴ The correct answer is 330 sheets and pages.

Additional Information

BODMAS Rule

Order of operations: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Unitary Method

Calculate the value of a single unit first, then multiply by the required number of units.

Simplification Shortcut

Multiplying by 15 is equivalent to multiplying by 10 and adding half of that result (15 × 22 = 220 + 110 = 330).

Shortcut Trick

Discount = 4000 − 280 = 3720

Discount % = (3720 ÷ 4000) × 100

Discount % = 372 ÷ 4 = 93%

∴ The correct answer is 93%.

Alternate Method

Given:

Marked Price (MP) = ₹4,000

Selling Price (SP) = ₹280

Formula Used:

Discount % = [(MP − SP) ÷ MP] × 100

Calculations:

⇒ Discount = 4000 − 280

⇒ Discount = 3720

⇒ Discount % = (3720 ÷ 4000) × 100

⇒ Discount % = 372000 ÷ 4000

⇒ Discount % = 93%

∴ The correct answer is 93%.

Additional Information

Marked Price (MP)

The price printed on the item or label, also known as the List Price or Tag Price.

Successive Discounts

When multiple discounts are applied, the Effective Discount is calculated as d₁ + d₂ − (d₁ × d₂)/100.

Profit and Discount Relation

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

Shortcut Trick

Let the number be 100%.

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

⇒ 72 = 100% − 20% = 80%

Directly, 80% of the number = 72.

∴ The correct answer is 72.

Alternate Method

Given: 20% of a number added to 72 results in the number itself.

Formula: Percentage × Value = Result

Let the number be x.

⇒ 20% of x + 72 = x

⇒ (20/100) × x + 72 = x

⇒ 0.2x + 72 = x

⇒ 72 = x − 0.2x

⇒ 72 = 0.8x

⇒ 80% of the number = 0.8x = 72

∴ The correct answer is 72.

Additional Information

Percentage Change

To find a percentage of a number, multiply the number by the percent expressed as a fraction or decimal: x × (p/100).

Complementary Percentages

If p% of a value is removed, the remaining part is (100 − p)% of that value.

Fraction Conversion

Common fractions: 20% = 1/5, 40% = 2/5, 60% = 3/5, 80% = 4/5. Using fractions often speeds up calculation.

Shortcut Trick

Breadth = Area ÷ Length = 195 ÷ 15 = 13 cm

Perimeter = 2 × (Length + Breadth)

⇒ Perimeter = 2 × (15 + 13) = 2 × 28 = 56 cm

∴ The correct answer is 56 cm.

Alternate Method

Given:

Area of Rectangle = 195 cm²

Length (L) of Rectangle = 15 cm

Formula Used:

1. Area of Rectangle = Length × Breadth
2. Perimeter of Rectangle = 2 × (Length + Breadth)

Calculations:

⇒ Area = Length × Breadth

⇒ 195 = 15 × Breadth

⇒ Breadth = 195 ÷ 15

⇒ Breadth = 13 cm

Now, calculate the Perimeter:

⇒ Perimeter = 2 × (Length + Breadth)

⇒ Perimeter = 2 × (15 + 13)

⇒ Perimeter = 2 × 28

⇒ Perimeter = 56 cm

∴ The correct answer is 56 cm.

Additional Information

Diagonal of a Rectangle

The distance between opposite vertices is calculated using Pythagoras theorem: d = √(L² + B²).

Properties of a Square

A square is a special rectangle where Length = Breadth. Area = side² and Perimeter = 4 × side.

Relationship between Area and Perimeter

For a fixed area, a square has the minimum possible perimeter among all rectangles.

Shortcut Trick

By using the standard algebraic identity, if x³ + y³ + z³ = 3xyz, then x + y + z = 0.

Let x = a − 1, y = a − 4, and z = a − 7.

⇒ (a − 1) + (a − 4) + (a − 7) = 0

⇒ 3a − 12 = 0

⇒ 3a = 12

⇒ a = 4

∴ The correct answer is 4.

Alternate Method

Given:

(a − 1)³ + (a − 4)³ + (a − 7)³ = 3(a − 1)(a − 4)(a − 7)

Formula Used:

The algebraic factorization identity is:

x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx)

For the expression to equal zero (i.e., x³ + y³ + z³ = 3xyz), one of the following conditions must be satisfied:

1. x + y + z = 0

2. x = y = z

Calculations:

Let x = a − 1, y = a − 4, and z = a − 7.

Applying Case 1:

⇒ (a − 1) + (a − 4) + (a − 7) = 0

⇒ a + a + a − 1 − 4 − 7 = 0

⇒ 3a − 12 = 0

⇒ 3a = 12

⇒ a = 12 ÷ 3

⇒ a = 4

Applying Case 2:

⇒ a − 1 = a − 4

⇒ −1 = −4 (which is an inconsistent equation and has no solution)

Hence, the only possible real value is a = 4.

∴ The correct answer is 4.

Additional Information

Algebraic Identity for Cubes

The formula x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx) is a key polynomial identity used to simplify cubic relations.

Conditional Identity

If the sum of three real variables is zero (x + y + z = 0), then the sum of their cubes is directly three times their product, written as x³ + y³ + z³ = 3xyz.

Sum of Squares Equivalence

The quadratic part of the identity can be represented as 0.5 × [(x − y)² + (y − z)² + (z − x)²], which is always positive for distinct real values of x, y, and z.

Shortcut Trick

Aman total days = 13 × 6 = 78 days.

Chaman total days = 19 × 7 = 133 days.

Combined days = (A × C) ÷ (A + C) = (78 × 133) ÷ (78 + 133) = 10374 ÷ 211.

Combined days ≈ 49.165... days.

∴ The correct answer is 49.2 days.

Alternate Method

Given:

Aman does 1/6 work in 13 days.

Chaman does 1/7 work in 19 days.

Formula Used:

Total Work = Time taken to complete 1 unit of work.

Combined Time = (Total Work) ÷ (Combined Efficiency).

Calculations:

⇒ Aman total days = 13 ÷ (1/6) = 78 days.

⇒ Chaman total days = 19 ÷ (1/7) = 133 days.

⇒ Combined Efficiency = 133 + 78 = 211 units/day.

⇒ Time taken = 10374 ÷ 211 = 49.1658... days.

⇒ Rounded to one decimal place = 49.2 days.

∴ The correct answer is 49.2 days.

Additional Information

Fractional Work Rule

If a person completes 1/n part of work in D days, the total time to complete the full work is D × n days.

Combined Work Formula

For two people A and B completing work in x and y days respectively, time together = (x × y) ÷ (x + y).

Efficiency and Time

Efficiency is inversely proportional to time. If the ratio of time taken is a:b, then the ratio of efficiency is b:a.

Shortcut Trick

Effective Compound Interest rate for 2 years at 10% = 10 + 10 + (10 × 10)/100 = 21%.

Total Amount percentage = 100% (Principal) + 21% (Interest) = 121%.

Required Amount = 121% of ₹7,700 = 1.21 × 7700 = ₹9,317.

∴ The correct answer is ₹9,317.

Alternate Method

Given:

Principal (P) = ₹7,700

Rate (R) = 10% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 7700 × (1 + 10/100)²

⇒ A = 7700 × (110/100) × (110/100)

⇒ A = 7700 × (11/10) × (11/10)

⇒ A = 77 × 121

⇒ A = 9317

∴ The correct answer is ₹9,317.

Additional Information

Compound Interest (CI) Formula

The interest earned is calculated as CI = Amount − Principal = P[(1 + R/100)ⁿ − 1].

Effective Rate for 2 Years

For a rate R%, the effective CI rate for 2 years is given by the successive percentage formula: (2R + R²/100)%.

CI vs SI Difference

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

Shortcut Trick

Ratio of Red : Green = 14 : 8 = 7 : 4

Ratio of Pink : Red = 2 : 6 = 1 : 3

To find Green : Pink, use the common term Red: (Green/Red) × (Red/Pink) = (4/7) × (3/1) = 12/7

∴ The correct answer is 12 : 7.

Alternate Method

Given:

Red (R) : Green (G) = 14 : 8 = 7 : 4

Pink (P) : Red (R) = 2 : 6 = 1 : 3

Calculations:

⇒ To combine the ratios, make the Red (R) value equal in both ratios.

⇒ LCM of 7 (from R:G) and 3 (from P:R) is 21.

⇒ R : G = (7 × 3) : (4 × 3) = 21 : 12

⇒ P : R = (1 × 7) : (3 × 7) = 7 : 21

⇒ Combined Ratio Green : Red : Pink = 12 : 21 : 7

⇒ Required Ratio Green : Pink = 12 : 7

∴ The correct answer is 12 : 7.

Additional Information

Compound Ratio

The ratio of the product of antecedents to the product of consequents of two or more ratios is called the compound ratio.

Inverse Ratio

If a : b is a ratio, then its inverse ratio is b : a. In this problem, Pink : Red was inverted to help solve for Green : Pink.

Scaling Ratios

To compare or combine ratios, always find a common bridge element and multiply the ratios by factors to make that element equal across all sets.

Shortcut Trick

Identify complementary angles: 80° + 10° = 90° and 57° + 33° = 90°.

Apply Complementary Ratio property: sin 80° = cos 10° and sin 57° = cos 33°.

Expression becomes: (1) − (sin 57° × (1/cos 33°)) + 2.

Since sin 57° = cos 33°, the middle term is 1.

⇒ 1 − 1 + 2 = 2.

∴ The correct answer is 2.

Alternate Method

Given: sin 80°/cos 10° − sin 57° sec 33° + 2

Formula Used: sin(90° − θ) = cos θ and sec θ = 1/cos θ

⇒ sin 80° = sin(90° − 10°) = cos 10°

⇒ sec 33° = 1 / cos 33°

⇒ sin 57° = sin(90° − 33°) = cos 33°

⇒ Term 1: cos 10° / cos 10° = 1

⇒ Term 2: cos 33° × (1 / cos 33°) = 1

⇒ Final Calculation: 1 − 1 + 2

⇒ 0 + 2 = 2

∴ The correct answer is 2.

Additional Information

Complementary Angle Identities

Sin and Cos are co-functions: sin θ = cos(90° − θ). Similarly, tan θ = cot(90° − θ) and sec θ = cosec(90° − θ).

Reciprocal Identities

Standard reciprocals are: cosec θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.

Trigonometric Table Values

Always remember sin 30° = 1/2, sin 45° = 1/√2, and sin 60° = √3/2 for quick substitutions.

Shortcut Trick

Weighted Average Speed = (v₁t₁ + v₂t₂) ÷ (t₁ + t₂)

Ratio of time = 80 : 60 = 4 : 3

Average Speed = (20 × 4 + 55 × 3) ÷ (4 + 3)

Average Speed = (80 + 165) ÷ 7 = 245 ÷ 7 = 35 m/min

∴ The correct answer is 35 m/min.

Alternate Method

Given:

Speed₁ (v₁) = 20 m/min, Time₁ (t₁) = 80 min

Speed₂ (v₂) = 55 m/min, Time₂ (t₂) = 60 min

Formula Used:

Distance = Speed × Time

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Distance₁ = 20 × 80 = 1600 m

⇒ Distance₂ = 55 × 60 = 3300 m

⇒ Total Distance = 1600 + 3300 = 4900 m

⇒ Total Time = 80 + 60 = 140 min

⇒ Average Speed = 4900 ÷ 140

⇒ Average Speed = 35 m/min

∴ The correct answer is 35 m/min.

Additional Information

Average Speed (Equal Distances)

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

Unit Conversion

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

Relative Speed

For two objects moving in the same direction, relative speed = S₁ − S₂. In opposite directions, relative speed = S₁ + S₂.

Shortcut Trick

Speed is constant, so Time is directly proportional to Distance.

Time for pole (1533 m) = 73 seconds.

Time for platform (1533 + 420 m = 1953 m) = 73 × (1953 ÷ 1533) seconds.

Required Time = 73 × (21 ÷ 17) − No, simply divide 1953 by speed (1533 ÷ 73 = 21 m/s).

Required Time = 1953 ÷ 21 = 93 seconds.

∴ The correct answer is 93 seconds.

Alternate Method

Given: Length of train (L_t) = 1533 m, Time to cross pole (T₁) = 73 seconds, Length of platform (L_p) = 420 m.

Formula Used: Speed = Distance ÷ Time; Time = (L_t + L_p) ÷ Speed

⇒ Speed of train = Length of train ÷ Time to cross pole

⇒ Speed = 1533 ÷ 73 = 21 m/s

⇒ Total distance to cross platform = Length of train + Length of platform

⇒ Total distance = 1533 + 420 = 1953 m

⇒ Time taken = Total distance ÷ Speed

⇒ Time taken = 1953 ÷ 21 = 93 seconds

∴ The correct answer is 93 seconds.

Additional Information

Crossing a stationary object without length

When a train crosses a pole, tree, or person, the distance traveled is equal to the length of the train itself.

Crossing a stationary object with length

When a train crosses a platform, bridge, or tunnel, the total distance covered is the sum of the length of the train and the length of the object.

Speed conversion

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

Shortcut Trick

Profit Ratio = Pushpraj : Himani = 2100 : (5600 − 2100) = 2100 : 3500 = 3 : 5.

Since the time period is same, Investment Ratio = Profit Ratio = 3 : 5.

Total units = 3 + 5 = 8 units → ₹17,800.

Pushpraj's investment (3 units) = (17800 ÷ 8) × 3 = 2225 × 3 = ₹6,675.

∴ The correct answer is ₹6,675.

Alternate Method

Given:

Total Investment = ₹17,800

Total Profit = ₹5,600

Pushpraj's Profit = ₹2,100

Formula Used:

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

Calculations:

⇒ Himani's Profit = Total Profit − Pushpraj's Profit

⇒ Himani's Profit = 5600 − 2100 = ₹3,500

⇒ Ratio of Profit (Pushpraj : Himani) = 2100 : 3500 = 3 : 5

⇒ Since time is the same, Ratio of Investment = 3 : 5

⇒ Pushpraj's Investment = [3 ÷ (3 + 5)] × 17800

⇒ Pushpraj's Investment = (3 ÷ 8) × 17800

⇒ Pushpraj's Investment = 3 × 2225 = ₹6,675

∴ The correct answer is ₹6,675.

Additional Information

Basic Partnership Rule

Profit is distributed in the ratio of the product of capital and time period (P = C × T).

Active vs Sleeping Partner

An active partner manages the business and may receive a salary or extra commission before the remaining profit is shared.

Ratio of Capitals

If profits and time periods are known, C₁ : C₂ = (P₁/T₁) : (P₂/T₂).

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

Percentage = (1/20 ÷ 10/70) × 100

Percentage = (1/20 × 70/10) × 100

Percentage = (70/200) × 100 = 70/2 = 35%

∴ The correct answer is 35%.

Alternate Method

Given: First fraction = 1/20, Second fraction = 10/70

Formula: Percentage = (A/B) × 100

⇒ Simplify the second fraction: 10/70 = 1/7

⇒ Required Percentage = (1/20 ÷ 1/7) × 100

⇒ Percentage = (1/20 × 7/1) × 100

⇒ Percentage = (7/20) × 100

⇒ Percentage = 7 × 5 = 35%

∴ The correct answer is 35%.

Additional Information

Fraction to Percentage

To convert any fraction a/b to a percentage, multiply the fraction by 100 and add the % symbol.

Percentage Change

Percentage Change = [(New Value − Old Value) ÷ Old Value] × 100. This measures the relative increase or decrease.

Base Change Rule

If A is x% of B, then B is [10000 ÷ x]% of A. This is useful for reversing the comparison base.

Shortcut Trick

To find prime numbers, check numbers not divisible by 2, 3, 5, 7, 11, or 13.

Between 146 and 196, the candidates are: 149, 151, 157, 163, 167, 173, 179, 181, 191, 193.

There are exactly 10 such numbers in this range.

∴ The correct answer is 10.

Alternate Method

Given: Range from 146 to 196.

Formula Used: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Step-by-step verification of odd numbers in the range:

⇒ 147 (3×49), 149 (Prime)

⇒ 151 (Prime), 153 (3×51), 155 (5×31), 157 (Prime), 159 (3×53)

⇒ 161 (7×23), 163 (Prime), 165 (5×33), 167 (Prime), 169 (13²)

⇒ 171 (3×57), 173 (Prime), 175 (5×35), 177 (3×59), 179 (Prime)

⇒ 181 (Prime), 183 (3×61), 185 (5×37), 187 (11×17), 189 (3×63)

⇒ 191 (Prime), 193 (Prime), 195 (5×39)

⇒ Total Prime Numbers = {149, 151, 157, 163, 167, 173, 179, 181, 191, 193} = 10 numbers.

∴ The correct answer is 10.

Additional Information

Prime Number Identification

To check if N is prime, test divisibility by primes up to √N. For N ≈ 196, √N = 14.

Co-Prime Numbers

Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

Twin Primes

Pairs of prime numbers with a difference of 2, such as (149, 151), (179, 181), and (191, 193).

Shortcut Trick Let the three pieces be in Arithmetic Progression (AP) because the differences between consecutive pieces are equal. Let the pieces be a − d, a, a + d. Sum = 3a = 24 ⇒ a = 8. Condition: Largest = 3 × Smallest ⇒ 8 + d = 3(8 − d). 8 + d = 24 − 3d ⇒ 4d = 16 ⇒ d = 4. Largest piece = 8 + 4 = 12. ∴ The correct answer is 12 metres. Alternate Method

Given: Total length = 24 m. Let lengths be x, y, z where x < y < z. Largest piece (z) = 3x. Difference condition: z − y = y − x. ⇒ 2y = z + x ⇒ 2y = 3x + x = 4x ⇒ y = 2x Calculations: ⇒ x + y + z = 24 ⇒ x + 2x + 3x = 24 ⇒ 6x = 24 ⇒ x = 4 m ⇒ Largest piece (z) = 3 × 4 = 12 m ∴ The correct answer is 12 metres. Additional Information Arithmetic Progression (AP) A sequence where the difference between any two consecutive terms is constant. For three terms in AP, the middle term is the arithmetic mean of the other two: 2b = a + c. Sum of Pieces When a single object is divided, the sum of the individual parts must always equal the initial total value.

Shortcut Trick

To find the greatest possible length of each piece, calculate the HCF (Highest Common Factor) of the given lengths.

Lengths are 57 m and 399 m.

Check if the smaller number divides the larger number: 399 ÷ 57 = 7.

Since 57 perfectly divides 399, the HCF is 57.

∴ The correct answer is 57 m.

Alternate Method

Given: Length of first stick = 57 m, Length of second stick = 399 m.

Formula Used: Greatest common length = HCF(Length₁, Length₂)

Calculations:

⇒ Prime factorization of 57 = 3 × 19

⇒ Prime factorization of 399 = 3 × 7 × 19

⇒ Common factors = 3 × 19

⇒ HCF = 57

∴ The correct answer is 57 m.

Additional Information

HCF (Highest Common Factor)

Also known as GCD, it is the largest number that divides two or more numbers without leaving a remainder.

LCM (Least Common Multiple)

The smallest positive integer that is divisible by both numbers. Relationship: HCF × LCM = Product of two numbers.

Division Method for HCF

Divide the larger number by the smaller number; if remainder is 0, the smaller number is the HCF.

Persons: A, B, C, D, W, X, Y

All face North

Linear arrangement (left → right)

No one sits to the left of D

Four persons between D and C

Three persons to the right of Y

X sits immediate left of A

W not neighbour of Y

Final arrangement:

⇒ Y sits third to the left of W

Hence the right answer is Option 2.

Series:

2 8 1 7 3 6 6 6 1 5 5 3 4 1 5 5 3 9 7 7 7

Condition:

Digit must be even

Immediately preceded by an even digit

Immediately followed by an odd digit

Check pattern → Even, Even, Odd

2 8 1 → even, even, odd ✔

⇒ 8 satisfies condition

3 6 6 → odd, even, even ✖

6 6 6 → even, even, even ✖

6 6 1 → even, even, odd ✔

⇒ Middle 6 satisfies condition

3 4 1 → odd, even, odd ✖

All other combinations do not satisfy

⇒ Total such digits = 2

∴ Required number of digits = 2.

Hence the correct answer  is Option 2.

Statement 1: Some pillows are mats

Statement 2: Some mats are cushions

Syllogism rule (Some + Some → No definite conclusion)

From Statement 1:

Check Conclusion I:

Some pillows are cushions

⇒ Not definite (pillows part of mats may be different from mats part of cushions) ✖

Check Conclusion II:

All cushions are mats

⇒ Given only "Some mats are cushions"

⇒ Cannot conclude "All cushions are mats" ✖

∴ Neither conclusion I nor II follows.

Series: 15, 17, 21, 23, 27, ?

Alternating pattern (+2, +4)

15 → 17   (+2)

17 → 21   (+4)

21 → 23   (+2)

23 → 27   (+4)

⇒ Next pattern = +2

27 + 2 = 29

∴ Missing number = 29.

Hence the Correct answer is Option 3.

Number = 3241586

Total digits = 7

Arrange digits in ascending order and compare positions

Original number:

3   2   4   1   5   8   6

Positions:

1   2   3   4   5   6   7

Ascending order:

1   2   3   4   5   6   8

Positions:

1   2   3   4   5   6   7

Compare positions:

Position 2 → 2 = 2 ✔

Position 5 → 5 = 5 ✔

All other positions change

⇒ Unchanged digits = 2

∴ Number of digits whose position remains unchanged = 2.

Hence the correct answer is Option 2.

AUTO → 2683

TOUR → 8136

Find code for R

Common letter → Common digit

AUTO → A U T O → 2 6 8 3

TOUR → T O U R → 8 1 3 6

Common letters in both words:

T, O, U

Common digits in both codes:

8, 3, 6

Match them:

T → 8

U → 6

O → 3

Remaining digit in TOUR = 1

⇒ R → 1

∴ Code for R = 1.

Expression: $104 - 176 \times 16 \div 9 + 192$

Interchange:

• ↔ −

× ↔ ÷

Apply operator interchange then follow BODMAS

After interchange:

104 + 176 ÷ 16 × 9 − 192

First ÷ and ×

176 ÷ 16 = 11

11 × 9 = 99

Now expression:

104 + 99 − 192

104 + 99 = 203

203 − 192 = 11

∴ Required value = 11.

Hence the correct answer is Option 4.

Persons: A, B, C, D, E, F, G

All face centre

Circular arrangement (facing centre → left = clockwise, right = anticlockwise)

B sits immediate right of C

3 persons between B and A from left of B

3 persons between C and D from left of C

G sits immediate right of F

Final arrangement

∴ Number of people between A and C from right of C = 3.

Hence the correct answer is Option 3.

Series:

2 3 8 1 7 9 1 7 7 5 5 3 7 8 6 9 6 9 2 5 1 9 1

Condition:

Digit must be even

Immediately preceded by an even digit

Immediately followed by an odd digit

Check pattern → Even, Even, Odd

2 3 8 → even, odd, even ✖

3 8 1 → odd, even, odd ✖

7 8 6 → odd, even, even ✖

8 6 9 → even, even, odd ✔

⇒ Digit 6 satisfies condition

6 9 6 → even, odd, even ✖

9 6 9 → odd, even, odd ✖

9 2 5 → odd, even, odd ✖

All other combinations do not satisfy

⇒ Total such digits = 1

∴ Required number of digits = 1.

Hence the correct answer is Option 1.

Series:

5 $5 7$ $ @ & * € 3 # 8 # $ 5 © 7 £ 4 & *

Condition:

Number must be immediately preceded by a number

Number must be immediately followed by a symbol

Check pattern → Number, Number, Symbol

5 7 $ → number, number, symbol ✔

⇒ Total such numbers = 1

∴ Required number of numbers = 1.

Hence the correct answer is Option 2.

Given:

From A → 10 km West

Right turn → 8 km North

Right turn → 14 km East

Right turn → 12 km South

Right turn → 4 km West

∴ Required shortest distance = 4 km towards North.

Hence the correct answer is Option 2.

Persons: M, N, O, P, W, X, Y

All face centre

Circular arrangement (facing centre → right = anticlockwise, left = clockwise)

2 persons between X and W from right of X

2 persons between W and M from right of W

3 persons between X and N from right of X

P immediate left of Y

Final circular arrangement

∴ Y sits third to the left of O.

Hence the correct answer is Option 3.

Series: ECO, HFN, KIM, ?, QOK

Alphabet position pattern (+3)

E C O

H F N

K I M

? ? ?

Q O K

Check 1st letters:

E → H → K → ? → Q

⇒ +3 pattern

E(5) +3 = H(8)

H(8) +3 = K(11)

K(11) +3 = N(14)

N(14) +3 = Q(17)

1st letter = N

Check 2nd letters:

C → F → I → ? → O

⇒ +3 pattern

C(3) +3 = F(6)

F(6) +3 = I(9)

I(9) +3 = L(12)

L(12) +3 = O(15)

2nd letter = L

Check 3rd letters:

O → N → M → ? → K

⇒ −1 pattern

O(15) −1 = N(14)

N(14) −1 = M(13)

M(13) −1 = L(12)

L(12) −1 = K(11)

3rd letter = L

⇒ Missing cluster = NLL

∴ Required letter cluster = NLL.

Hence the correct answer is Option 2.

Series: 1211, 1212, 1215, 1210, 1219, 1208, ?, 1206, 1227

Alternating Series Logic:

Odd position terms: +4 pattern

Even position terms: -2 pattern

 

⇒ ? = 1223

∴ The number that should come in place of the question mark is 1223.

Hence the correct answer is Option 2.

From A → 5 km West

Right turn → 4 km North

Right turn → 3 km East

Left turn → 4 km North

Right turn → 2 km East

To reach A from P:

⇒ Drive 8 km South

∴ Required shortest distance = 8 km towards South.

Hence the correct answer is Option 3.

Given:

C = D √ E ? F @ L

= means Son

? means Daughter

@ means Father

√ means Sister

Decode relation from right to left

F @ L

⇒ F is father of L

E ? F

⇒ E is daughter of F

D √ E

⇒ D is sister of E

⇒ D and E are children of F

C = D

⇒ C is son of D

⇒ D is sister of L

⇒ C is child of L’s sister

⇒ C is nephew of L

∴ C is nephew of L.

Letter pairs:

RX – WM

YE – DT

KQ – QE

IO – ND

Alphabet positions

A = 1, B = 2, … , Z = 26

Check shift from first pair to second pair

R(18) → W(23) = +5

X(24) → M(13) = −11

2) YE – DT

Y(25) → D(4) = +5 (cyclic)

E(5) → T(20) = +15

3) IO – ND

I(9) → N(14) = +5

O(15) → D(4) = −11

4) KQ – QE

K(11) → Q(17) = +6

Q(17) → E(5) = −12

⇒ First, second and fourth pairs mostly follow pattern near +5 shift in first letter, but KQ – QE breaks the pattern clearly.

∴ KQ – QE is the odd pair.

Hence the correct answer is Option 3.

Rank from top = 24

Rank from bottom = 56

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

⇒ Total students = 24 + 56 − 1

⇒ Total students = 80 − 1

⇒ Total students = 79

∴ Total number of students in the class = 79.

Hence the correct answer is Option 1.

Series: GRK 45, KMG 54, OHC 63, SCY 72, ?

Alphabet position pattern and number pattern

A = 1, B = 2, … , Z = 26

Numbers pattern:

45 → 54 → 63 → 72

⇒ +9 each step

⇒ Next = 72 + 9 = 81

Letters pattern:

First letters:

G(7) → K(11) → O(15) → S(19)

⇒ +4 each step

⇒ Next = W(23)

Second letters:

R(18) → M(13) → H(8) → C(3)

⇒ −5 each step

⇒ Next = X(24)

Third letters:

K(11) → G(7) → C(3) → Y(25)

⇒ −4 each step (cyclic)

⇒ Next = U(21)

Next term:

WXU 81

∴ Missing cluster = WXU 81.

Hence the correct answer is Option 1.

Statement 1: Some shade are roof

Statement 2: All ceiling are roof

Basic syllogism logic

Check Conclusion I:

Some shade are ceiling

⇒ Not definite because shade may or may not overlap with ceiling

⇒ Conclusion I does not follow

Check Conclusion II:

All roof are ceiling

⇒ Reverse of given statement

⇒ Only given: All ceiling are roof

⇒ Conclusion II does not follow

∴ Neither conclusion I nor II follows.

Hence the correct answer is Option 3.

Rank from top = 9

Rank from bottom = 23

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

⇒ Total students = 9 + 23 − 1

⇒ Total students = 32 − 1

⇒ Total students = 31

∴ Total number of students in the class = 31.

Hence the correct answer is Option 2.

nice large land → cd ak pi

land is fertile → rf pc ak

cost is high → si ig pc

Common word → Common code

Compare sentence 1 and 2

Common word = land

Common code = ak

⇒ land = ak

Compare sentence 2 and 3

Common word = is

Common code = pc

⇒ is = pc

Sentence 2: land is fertile → rf pc ak

Remove codes of land (ak) and is (pc)

⇒ Remaining code = rf

∴ fertile = rf.

Hence the correct answer is Option 1.