RRB Group D Official Paper (Held On: 02 Dec, 2025 Shift 3) (Previous Year Paper)

The correct answer is Copper sulphate solution.

Key Points

• Tyndall effect refers to the scattering of light by particles in a colloid or a very fine suspension. It is visible when the particles are large enough to scatter light but small enough to remain suspended.
• Copper sulphate solution is a true solution where the solute particles are completely dissolved in the solvent. The particle size is extremely small (less than 1 nm), and hence, it does not scatter light. Therefore, it does not show Tyndall effect.
• Emulsions, such as oil in water, are colloidal systems where the dispersed phase scatters light due to larger particle sizes (1-1000 nm). Hence, they exhibit Tyndall effect.
• Milk is a colloidal suspension of fat globules in water, making it capable of scattering light. Thus, milk shows Tyndall effect.
• Starch solution is a colloidal suspension, and its particles scatter light, demonstrating the Tyndall effect. Hence, it shows Tyndall effect.
• Thus, the correct answer is Copper sulphate solution.

Additional Information

• Tyndall Effect:
• The phenomenon is named after John Tyndall, who studied the scattering of light by particles in 1869.
• It occurs when light passes through a colloidal system or a suspension, and the scattered light makes the path of the beam visible.
• It is commonly observed in phenomena like sunlight passing through mist or a dusty room.
• True solutions, like salt or sugar dissolved in water, do not exhibit Tyndall effect because their particles are too small to scatter light.
• Colloids:
• A colloid is a mixture in which one substance is dispersed evenly throughout another substance.
• Particle size in colloids ranges from 1 nm to 1000 nm, which is large enough to scatter light but small enough to remain suspended.
• Examples include milk, fog, and gelatin.
• True Solutions:
• True solutions are homogeneous mixtures where the solute is completely dissolved in the solvent.
• Particle size in true solutions is less than 1 nm, making it impossible for them to scatter light.
• Examples include salt dissolved in water and sugar solution.
• Applications of Tyndall Effect:
• Used in distinguishing colloidal solutions from true solutions.
• Helpful in determining the size and nature of particles in solutions.
• Applied in nephelometry and turbidimetry for measuring the concentration of suspended particles.

Important Points

• Key Characteristics of Colloids:
• Colloids exhibit properties such as Brownian movement, charge stability, and Tyndall effect.
• They are widely used in industries like pharmaceuticals, food processing, and cosmetics for creating stable mixtures.
• Colloids are classified as sols, gels, emulsions, foams, etc., based on the state of the dispersed phase and the dispersion medium.
• Importance of Particle Size:
• Particle size determines the ability of a system to exhibit the Tyndall effect.
• Larger particles (>1000 nm) settle due to gravity (as in suspensions), while smaller particles (<1 nm) do not scatter light (as in true solutions).
• Scientific Applications:
• Tyndall effect is used in optical instruments to study particle properties.
• It is applied in environmental sciences to monitor airborne pollutants and water quality.

The correct answer is In stethoscopes, the sound of the patient’s heartbeat reaches the doctor’s ears by multiple reflection of sound..

Key Points

• Principle of Multiple Reflection: The stethoscope operates on the principle of multiple reflection of sound waves. The sound of the heartbeat is captured by the diaphragm of the stethoscope and is reflected multiple times inside the hollow tubes, enhancing its clarity and allowing it to reach the doctor’s ears effectively.
• Acoustic Tubes: The stethoscope contains acoustic tubes that guide the sound from the chest piece to the doctor’s ears. These tubes are designed to minimize sound loss and ensure that the sound waves travel efficiently through the device.
• Diaphragm and Bell Mechanism: The chest piece of the stethoscope consists of a diaphragm (for high-pitched sounds) and a bell (for low-pitched sounds). The diaphragm detects vibrations caused by the heartbeat, and these vibrations are converted into sound waves.
• Amplification without Electricity: Unlike some modern electronic stethoscopes, the traditional stethoscope does not use electricity to amplify sound. Instead, it relies on the natural amplification provided by the design of the acoustic chambers and the process of multiple reflections.
• Sound Clarity: The multiple reflections help in amplifying and clarifying the sound, making it easier for doctors to detect abnormalities in the heartbeat, such as murmurs or irregular rhythms.

Additional Information

• Parts of a Stethoscope:
• Chest Piece: Contains the diaphragm and bell, used to capture sound.
• Acoustic Tubes: Hollow tubes that transmit sound to the doctor’s ears through multiple reflections.
• Ear Tubes: Metal tubes that connect to the ear tips and deliver sound to the doctor’s ears.
• Ear Tips: Soft tips that fit into the doctor’s ears for comfort and to block external noise.
• Modern Enhancements: Some modern stethoscopes include electronic amplification and noise cancellation features, but traditional acoustic stethoscopes remain widely used due to their simplicity and reliability.
• Importance in Medicine: Stethoscopes are essential for diagnosing conditions such as heart murmurs, arrhythmias, and respiratory issues. They allow doctors to listen to internal sounds without invasive procedures.
• History: The stethoscope was invented by René Laennec in 1816. Initially, it was a simple wooden tube, but it has since evolved into the modern design we see today.
• Applications: Apart from detecting heart and lung sounds, stethoscopes are also used in blood pressure measurement (in conjunction with a sphygmomanometer).

Important Points

• Option Analysis:
• Option 1: The statement that sound travels directly without reflection is incorrect because sound undergoes multiple reflections within the stethoscope.
• Option 2: The stethoscope does not use electricity to amplify sound in its traditional form. This is incorrect for non-electronic stethoscopes.
• Option 3: Sound is not carried by moving air but by vibrations transmitted through the stethoscope’s tubes. Hence, this is incorrect.
• Option 4: This is correct because the sound of the heartbeat reaches the doctor’s ears through multiple reflections inside the stethoscope.
• Stethoscope Care: Proper maintenance, such as cleaning the diaphragm and ear tips, ensures accurate performance and hygiene.
• Advanced Applications: In modern medical practices, stethoscopes are sometimes integrated with digital recording and analysis tools for detailed diagnosis.

The correct answer is Bus.

Key Points

• Force required to stop a moving vehicle is directly proportional to the vehicle's mass and its velocity. Since all vehicles are moving at the same velocity (50 km/hr), the determining factor here is the mass of the vehicle.
• A bus has significantly more mass compared to a bicycle, car, or scooter. Hence, it will require more force to bring it to a stop.
• The equation for force is derived from Newton's second law of motion:
• F = ma, where F is force, m is mass, and a is acceleration (or deceleration in this case).
• A larger mass (bus) will result in a greater force being required to achieve the same deceleration.
• The stopping distance and braking force also depend on the kinetic energy of the vehicle, which is given by the formula: K.E. = 1/2 mv². Since the velocity (v) is constant for all vehicles, the kinetic energy is directly proportional to the mass. The bus, having the highest mass, possesses the highest kinetic energy and hence requires the most force to stop.
• Frictional forces between the tires and the road surface, as well as the efficiency of the braking system, also play a role. However, the primary determinant in this scenario remains the mass of the vehicle.

Additional Information

• Newton's Second Law of Motion:
• This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
• In the context of braking, negative acceleration (deceleration) is applied to bring the vehicle to a stop. A heavier vehicle like a bus requires more force to achieve the same deceleration as a lighter vehicle.
• Kinetic Energy:
• Kinetic energy is the energy possessed by a body due to its motion and is given by the formula K.E. = 1/2 mv².
• As the mass of the vehicle increases, its kinetic energy increases proportionally, making it harder to stop the vehicle. This is why a bus, being the heaviest among the given options, requires the most force to stop.
• Factors Affecting Braking Force:
• The mass of the vehicle: Heavier vehicles require more force to stop.
• The velocity of the vehicle: Higher speeds result in greater kinetic energy, requiring more force for deceleration.
• The condition of the brakes: Well-maintained brakes can generate higher frictional force, aiding in quicker stopping.
• Road conditions: Slippery or uneven surfaces can reduce the effectiveness of braking.
• Examples:
• In real-world scenarios, heavy vehicles like buses and trucks require longer stopping distances and more robust braking systems compared to lighter vehicles such as bicycles or scooters.
• Highway safety measures often account for the increased stopping distance of heavy vehicles by providing longer braking zones.

Important Points

• Role of Braking Systems: Advanced braking systems like Anti-lock Braking System (ABS) are used in modern vehicles to prevent skidding and reduce stopping distance, especially in heavy vehicles like buses.
• Momentum: The momentum of a vehicle is the product of its mass and velocity (p = mv). A bus has a greater momentum than a bicycle, car, or scooter at the same velocity, requiring more force to stop.
• Energy Dissipation: Brakes convert the kinetic energy of a moving vehicle into heat energy. A bus, with higher kinetic energy, generates more heat during braking, requiring a more efficient braking system.
• Real-life Application: The design of braking systems for heavy vehicles takes into account their higher mass and momentum. This is why buses and trucks are equipped with air brakes or hydraulic brakes, which provide greater stopping power.

The correct answer is Individuals who have multiple unprotected sexual partners.

Key Points

• HIV (Human Immunodeficiency Virus) is primarily transmitted through unprotected sexual intercourse, sharing of needles, and from mother to child during pregnancy, childbirth, or breastfeeding.
• Individuals who have multiple unprotected sexual partners are at significantly higher risk of contracting HIV due to increased exposure to potential carriers of the virus. This applies to heterosexual, homosexual, or bisexual relationships.
• Unprotected sexual activities can result in direct exposure to infected bodily fluids such as semen, vaginal fluids, or blood, which are primary carriers of the HIV virus.
• While maintaining personal hygiene and regular exercise are beneficial for overall health, they do not influence the risk of HIV infection. Hence, these factors are not directly related to HIV risk.
• Using protective measures like condoms during intercourse and undergoing regular health check-ups can significantly reduce the risk of HIV infection for individuals with multiple sexual partners.

Additional Information

• HIV Transmission Modes:
• Sexual Contact: The most common route of HIV transmission globally is through unprotected vaginal or anal intercourse.
• Blood Transmission: Sharing needles or syringes contaminated with infected blood can lead to HIV transmission, especially among intravenous drug users.
• Mother-to-Child: HIV can be transmitted from an infected mother to her child during pregnancy, childbirth, or breastfeeding.
• Infected Blood Products: Receiving contaminated blood or blood products during transfusions is another potential source of HIV infection, though this has decreased significantly with modern blood screening techniques.
• Prevention Strategies:
• Practice safe sex by using condoms consistently and correctly.
• Regular HIV testing, especially for individuals with multiple sexual partners or those engaging in high-risk behaviors, helps in early detection and treatment.
• Utilize Pre-Exposure Prophylaxis (PrEP), a medication for individuals at high risk of HIV infection, to prevent transmission.
• Avoid sharing needles and syringes, and ensure that tattoo or piercing equipment is sterilized properly.
• Global Statistics:
• According to the World Health Organization (WHO), approximately 38.4 million people were living with HIV globally as of 2021.
• Sub-Saharan Africa remains the most heavily affected region, accounting for nearly two-thirds of the global HIV cases.
• Early diagnosis and access to antiretroviral therapy (ART) have significantly reduced the mortality rate associated with HIV/AIDS.
• Important Points:
• HIV cannot be transmitted through casual contact such as hugging, shaking hands, or sharing food and water.
• Maintaining a healthy immune system through proper nutrition and lifestyle choices can support overall health but does not prevent HIV infection.
• Individuals in high-risk categories, such as sex workers, intravenous drug users, and men who have sex with men, should receive targeted education and prevention resources.

The correct answer is Parenchyma.

Key Points

• Parenchyma is the most common and widely distributed simple permanent tissue in plants.
• It consists of thin-walled, living cells that are loosely arranged, leaving large intercellular spaces.
• This tissue performs various functions such as photosynthesis, storage of food, and secretion.
• Parenchyma cells are found in almost all parts of the plant, including the cortex, pith, mesophyll of leaves, and vascular tissues.
• They can be specialized into forms like chlorenchyma (for photosynthesis) and aerenchyma (for buoyancy in aquatic plants).
• Their ability to divide and differentiate also makes them crucial for wound healing and regeneration in plants.

Additional Information

• Sclerenchyma:
• This tissue consists of dead cells with very thick walls, hardened due to lignin.
• Sclerenchyma provides mechanical support and rigidity to plants.
• It is commonly found in regions like the seed coat, nut shells, and around vascular bundles.
• Types of sclerenchyma include fibers and sclereids, which vary in shape and function.
• Aerenchyma:
• A specialized form of parenchyma tissue found in aquatic plants.
• It contains large air spaces between cells, aiding in buoyancy and facilitating gas exchange.
• This tissue is crucial for plants growing in waterlogged conditions.
• Examples include hydrophytes like water hyacinth and lotus.
• Chlorenchyma:
• A type of parenchyma tissue rich in chloroplasts, enabling it to perform photosynthesis.
• It is present in the green parts of plants, such as leaves and stems.
• Chlorenchyma plays a vital role in the production of food and energy for plants.
• Parenchyma:
• Parenchyma cells are living and have the ability to divide, which helps in plant growth and repair.
• This tissue is versatile and adapts to various functions such as storage, photosynthesis, secretion, and transport.
• Parenchyma can transform into other cell types during secondary growth, showcasing its plasticity.
• Examples include the storage parenchyma in potato tubers and photosynthetic parenchyma in leaf mesophyll.
• Important Points:
• Permanent tissues are classified into simple and complex types based on their structure and function.
• Simple permanent tissues consist of similar types of cells, while complex tissues like xylem and phloem consist of different cell types.
• Simple tissues include parenchyma, collenchyma, and sclerenchyma, with parenchyma being the most abundant.
• The intercellular spaces in parenchyma aid in gaseous exchange and nutrient storage.

The correct answer is Honey.

Key Points

• Apiaries, also known as bee farms, are places where beekeeping is conducted to maintain colonies of honeybees for the production of honey and other bee-related products like beeswax.
• Honey production involves honeybees collecting nectar from flowers, which is then enzymatically transformed into honey and stored in honeycombs.
• Apiaries are critical for maintaining healthy bee populations, which are essential for pollination. Pollination directly affects the growth of various crops and ecosystems.
• Honey is a natural sweetener, rich in antioxidants, and has been used for centuries for its nutritional and medicinal properties.
• Apiaries are also an environmentally friendly and sustainable way to produce honey, contributing to biodiversity and the health of plant ecosystems.

Additional Information

• Characteristics of Honey:
• Honey is composed of natural sugars like fructose and glucose, along with vitamins, minerals, and antioxidants.
• It exhibits antibacterial and antifungal properties, making it a popular choice for traditional medicine.
• Honey is used as an energy booster, digestive aid, and natural remedy for wounds and sore throats.
• Importance of Beekeeping:
• Beekeeping contributes to pollination, which is vital for the reproduction of many flowering plants and crops.
• It supports the production of other agricultural products by increasing crop yields and improving quality.
• Beekeeping generates employment opportunities and provides an additional source of income to rural communities.
• Other Products from Apiaries:
• Beeswax: Used in the production of candles, cosmetics, and polishes.
• Royal Jelly: A nutrient-rich substance produced by worker bees, known for its health benefits.
• Propolis: A resin-like material used for its medicinal properties and in health supplements.
• Comparison with Other Options:
• Poultry Egg: Obtained from poultry farming, not related to apiaries.
• Fish: Produced through aquaculture or fishing, unrelated to beekeeping.
• Milk: Derived from dairy farming, involving animals like cows, buffaloes, and goats, and not associated with apiaries.

Important Points

• Global Honey Production:
• China is the largest producer of honey globally, followed by countries like Turkey and India.
• India is one of the major exporters of honey, particularly to countries like the USA, Saudi Arabia, and the UAE.
• Role of Honeybees in Agriculture:
• Honeybees are responsible for the pollination of approximately 70% of the world’s crops.
• Key crops like apples, almonds, and blueberries rely heavily on bee pollination.
• Threats to Honeybee Populations:
• Factors like pesticide usage, habitat loss, and climate change pose significant threats to honeybee populations.
• Efforts like organic farming and establishing more apiaries can help mitigate these threats.

The correct answer is Uniform circular motion.

Key Points

• Uniform circular motion refers to the motion of an object traveling along a circular path with a constant speed.
• Even though the speed remains constant, the direction of motion continuously changes, resulting in a change in velocity. This change in velocity means that the object is accelerating, as acceleration is defined as the rate of change of velocity.
• The acceleration in uniform circular motion is directed towards the center of the circular path, and this is called centripetal acceleration.
• The force responsible for maintaining the circular motion is called the centripetal force, which acts perpendicular to the velocity of the object and towards the center of the circle.
• Examples of uniform circular motion include the motion of a satellite in orbit, the movement of a car around a circular track at constant speed, and the rotation of a fan blade.

Additional Information

• Non-linear motion:
• This refers to motion that does not follow a straight line. It can involve changes in direction or curved paths. Examples include projectile motion and motion along parabolic or elliptical paths.
• It is not specific to circular motion and does not necessarily involve constant speed. Hence, this option is incorrect in the context of the question.
• Linear motion:
• Linear motion refers to motion in a straight line, such as a car moving on a straight road or a free-falling object under gravity.
• Since the question specifies motion along a circular path, this option is incorrect as circular motion is not linear.
• Non-uniform circular motion:
• Non-uniform circular motion occurs when an object moves along a circular path but with a varying speed.
• This is different from uniform circular motion, where the speed remains constant. Hence, this option is also incorrect in the context of the question.
• Characteristics of Uniform Circular Motion:
• The magnitude of velocity (speed) is constant.
• The direction of velocity changes continuously as the object moves along the circular path.
• The force responsible for maintaining the circular motion (centripetal force) is always directed towards the center of the circle.
• The acceleration acting on the object (centripetal acceleration) is also directed towards the center of the circle and is perpendicular to the velocity.

Important Points

• Examples of Uniform Circular Motion:
• The motion of planets around the sun in circular orbits.
• The movement of electrons around the nucleus in an atom (in simplified models).
• A vehicle moving at constant speed around a circular track.
• The rotation of a ceiling fan or a merry-go-round at a constant speed.
• Equations for Uniform Circular Motion:
• The centripetal force is given by: F = (mv²)/r, where m is the mass of the object, v is its speed, and r is the radius of the circular path.
• The centripetal acceleration is given by: a = v²/r.
• The time period of revolution (T) is the time taken to complete one full circle and is related to the speed and radius by T = 2πr/v.

The correct answer is Ethanol burns with a clean, smokeless flame.

Key Points

• Ethanol burns with a clean, smokeless flame: This is the most significant property that makes ethanol a better alternative fuel compared to petrol. When ethanol combusts, it produces fewer harmful emissions such as carbon monoxide and particulates. This ensures cleaner air and reduces environmental pollution.
• Renewable Source: Ethanol is produced from biomass such as corn, sugarcane, and other plant materials, making it a renewable source of energy. In contrast, petrol is derived from fossil fuels, which are non-renewable and depleting resources.
• Lower Greenhouse Gas Emissions: The combustion of ethanol results in lower greenhouse gas emissions compared to petrol. This contributes to mitigating climate change and achieving sustainability goals.
• High Octane Number: Ethanol has a higher octane number than petrol, which helps improve engine performance and efficiency while reducing the likelihood of engine knocking.
• Biodegradability: Ethanol is biodegradable, meaning that any accidental spills or leaks cause minimal environmental damage compared to the harmful effects of petrol spills.

Additional Information

• Properties of Ethanol:
• Ethanol is a colorless, volatile, and flammable liquid. It is commonly used as a biofuel and blending agent with petrol.
• It has a lower energy content per liter compared to petrol, meaning vehicles running on ethanol may need more fuel to cover the same distance.
• Comparison of Ethanol and Petrol:
• Environmental Impact: Ethanol produces fewer toxic emissions, including carbon monoxide and unburned hydrocarbons, whereas petrol combustion releases more pollutants.
• Sustainability: Ethanol is derived from renewable sources like plants, whereas petrol is a fossil fuel derived from crude oil, a finite resource.
• Economic Benefits: The production of ethanol can boost the agricultural sector, providing farmers with additional income through the cultivation of ethanol-producing crops like sugarcane and corn.
• Applications of Ethanol as a Fuel:
• Ethanol is widely used as a fuel additive to petrol (e.g., E10, E85 blends) to reduce harmful emissions and improve engine efficiency.
• It is also used in flexible-fuel vehicles (FFVs) that can run on high ethanol blends or even pure ethanol.
• Challenges in Ethanol Usage:
• The energy density of ethanol is lower than that of petrol, which can lead to reduced mileage.
• Large-scale ethanol production may lead to food vs. fuel conflicts, as crops like corn and sugarcane are used for both food and ethanol production.
• Storage and transportation of ethanol require special handling due to its hygroscopic nature (it absorbs water from the environment).

Important Points

• Statement 1: Ethanol is a solid fuel – This is incorrect. Ethanol is a liquid fuel, not a solid one.
• Statement 2: Ethanol has a higher boiling point than petrol – This is incorrect. Ethanol’s boiling point (78.5°C) is lower than petrol’s boiling range (typically 95–200°C).
• Statement 3: Ethanol burns with a clean, smokeless flame – This is correct. Ethanol combustion produces fewer particulates, making it a cleaner alternative.
• Statement 4: Ethanol produces harmful smoke when burned – This is incorrect. Ethanol produces significantly less harmful smoke compared to petrol.

The correct answer is 180 u.

Key Points

• The molecular formula of glucose is C6H12O6, which represents six carbon atoms, twelve hydrogen atoms, and six oxygen atoms.
• The atomic masses of the elements involved are as follows: Carbon (C) = 12 u, Hydrogen (H) = 1 u, and Oxygen (O) = 16 u.
• To calculate the molecular mass of glucose, multiply the atomic mass of each element by the number of atoms of that element in the molecule:
• Carbon: 12 u × 6 = 72 u
• Hydrogen: 1 u × 12 = 12 u
• Oxygen: 16 u × 6 = 96 u
• Add these values to find the molecular mass of glucose:
• Molecular mass = 72 u + 12 u + 96 u = 180 u
• Hence, the molecular mass of glucose is 180 u.

Additional Information

• Glucose:
• Glucose is a simple sugar and a type of carbohydrate with the molecular formula C6H12O6.
• It is an important source of energy for living organisms and is commonly found in fruits and honey.
• In plants, glucose is produced during photosynthesis, using sunlight, carbon dioxide, and water.
• Molecular Mass:
• The molecular mass of a compound is the sum of the atomic masses of all the atoms in its molecular formula.
• It is expressed in atomic mass units (u).
• Molecular mass is a key property used to understand the behavior and reactions of molecules.
• Atomic Mass:
• The atomic mass of an element is the weighted average mass of its isotopes, measured in atomic mass units (u).
• For example, the atomic masses of Carbon, Hydrogen, and Oxygen are approximately 12 u, 1 u, and 16 u, respectively.
• Applications of Glucose:
• Glucose is widely used in the food and beverage industry as a sweetener.
• It is also used in the pharmaceutical industry as an energy supplement and in intravenous solutions.
• In biological processes, glucose is a primary substrate for cellular respiration, providing energy (ATP).
• Importance of Molecular Mass in Chemistry:
• Molecular mass is essential for calculating chemical quantities, such as moles and molarity.
• It helps in determining the stoichiometry of chemical reactions.
• Understanding molecular mass is crucial for fields like biochemistry, medicine, and materials science.

The correct answer is Acromegaly.

Key Points

• Acromegaly is a hormonal disorder that develops when the pituitary gland produces excess growth hormone (GH) in adulthood.
• Unlike in children, where excess GH causes gigantism, in adults, it leads to abnormal growth of bones and tissues, especially in the hands, feet, and face.
• The primary cause of acromegaly is a benign tumor (adenoma) in the pituitary gland, which secretes excessive growth hormone.
• Symptoms include enlarged hands and feet, changes in facial features (e.g., protruding jaw and brow), joint pain, thickened skin, and sometimes complications like diabetes and hypertension.
• Diagnosis is made through blood tests for elevated levels of insulin-like growth factor-1 (IGF-1) and imaging studies like an MRI to locate the tumor.
• Treatment options include surgery to remove the tumor, medications to block GH production, and radiation therapy.

Additional Information

• Graves’ Disease:
• It is an autoimmune disorder that causes hyperthyroidism, where the thyroid gland produces excessive thyroid hormones.
• Symptoms include bulging eyes (exophthalmos), weight loss, rapid heartbeat, and irritability.
• It primarily affects women and is often treated with antithyroid medications, radioactive iodine therapy, or surgery.
• Addison’s Disease:
• This is a rare disorder where the adrenal glands do not produce enough cortisol and aldosterone.
• Symptoms include fatigue, weight loss, low blood pressure, and hyperpigmentation of the skin.
• Treatment involves hormone replacement therapy to correct the levels of deficient hormones.
• Dwarfism:
• Dwarfism is a condition characterized by short stature, typically resulting from a genetic mutation or hormonal deficiencies such as growth hormone deficiency in childhood.
• It contrasts with acromegaly, as it occurs when growth hormone levels are insufficient during growth years.
• Management depends on the type of dwarfism and may include growth hormone therapy in certain cases.
• Growth Hormone (GH):
• Produced by the anterior pituitary gland, GH plays a crucial role in regulating growth, metabolism, and cell regeneration.
• GH levels are highest during childhood and adolescence, promoting the growth of bones and tissues.
• Both over-secretion (acromegaly/gigantism) and under-secretion (dwarfism) of GH can lead to significant health issues.

Important Points

• Gigantism vs. Acromegaly:
• Gigantism occurs in children when excess GH secretion happens before the fusion of growth plates, leading to abnormal height and proportional growth.
• Acromegaly occurs in adults after the growth plates have fused, leading to thickening of bones and tissues rather than increased height.
• Complications of Acromegaly:
• Untreated acromegaly can lead to severe complications such as cardiovascular diseases, diabetes mellitus, sleep apnea, and arthritis.
• Early diagnosis and treatment are essential to prevent these complications and improve the quality of life.
• Diagnostic Tests for Hormonal Disorders:
• Blood tests to measure levels of hormones like GH, IGF-1, cortisol, and thyroid hormones are critical for diagnosing related disorders.
• Imaging techniques like MRI or CT scans are used to detect pituitary tumors or adrenal gland abnormalities.

The correct answer is Virtual and erect image.

Key Points

• A convex mirror always forms a virtual and erect image, irrespective of the position of the object.
• The image formed by a convex mirror is always diminished, meaning it is smaller than the actual object.
• A convex mirror diverges the reflected light rays, making them appear to originate from a single point behind the mirror. This is why the image is virtual and cannot be projected on a screen.
• Unlike concave mirrors, a convex mirror does not produce real images because the reflected rays never actually meet but appear to diverge from a point behind the mirror.
• Convex mirrors are widely used in vehicles as rearview mirrors due to their ability to provide a wider field of view and erect images.

Additional Information

• Characteristics of Convex Mirror:
• Convex mirrors have a reflecting surface that bulges outward.
• They are also known as diverging mirrors because they cause parallel rays of light to spread apart after reflection.
• The focal length of a convex mirror is positive, and the focus is virtual and located behind the mirror.
• The image formed by a convex mirror is always located behind the mirror and is smaller than the object.
• Applications of Convex Mirrors:
• Used as rearview mirrors in vehicles to provide a wide field of view, helping drivers see traffic behind them.
• Installed in parking areas and driveways for better visibility and to prevent accidents.
• Used in security and surveillance systems for monitoring large areas.
• Other Types of Mirrors:
• Concave Mirror: Forms real or virtual images depending on the object's position. Images can be magnified or inverted.
• Plane Mirror: Always forms virtual and erect images that are the same size as the object.
• Key Equations of Mirrors:
• The mirror formula is given by: 1/f = 1/v + 1/u, where f is the focal length, v is the image distance, and u is the object distance.
• The magnification (M) is calculated using: M = h'/h = -v/u, where h' is the height of the image, and h is the height of the object.

Important Points

• Option 1: Magnified image: A convex mirror never forms a magnified image, as the image is always diminished. This option is incorrect.
• Option 2: Virtual and erect image: This is the only correct statement, as convex mirrors always form virtual, erect, and diminished images.
• Option 3: Real and inverted image: Real images are not formed by convex mirrors because the reflected rays diverge and never actually meet. This option is incorrect.
• Option 4: Real and erect image: Convex mirrors do not produce real images, making this option incorrect.

The correct answer is sulphide.

Key Points

• Roasting is a metallurgical process in which a sulphide ore is heated in the presence of excess air to convert it into its oxide form. This is a necessary step before the ore can be reduced to the pure metal.
• The primary purpose of roasting is to eliminate sulphur from the ore by converting it into gaseous sulphur dioxide (SO₂). For example, zinc sulphide (ZnS) is roasted to form zinc oxide (ZnO).
• Sulphide ores are particularly suitable for roasting because they release harmful sulphur compounds during heating, which must be oxidized into a less harmful form.
• Roasting is commonly used for sulphide ores like zinc blende (ZnS), cinnabar (HgS), and galena (PbS). For example:
• ZnS + O₂ → ZnO + SO₂
• PbS + O₂ → PbO + SO₂
• Unlike sulphide ores, carbonate, chloride, and sulphite ores are not primarily treated using roasting. They require different processes such as calcination or leaching.

Additional Information

• Roasting vs. Calcination:
• Roasting is carried out in the presence of excess air to oxidize sulphide ores into oxides. It also helps remove volatile impurities such as sulphur dioxide.
• Calcination, on the other hand, is done in the absence or limited supply of air and is used for carbonate and hydroxide ores to drive off carbon dioxide or water.
• Processes for Other Ores:
• Carbonate ores: These are typically treated by calcination. For example, calcium carbonate (CaCO₃) is heated to produce calcium oxide (CaO) and carbon dioxide (CO₂).
• Chloride ores: These are usually treated by leaching or electrolysis. For instance, sodium chloride (NaCl) is used in the electrolysis process to produce sodium and chlorine.
• Sulphite ores: Sulphite ores are rare and are generally processed differently, often involving chemical methods to extract the desired metal.
• Environmental Impact:
• The roasting process releases sulphur dioxide (SO₂), which is a major contributor to air pollution and acid rain. Modern roasting facilities employ gas scrubbers to capture and convert SO₂ into useful by-products like sulphuric acid.
• Common Metals Extracted via Roasting:
• Zinc: Extracted from zinc blende (ZnS).
• Lead: Extracted from galena (PbS).
• Copper: Extracted from copper pyrites (CuFeS₂).

Important Points

• Thermodynamics of Roasting: Roasting is an exothermic process, and the heat generated helps sustain the reaction.
• Reactions in Roasting: In addition to oxide formation, other reactions such as sulphation (formation of sulphates) and chlorination (formation of metal chlorides) can also occur depending on the conditions.
• Industrial Use: The sulphur dioxide generated during roasting is often converted into sulphuric acid, a valuable industrial chemical.

The correct answer is Burning in air.

Key Points

• Chemical properties describe the ability of a substance to undergo chemical changes, resulting in the formation of new substances. Burning in air involves a chemical reaction where the substance reacts with oxygen to produce new compounds, such as carbon dioxide and water. Hence, this is a chemical property.
• Boiling point is a physical property as it involves a phase change from liquid to gas without altering the chemical composition of the substance. Hence, this is not a chemical property.
• Melting point, like boiling point, is a physical property that signifies the temperature at which a solid becomes a liquid. It does not involve a chemical change. Hence, this is not a chemical property.
• Fluidity refers to the ability of a substance to flow, which is related to its physical characteristics like viscosity and state of matter. It does not involve a chemical reaction. Hence, this is not a chemical property.
• Burning in air is a chemical property because it involves oxidation, a chemical reaction resulting in the formation of new substances. For example, burning wood produces ash, carbon dioxide, and water vapor, which are chemically different from the original wood.

Additional Information

• Definition of chemical properties: Chemical properties are characteristics of a substance that become evident during a chemical reaction. Examples include flammability, reactivity with acids, and oxidation states.
• Examples of chemical properties:
• Flammability: The ability of a substance to burn in the presence of oxygen, such as wood or gasoline.
• Reactivity with acids: Substances like metals reacting with acids to produce hydrogen gas.
• Oxidation: The ability of a substance to react with oxygen, like rusting of iron.
• Physical properties vs. chemical properties:
• Physical properties can be observed without changing the substance's chemical identity, e.g., color, boiling point, and melting point.
• Chemical properties involve observing how a substance reacts chemically, leading to the formation of new substances.
• Combustion process: Combustion is a chemical reaction where a substance reacts with oxygen to release energy in the form of heat and light. For instance, burning hydrocarbons produces carbon dioxide and water.
• Industrial significance: Understanding chemical properties is crucial in industries like pharmaceuticals, materials science, and environmental studies for developing new compounds and understanding reactions.

Important Points

• Burning in air: This process is an example of oxidation, which is a type of chemical reaction. It is exothermic, releasing energy.
• Boiling and melting points: These are physical properties that are used to identify substances but do not involve chemical changes.
• Fluidity: It is related to the physical state and viscosity of a substance and does not involve chemical reactivity.
• Understanding the distinction between physical and chemical properties is essential for scientific studies and industrial applications, as it helps predict the behavior of substances under different conditions.

The correct answer is Sexual reproduction.

Key Points

• Sexual reproduction is a biological process in which both male and female sexes are involved to produce offspring.
• It involves the fusion of specialized cells known as gametes (male sperm and female egg), leading to the formation of a zygote.
• This process ensures genetic variation, as the offspring inherit characteristics from both parents, promoting adaptability and evolution.
• Sexual reproduction occurs in humans, animals, and many plants. In plants, the male and female gametes are produced in different structures, such as stamens and carpels.
• Unlike asexual reproduction methods (like budding, spore formation, fragmentation), sexual reproduction necessarily requires the involvement of two parents of opposite sexes.

Additional Information

• Budding:
• Budding is an asexual reproduction method where a new organism grows from a bud or an outgrowth on the parent body.
• It is commonly seen in organisms like yeast and hydra. In this process, only one parent is involved, and the offspring is genetically identical to the parent.
• Spore Formation:
• Spore formation is another asexual reproduction method where organisms produce spores that develop into new individuals.
• It is common in fungi, algae, and bacteria. Spores are highly resistant structures designed for survival under unfavorable conditions.
• Fragmentation:
• Fragmentation is an asexual reproduction process where an organism breaks into fragments, and each fragment develops into a new organism.
• This method is prevalent in sponges, algae, and flatworms. It requires only one parent and produces genetically identical offspring.
• Advantages of Sexual Reproduction:
• Sexual reproduction provides genetic diversity, helping species survive and adapt to changing environments.
• It reduces the risk of genetic disorders caused by mutation accumulation, unlike asexual reproduction.
• It facilitates evolution and natural selection, ensuring the survival of the fittest.

Important Points

• Examples of Sexual Reproduction:
• In humans, the male gamete (sperm) and female gamete (egg) fuse during fertilization to form a zygote.
• In flowering plants, the pollen grain (male gamete) fertilizes the ovule (female gamete) to form seeds.
• In animals like frogs, external fertilization occurs in water, where sperm and eggs are released simultaneously.
• Key Components of Sexual Reproduction:
• Meiosis: A special type of cell division that produces haploid gametes.
• Fertilization: The fusion of male and female gametes to form a diploid zygote.
• Genetic Recombination: Exchange of genetic material during meiosis ensures variation.
• Evolutionary Perspective:
• Sexual reproduction plays a crucial role in the adaptive evolution of species.
• It introduces new gene combinations, helping organisms cope with environmental challenges.

The correct answer is Molecular movement is only found in animals, not plants.

Key Points

• Statement 1: Growth-related movement in plants, like bending toward light, is a form of movement. Plants exhibit phototropism, which is their ability to grow toward light due to the action of auxin hormones. This is a correct statement.
• Statement 2: Living organisms show internal or molecular movement, even when external movement is not visible. Internal processes such as cellular respiration, cytoplasmic streaming, and molecular transport are universal to living organisms. This statement is correct.
• Statement 3: Molecular movement is only found in animals, not plants. This is incorrect because plants also exhibit molecular movement, including nutrient transport through xylem and phloem, photosynthesis, and cellular processes. Hence, this is the correct option.
• Statement 4: Movement is a characteristic of living organisms. Movement, whether external (locomotion) or internal (molecular and cellular processes), is indeed a defining characteristic of living organisms. This statement is correct.

Additional Information

• Plant Movements:
• Phototropism: Movement of plants toward light due to auxin distribution.
• Geotropism: Movement of roots toward gravity, aiding in nutrient absorption.
• Nastic Movements: Non-directional responses to stimuli such as touch (e.g., Mimosa pudica).
• Hydrotropism: Movement of roots toward water sources for survival and growth.
• Molecular Movement in Plants:
• Photosynthesis: Molecular movement of carbon dioxide, water, and glucose within plant cells.
• Xylem and Phloem Transport: Movement of nutrients, water, and food across plant tissues.
• Cytoplasmic Streaming: Movement of organelles and molecules within cells for efficient functioning.
• Stomatal Regulation: Molecular movement of gases like oxygen and carbon dioxide through stomata.
• Characteristics of Living Organisms:
• Respiration: Involves molecular movement of oxygen and carbon dioxide.
• Growth: Cellular and molecular processes aid in development.
• Reproduction: Molecular movement in gamete formation and fertilization.
• Excretion: Movement of waste molecules from cells to the outside environment.

Important Points

• Types of Movement in Living Organisms:
• External Movement: Includes locomotion like walking, swimming, or flying.
• Internal Movement: Includes processes like blood circulation, digestion, and cellular transport.
• Role of Hormones:
• Auxins: Promote growth-related movements in plants.
• Adrenaline: Facilitates movement during stress in animals.
• Examples of Plant Molecular Movement:
• Transpiration: Movement of water vapor out of stomata.
• Nutrient Uptake: Absorption of minerals and water through roots.

The correct answer is Contract and make the lens thicker.

Key Points

• The ciliary muscles are a part of the eye responsible for adjusting the shape of the lens to focus on objects at different distances. This process is called accommodation.
• When a person looks at a nearby object, the ciliary muscles contract, reducing the tension on the suspensory ligaments (zonules of Zinn).
• This relaxation of the suspensory ligaments allows the lens to become thicker and more convex, increasing its refractive power to focus light precisely on the retina.
• The ability of the lens to become thicker is crucial for focusing on objects that are close to the eye, such as while reading or looking at a smartphone screen.
• This process ensures clear vision for nearby objects by increasing the focal length of the lens. Hence, the statement "Contract and make the lens thicker" is correct.

Additional Information

• Function of Ciliary Muscles:
• The ciliary muscles are smooth muscles within the eye located around the lens and are controlled by the autonomic nervous system.
• They are responsible for controlling the lens curvature to enable focusing on objects at varying distances.
• When focusing on distant objects, the ciliary muscles relax, allowing the lens to become thinner and less convex.
• When focusing on nearby objects, the ciliary muscles contract, allowing the lens to become thicker and more convex.
• Accommodation Mechanism:
• The process of accommodation refers to the eye's ability to adjust the lens shape to focus light from objects at different distances.
• It involves the ciliary muscles, suspensory ligaments, and the lens working together.
• For nearby objects, the lens becomes thicker to increase its refractive power, and for distant objects, the lens becomes thinner to reduce refractive power.
• Explanation of Incorrect Options:
• Relax and make the lens thicker: This is incorrect because relaxation of the ciliary muscles causes the lens to become thinner, not thicker.
• Contract and make the lens thinner: This is incorrect because contraction of the ciliary muscles allows the lens to become thicker, not thinner.
• Relax and make the lens thinner: This is correct only for distant vision and not for nearby objects.
• Relevance of Accommodation:
• The process of accommodation is essential for day-to-day activities such as reading, writing, and recognizing objects at different distances.
• Issues with accommodation can lead to conditions such as presbyopia, where the lens loses flexibility with age, or myopia and hyperopia, where the eye's focusing ability is impaired.

Important Points

• The near point of the human eye (the closest distance at which the eye can focus on an object) is approximately 25 cm for a normal young adult.
• The far point (the farthest distance at which the eye can focus) is theoretically infinity for a normal eye.
• Accommodation diminishes with age due to the loss of elasticity in the lens, a condition known as presbyopia, which typically begins around the age of 40 years.
• Accommodation is controlled by the parasympathetic nervous system, which activates the contraction of ciliary muscles.
• Some optical devices, such as reading glasses or contact lenses, are used to correct accommodation defects like presbyopia or myopia.

The correct answer is The gas is carbon dioxide; the solid is either a carbonate or bicarbonate..

Key Points

• The reaction of a white solid with dilute hydrochloric acid (HCl) producing a gas that turns lime water milky confirms the gas is carbon dioxide (CO₂). This is a characteristic test for carbon dioxide.
• Lime water (calcium hydroxide solution) reacts with carbon dioxide to form calcium carbonate (CaCO₃), which appears as a milky precipitate. This confirms the presence of carbon dioxide. Hence, this statement is correct.
• The white solid reacting with dilute HCl to release carbon dioxide indicates that the solid is either a carbonate (CO₃^2-) or a bicarbonate (HCO₃^-). Both compounds react with acids to release CO₂. Hence, this statement is correct.
• While the gas is confirmed as carbon dioxide, the exact identity of the solid (e.g., calcium carbonate, sodium bicarbonate, etc.) cannot be determined solely from this reaction. Further tests would be required to specify the compound. Hence, the statement about the solid being calcium carbonate alone is incorrect.
• The reaction does not involve hydrogen gas or any reactive metal. Hydrogen is released when acids react with metals, but this is not the case here. Hence, this statement is incorrect.

Additional Information

• Carbonates and Bicarbonates:
• Carbonates (e.g., CaCO₃, Na₂CO₃) and bicarbonates (e.g., NaHCO₃) are compounds containing carbonate ions (CO₃^2-) or bicarbonate ions (HCO₃^-).
• When these compounds react with acids, they release carbon dioxide gas according to the reactions:
• Carbonate: CO₃^2- + 2H⁺ → CO₂ + H₂O
• Bicarbonate: HCO₃^- + H⁺ → CO₂ + H₂O
• Lime Water Test:
• The lime water test is a standard chemical test for the identification of carbon dioxide gas.
• When carbon dioxide is bubbled through lime water (Ca(OH)₂ solution), it reacts to form calcium carbonate (CaCO₃), which is insoluble in water and appears as a milky precipitate:
• Ca(OH)₂ + CO₂ → CaCO₃ + H₂O
• If excess CO₂ is passed, the milkiness disappears as calcium carbonate reacts with water and carbon dioxide to form calcium bicarbonate, which is soluble in water:
• CaCO₃ + H₂O + CO₂ → Ca(HCO₃)₂
• Difference Between Carbonates and Bicarbonates:
• Carbonates contain CO₃^2- ions, while bicarbonates contain HCO₃^- ions.
• Both types of compounds react with acids to release CO₂ gas, but bicarbonates require only one H⁺ ion per molecule, while carbonates require two.
• Common Applications:
• Calcium carbonate (CaCO₃) is used in construction (as limestone), manufacturing (as a raw material for cement), and as an antacid in medicine.
• Sodium bicarbonate (NaHCO₃), commonly known as baking soda, is widely used in baking, cleaning, and as an antacid.
• Key Experiment Details:
• The reaction between acids and carbonates/bicarbonates is an example of a gas-evolution reaction.
• Such reactions are often used in laboratories to identify unknown compounds.

The correct answer is Trophic level.

Key Points

• Each step in a food chain represents a trophic level. It defines the position of an organism in the flow of energy and nutrients in an ecosystem.
• The first trophic level consists of producers, such as plants, which convert solar energy into chemical energy through photosynthesis.
• The second trophic level includes primary consumers, typically herbivores, which feed on producers for energy.
• Secondary consumers occupy the third trophic level and are carnivores or omnivores that eat primary consumers.
• The fourth trophic level and beyond consist of tertiary consumers, which are predators feeding on secondary consumers. Energy transfer diminishes as it progresses through the levels due to the 10% energy rule.

Additional Information

• Food Chain:
• A food chain is a sequence of organisms through which energy and nutrients flow, starting with producers and ending with top predators.
• Examples: Grass → Grasshopper → Frog → Snake → Hawk.
• Food chains are linear and show a single pathway of energy flow.
• Trophic Levels:
• Trophic levels are hierarchical steps in a food chain that represent an organism's position in the ecosystem's energy flow.
• There are usually 4 to 5 trophic levels in an ecosystem due to energy loss at each level.
• Energy transfer between levels follows the 10% energy rule, which states that only 10% of energy is passed on to the next level, while the rest is lost as heat.
• Stratosphere Level:
• The stratosphere is a layer of Earth’s atmosphere above the troposphere, extending from about 10 to 50 km altitude.
• It contains the ozone layer, which absorbs harmful ultraviolet radiation from the sun.
• It is unrelated to food chains or trophic levels.
• Troposphere Level:
• The troposphere is the lowest layer of Earth’s atmosphere, where weather phenomena occur and life exists.
• It extends up to about 10 km above sea level.
• It does not correspond to food chains or trophic levels.
• Ozone Layer:
• The ozone layer, located in the stratosphere, protects Earth from harmful ultraviolet (UV) radiation.
• It plays no role in food chains or the concept of trophic levels.

Important Points

• Energy Pyramid:
• It represents the flow of energy through trophic levels in a food chain.
• The base of the pyramid consists of producers, while the apex represents top predators.
• Energy decreases as it moves upward due to the 10% rule.
• Significance of Trophic Levels:
• They help in understanding energy distribution and ecosystem health.
• Any disruption in one level affects the entire food chain.

The correct answer is 16X J.

Key Points

• Kinetic Energy Formula: The kinetic energy (KE) of a body is calculated using the formula KE = (1/2)mv², where m is the mass of the body and v is its velocity.
• Initial Kinetic Energy: When the velocity of the body is v, the kinetic energy is given as ‘X’ joules. This implies X = (1/2)mv².
• Velocity Changes to 4v: If the velocity of the body increases to 4v, the new kinetic energy becomes KE = (1/2)m(4v)² = (1/2)m(16v²).
• Relation Between Old and New KE: The new kinetic energy is 16 times the initial kinetic energy. Therefore, the new kinetic energy equals 16X joules.
• Correct Option: The correct answer is 16X J, as the kinetic energy increases by a factor of 16 when the velocity is quadrupled.

Additional Information

• Kinetic Energy and Velocity Relationship: Kinetic energy is directly proportional to the square of the velocity. When velocity is increased, the kinetic energy increases quadratically (v²).
• Implications of Velocity Increase: For example, doubling the velocity increases the kinetic energy by a factor of 4, tripling it increases by a factor of 9, and quadrupling it increases by a factor of 16.
• Units of Kinetic Energy: The SI unit of kinetic energy is joules (J), where 1 joule is equivalent to the energy transferred when 1 newton of force moves an object by 1 meter.
• Examples of Kinetic Energy: Kinetic energy is observed in moving objects like vehicles, flowing water, flying airplanes, and falling objects. For instance, a faster-moving car has more kinetic energy than a slower one.
• Conservation of Energy: Kinetic energy is a part of the total mechanical energy, which is conserved in closed systems unless external forces like friction or air resistance act on the object.

Important Points

• Mathematical Derivation: KE = (1/2)m(4v)² simplifies to KE = (1/2)m × 16v² = 16 × (1/2)mv² = 16X.
• Energy Doubling Rule: If the velocity doubles, the kinetic energy increases fourfold. This principle is crucial in understanding energy dynamics in physics.
• Applications: The concept of kinetic energy is used in designing vehicles, calculating energy requirements for motion, and analyzing impacts during collisions.
• Practical Implications: In real-world scenarios, increased velocity leads to higher energy consumption and greater potential for damage in collisions.

The correct answer is Both low melting and boiling point.

Key Points

• Covalent bonding occurs when atoms share electrons to achieve stability, forming molecules. Carbon compounds exhibit covalent bonding, leading to specific physical properties.
• Carbon compounds generally have low melting and boiling points because covalent bonds are strong within molecules, but the intermolecular forces (like van der Waals forces) are weak. Hence, these compounds do not require much energy to change their state.
• Covalent compounds, including carbon-based molecules, are typically poor conductors of electricity because they lack free ions or electrons to carry the charge.
• The low melting and boiling points of carbon compounds are a result of their molecular structure and weak intermolecular forces, distinguishing them from ionic compounds with higher melting and boiling points.
• These properties make carbon compounds suitable for applications in organic chemistry, biochemistry, and materials science.

Additional Information

• Covalent Bonding:
• Covalent bonds are formed by the sharing of electrons between atoms to achieve a stable electron configuration (often a full outer shell).
• The strength of covalent bonds depends on the number of shared electron pairs; single bonds are weaker than double or triple bonds.
• Examples of covalent compounds include methane (CH₄), water (H₂O), and carbon dioxide (CO₂).
• Properties of Covalent Compounds:
• Low melting and boiling points: Due to weak intermolecular forces between molecules, covalent compounds require less energy to break these forces.
• Poor electrical conductivity: Covalent compounds do not have free ions or electrons, making them insulators.
• Solubility: Covalent compounds are typically soluble in nonpolar solvents like benzene but insoluble in polar solvents like water.
• Carbon Compounds:
• Carbon compounds form the basis of organic chemistry and are found in living organisms.
• Examples include hydrocarbons, alcohols, ketones, and carboxylic acids.
• Carbon's ability to form stable covalent bonds with itself and other elements (like hydrogen, oxygen, nitrogen) allows for a diverse range of compounds.
• Comparison with Ionic Compounds:
• Ionic compounds: High melting and boiling points due to strong electrostatic forces between ions.
• They generally conduct electricity in molten or dissolved states because of free-moving ions.
• Covalent compounds differ significantly due to weaker intermolecular forces and lack of ions.

Important Points

• Applications of Carbon Compounds:
• Medicinal Chemistry: Carbon-based compounds like drugs and antibiotics play a pivotal role in healthcare.
• Material Science: Polymers and plastics are carbon-based covalent compounds widely used in industry.
• Energy Sources: Hydrocarbons, such as methane and petroleum, serve as major energy resources.
• Biological Importance: Carbon compounds like carbohydrates, proteins, and nucleic acids are essential for life processes.
• Facts about Carbon:
• Carbon is the sixth element in the periodic table with an atomic number of 6.
• It is unique due to its ability to form long chains and rings through covalent bonding.
• Carbon exists in various allotropes, including diamond, graphite, and fullerenes, showcasing its versatility.

The correct answer is Option 1: Spore formation is a type of fragmentation process..

Key Points

• Fragmentation is a type of asexual reproduction where an organism splits into fragments, and each fragment develops into a new organism. This statement is correct.
• Spore formation is not a type of fragmentation but is an entirely different method of asexual reproduction. In spore formation, organisms produce spores that germinate under favorable conditions to form new individuals. Hence, Option 1 is incorrect.
• Fragmentation leads to the production of genetically identical offspring because it involves cloning the parent's genetic material. This statement is correct.
• Fragmentation occurs mainly in multicellular organisms such as algae, fungi, and certain plants. It is less common in animals. This statement is correct.
• In fragmentation, the organism often undergoes maturation before splitting into fragments that are capable of developing into fully functional individuals. This statement is correct.

Additional Information

• Fragmentation:
• Occurs in organisms like spirogyra, planaria, and certain fungi.
• It is a common reproduction method in aquatic environments where external environmental factors like water currents help in dispersing the fragments.
• Spore Formation:
• Spore formation is a reproductive method seen in fungi, bacteria, and certain plants like ferns and mosses.
• It involves the production of specialized cells called spores, which are resistant to unfavorable conditions and germinate when conditions improve.
• Asexual Reproduction:
• Involves a single parent and produces offspring that are genetically identical to the parent.
• Common types include binary fission, budding, fragmentation, and spore formation.
• Advantages of Fragmentation:
• Allows rapid reproduction in favorable environments.
• Ensures genetic uniformity, which can be beneficial in stable conditions.

Important Points

• Fragmentation vs Spore Formation:
• Fragmentation involves splitting of the organism's body, while spore formation involves the release of specialized reproductive cells.
• Fragmentation is common in multicellular organisms like spirogyra, whereas spore formation is seen in organisms like fungi and mosses.
• Examples:
• Fragmentation: Spirogyra (a type of algae), Planaria (a flatworm).
• Spore Formation: Rhizopus (a type of fungus), Ferns (plants).

The correct answer is NaHCO₃.

Key Points

• Baking soda is chemically known as sodium bicarbonate and has the chemical formula NaHCO₃. It is widely used in cooking, cleaning, and various household applications.
• It is a white crystalline compound that appears as a fine powder. It is slightly alkaline in nature and releases carbon dioxide when heated or mixed with an acid.
• When used in baking, baking soda acts as a leavening agent. It reacts with acidic components in the dough (e.g., buttermilk, vinegar) to produce carbon dioxide, causing the dough to rise.
• In addition to cooking, sodium bicarbonate is used for cleaning due to its mild abrasive nature and ability to neutralize odors.
• Baking soda is also utilized in medicine as an antacid to treat indigestion and heartburn by neutralizing stomach acid.

Additional Information

• Other Chemical Formulas Mentioned:
• NaCl (Sodium Chloride): Commonly known as table salt, NaCl is used in cooking, preserving food, and various industrial processes.
• Na₂CO₃ (Sodium Carbonate): Also known as washing soda, it is used in cleaning agents, glass manufacturing, and as a water softener.
• NaOH (Sodium Hydroxide): Commonly called caustic soda, it is a strong alkali used in soap making, paper production, and chemical manufacturing.
• Properties of NaHCO₃:
• It is a mild base and reacts with acids to release carbon dioxide gas.
• The decomposition of NaHCO₃ occurs at temperatures above 50°C, producing sodium carbonate (Na₂CO₃), water (H₂O), and carbon dioxide (CO₂).
• It is water-soluble and has a slightly alkaline pH of approximately 8.3 in solution.
• Applications:
• Cooking: Used in baking to make cakes and bread fluffy by releasing carbon dioxide gas during the reaction.
• Cleaning: Acts as a deodorizer and a mild abrasive for cleaning surfaces.
• Medical Use: Treats acid indigestion and heartburn by neutralizing excess stomach acid.
• Fire Extinguishers: Used in dry chemical fire extinguishers as it releases CO₂, which helps suppress flames.
• Environmental Benefits: Sodium bicarbonate is an environmentally friendly compound and is safe for use in households.

Important Points

• Safe Usage: Baking soda is generally safe to use but should be handled with care in large quantities as excessive ingestion can cause alkalosis.
• Storage: Sodium bicarbonate should be stored in a cool, dry place to prevent clumping and loss of efficacy.
• History: Sodium bicarbonate has been in use since ancient times and was first produced commercially in the late 18th century by French chemist Nicolas Leblanc.

The correct answer is 0.75 A.

Key Points

• The power formula in electrical engineering is derived from Ohm's Law and is given by the equation: P = V × I, where:
• P: Power in watts (W)
• V: Voltage in volts (V)
• I: Current in amperes (A)
• To find the current drawn by a device, the formula is rearranged as: I = P / V.
• In this case, the power rating of the device is 90 W, and the voltage is 120 V.
• Substituting these values into the formula, we get:
• I = 90 W / 120 V
• I = 0.75 A
• Hence, the device draws a current of 0.75 A.

Additional Information

• Power (P):
• It represents the rate at which electrical energy is consumed or converted into another form, such as heat, light, or mechanical energy.
• The SI unit of power is the watt (W).
• Voltage (V):
• Voltage is the potential difference between two points in an electrical circuit.
• It drives the flow of electric current, and its SI unit is the volt (V).
• Current (I):
• Electric current is the flow of electric charge through a conductor.
• The SI unit of current is the ampere (A).
• Ohm's Law:
• Ohm's Law states that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance.
• The formula is: V = I × R, where R is resistance in ohms (Ω).
• Common Applications:
• The calculation of current, power, and voltage is crucial in designing and using electrical devices like heaters, fans, and bulbs.
• For example, if a bulb is rated at 60 W and operates at 220 V, the current drawn can be calculated to ensure it works safely within circuit limits.

Important Points

• Electrical Safety:
• Understanding current and voltage ratings is critical to prevent overloading circuits or causing electrical fires.
• Devices should always be used within their specified voltage and current limits.
• Power Efficiency:
• Devices with higher efficiency consume less power for the same output, making them cost-effective and environmentally friendly.
• For example, LED lights consume less power compared to incandescent bulbs while providing the same brightness.
• Measurement Tools:
• Devices like multimeters are used to measure voltage, current, and resistance in circuits.
• They are essential tools for electricians and engineers to diagnose and repair electrical issues.

The correct answer is Double.

Key Points

• Gravitational Force Formula: The gravitational force between two objects is given by Newton's Law of Universal Gravitation: F = G × (m₁ × m₂) / r², where:
• F is the gravitational force.
• G is the gravitational constant (approximately 6.674 × 10⁻¹¹ N·m²/kg²).
• m₁ and m₂ are the masses of the two objects.
• r is the distance between the centers of the two objects.
• • This law states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• It is one of the fundamental laws in physics and helps explain phenomena such as planetary motion and tides.
• Gravitational Constant (G):
• The gravitational constant (G) has a fixed value of approximately 6.674 × 10⁻¹¹ N·m²/kg².
• It is a universal constant used in calculations of gravitational forces across the cosmos.
• Impact of Distance:
• The gravitational force is inversely proportional to the square of the distance between two objects. If the distance increases, the force decreases significantly.
• In this scenario, the distance remains constant, ensuring that only mass changes affect the force.
• Real-World Applications:
• Gravitational forces are crucial in understanding orbital mechanics, satellite launches, and space exploration.
• The moon's gravitational interaction with Earth also influences tides and stabilizes Earth's axial tilt.

Important Points

• Gravitational Force in Space: Gravitational force governs the motion of celestial bodies, including planets, moons, and stars.
• Tides on Earth: The gravitational interaction between the Earth and the moon causes tides, with the moon's gravity pulling ocean waters toward it.
• Doubling Mass Impact: Doubling the mass of any celestial object significantly alters gravitational interactions and may affect orbital stability.

The correct answer is ρ/a.

Key Points

• Resistance Formula: The resistance between two opposite faces of a cube is calculated using the formula: R = ρ × L/A, where:
• ρ is the resistivity of the material.
• L is the length of the cube (distance between the faces).
• A is the cross-sectional area of the face through which current flows.
• Dimensions of the Cube: For a cube of side a:
• The length L between two opposite faces is a.
• The cross-sectional area A is a² (area of the square face).
• Resistance Calculation: Substituting the cube dimensions into the resistance formula:
• R = ρ × a / a²
• R = ρ / a

Hence, the resistance between the opposite faces of the cube is ρ/a.

• Correct Option: Among the provided options, only ρ/a (Option 4) matches the calculated resistance.

Additional Information

• Resistivity (ρ):
• Resistivity is an intrinsic property of a material that quantifies its ability to resist the flow of electric current.
• It is measured in ohm-meters (Ω·m).
• Materials with low resistivity are good conductors (e.g., copper, silver), while those with high resistivity are insulators (e.g., rubber, glass).
• Resistance (R):
• Resistance is the opposition to the flow of electric current through a material.
• It depends on the material's resistivity, length of the conductor, and cross-sectional area.
• Unit of resistance is ohms (Ω).
• Factors Affecting Resistance:
• Length: Resistance increases with an increase in the length of the conductor.
• Cross-sectional Area: Resistance decreases with an increase in cross-sectional area.
• Material: Resistance depends on the material's resistivity (ρ).
• Applications of Resistance Calculations:
• Used in designing electrical circuits and components like resistors.
• Helps in determining the energy loss in transmission lines.
• Critical in material science for understanding the properties of conductive and insulating materials.

Important Points

• Special Case of a Cube:
• When the shape of the object is a regular cube, the symmetry simplifies the calculation of resistance.
• Resistance between opposite faces depends only on the resistivity and side length, as shown in the formula R = ρ/a.
• Practical Examples:
• Resistance measurements of semiconductor crystals shaped like cubes.
• Electrical resistance in small-scale materials used in research and development.

Shortcut Trick

Apply Successive Percentage formula: Net Change = a + b + (a × b)/100

Where a = +30% (Mark up) and b = −38% (Discount).

Net change = 30 − 38 + (30 × −38)/100 = −8 − 11.4 = −19.4%

Negative sign indicates a loss.

∴ The correct answer is 19.4%.

Alternate Method

Given: Mark-up = 30%, Discount = 38%

Formula: SP = CP × (100 + Mark-up%)/100 × (100 − Discount%)/100

⇒ Let Cost Price (CP) = 100

⇒ Marked Price (MP) = 100 + 30% of 100 = 130

⇒ Selling Price (SP) = 130 − 38% of 130

⇒ SP = 130 − (38/100 × 130) = 130 − 49.4 = 80.6

⇒ Loss = CP − SP = 100 − 80.6 = 19.4

⇒ Loss Percentage = (19.4 / 100) × 100 = 19.4%

∴ The correct answer is 19.4%.

Additional Information

Mark-up and Discount Relation

Profit/Loss% is the resultant of Mark-up% and Discount%. Formula: MP/CP = (100 + P%)/(100 − D%).

Successive Discount

If two discounts d₁ and d₂ are given, equivalent discount = d₁ + d₂ − (d₁ × d₂)/100.

Marked Price

MP is always calculated on CP, while Discount is always calculated on MP to arrive at the SP.

Given:

cos θ = 8 ÷ 17

Formula Used:

sin²θ + cos²θ = 1

sin θ = √(1 − cos²θ)

Calculation:

⇒ cos²θ = (8 ÷ 17)² = 64 ÷ 289

⇒ sin θ = √(1 − 64 ÷ 289)

⇒ sin θ = √(225 ÷ 289)

⇒ sin θ = 15 ÷ 17

∴ The correct answer is 15 ÷ 17.

Given:

17 × 684 ÷ 57 − (5/7) × 4.2 = x + 13 × 204 ÷ 17

Formula Used:

BODMAS rule

Calculation:

⇒ 17 × 684 ÷ 57 = 17 × 12 = 204

⇒ (5 ÷ 7) × 4.2 = (5 ÷ 7) × (42 ÷ 10) = 210 ÷ 70 = 3

⇒ LHS = 204 − 3 = 201

⇒ 13 × 204 ÷ 17 = 13 × 12 = 156

⇒ 201 = x + 156

⇒ x = 201 − 156

⇒ x = 45

∴ The correct answer is 45.

Shortcut Trick

When two equal halves of a distance are covered at speeds x and y, the Average Speed is the Harmonic Mean of the speeds.

Average Speed = (2 × x × y) ÷ (x + y)

Average Speed = (2 × 20 × 80) ÷ (20 + 80) = 3200 ÷ 100 = 32 km/hr.

∴ The correct answer is 32 km/hr.

Alternate Method

Given:

Speed of first half (v₁) = 20 km/hr

Speed of second half (v₂) = 80 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

Let the total distance be 160 km (LCM of 20 and 80 for easy calculation).

⇒ First half distance = 80 km, Second half distance = 80 km

⇒ Time for first half (t₁) = 80 ÷ 20 = 4 hours

⇒ Time for second half (t₂) = 80 ÷ 80 = 1 hour

⇒ Total Time = 4 + 1 = 5 hours

⇒ Average Speed = 160 ÷ 5 = 32 km/hr

∴ The correct answer is 32 km/hr.

Additional Information

General Average Speed Formula

For any distance and speed combination, use Average Speed = (d₁ + d₂ + ...) ÷ (t₁ + t₂ + ...).

Average Speed for Equal Time Intervals

If a body travels at speed v₁ for time t and speed v₂ for the same time t, Average Speed = (v₁ + v₂) ÷ 2.

Relative Speed

In opposite directions, relative speed = s₁ + s₂. In the same direction, relative speed = |s₁ − s₂|.

Shortcut Trick

Combined efficiency of A and B = 1/5

Efficiency of B alone = 1/55

Efficiency of A = 1/5 − 1/55 = 10/55 = 2/11

Time for A to do single work = 11/2 days

Time for A to do double the work = (11/2) × 2 = 11 days

∴ The correct answer is 11 days.

Alternate Method

Given: A + B = 5 days, B = 55 days.

Formula Used: Total Work = LCM of time taken; Efficiency = Total Work ÷ Time.

⇒ Efficiency of A = (Efficiency of A + B) − (Efficiency of B)

⇒ Efficiency of A = 11 − 1 = 10 units/day

⇒ Double the work = 55 × 2 = 110 units

⇒ Time taken by A alone = 110 ÷ 10 = 11 days

∴ The correct answer is 11 days.

Additional Information

Work and Efficiency Relationship

Time taken is inversely proportional to efficiency. If the amount of work is doubled, the time taken also doubles for a constant efficiency.

Combined Work Formula

If A takes 'x' days and B takes 'y' days, together they take (x × y) ÷ (x + y) days to complete the same work.

Shortcut Trick

Total age 20 years ago = 96 − (20 × 2) = 56 years.

Ratio 20 years ago = 3 : 1 (Total units = 4).

Son (20 yrs ago) = 56 ÷ 4 = 14; Father (20 yrs ago) = 14 × 3 = 42.

Present Ages: Son = 14 + 20 = 34; Father = 42 + 20 = 62.

Present Ratio = 62 : 34 = 31 : 17.

∴ The correct answer is 31 : 17.

Alternate Method

Given: Sum of present ages = 96. 20 years ago, Father = 3 × Son.

⇒ Let age of Son 20 years ago be x and Father be 3x.

⇒ Present age of Son = x + 20

⇒ Present age of Father = 3x + 20

⇒ (3x + 20) + (x + 20) = 96

⇒ 4x + 40 = 96 ⇒ 4x = 56 ⇒ x = 14

⇒ Father's present age = 3(14) + 20 = 42 + 20 = 62 years

⇒ Son's present age = 14 + 20 = 34 years

⇒ Required Ratio = 62 : 34 = 31 : 17

∴ The correct answer is 31 : 17.

Additional Information

Age Gap Consistency

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

Sum of Ages Shift

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

Ratio Application

Always define variables at the point in time where the ratio is given to simplify the linear equations.

Shortcut Trick

If the sum of three variables x + y + z = 0, then the identity x³ + y³ + z³ = 3xyz holds true.

Check: −15 + 25 + (−10) = 25 − 25 = 0.

Value = 3 × (−15) × 25 × (−10)

Value = 3 × 3750 = 11250.

∴ The correct answer is 11250.

Alternate Method

Given: x = −15, y = 25, z = −10

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

Calculations:

⇒ Calculate the sum of the given values: x + y + z = −15 + 25 + (−10)

⇒ x + y + z = 25 − 25 = 0

⇒ Substituting (x + y + z) = 0 into the algebraic identity:

⇒ x³ + y³ + z³ − 3xyz = 0 × (x² + y² + z² − xy − yz − zx)

⇒ x³ + y³ + z³ − 3xyz = 0

⇒ x³ + y³ + z³ = 3xyz

⇒ Value = 3 × (−15) × 25 × (−10)

⇒ Value = 3 × 3750 = 11250

∴ The correct answer is 11250.

Additional Information

Cubic Identity Condition

The expression x³ + y³ + z³ is equal to 3xyz if and only if x + y + z = 0 or x = y = z.

Standard Algebraic Identity

x³ + y³ + z³ − 3xyz = (x + y + z) × [1/2 × ((x − y)² + (y − z)² + (z − x)²)].

Sum of Squares Identity

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx), which is often used in conjunction with cubic identities.

Shortcut Trick

Equate the common teal tokens by multiplying the ratios.

Teal : Maroon = 15 : 10 = 3 : 2 (Multiplied by 11 → 33 : 22)

Lime : Teal = 19 : 11 (Multiplied by 3 → 57 : 33)

Ratio Lime : Maroon = 57 : 22

∴ The correct answer is 57 : 22.

Alternate Method

Given ratio of Teal (T) to Maroon (M) = 15 : 10 = 3 : 2

Given ratio of Lime (L) to Teal (T) = 19 : 11

To find the combined ratio, we make the value of Teal equal in both ratios.

⇒ LCM of Teal values (3 and 11) = 33

⇒ T : M = (3 × 11) : (2 × 11) = 33 : 22

⇒ L : T = (19 × 3) : (11 × 3) = 57 : 33

⇒ L : T : M = 57 : 33 : 22

⇒ L : M = 57 : 22

∴ The correct answer is 57 : 22.

Additional Information

Compound Ratio

When two or more ratios are multiplied term by term, the resulting ratio is called a compound ratio. If a:b and c:d are two ratios, the compound ratio is (ac):(bd).

Equating the Common Term

To combine ratios like A:B and B:C, identify the common element (B) and find the LCM of its numerical values to normalize the entire expression.

Shortcut Trick To determine if three numbers can form the sides of a triangle, check if the sum of the two smaller sides is greater than the largest side.

1. 8 + 2 = 10 (Not greater than 10) → Invalid
2. 8 + 4 = 12 (12 > 9) → Valid
3. 9 + 12 = 21 (Not greater than 22) → Invalid
4. 5 + 6 = 11 (Not greater than 11) → Invalid ∴ The correct answer is {8 cm, 9 cm, 4 cm}.

Alternate Method Given: Four sets of potential side lengths for a triangle. Property Used: According to the Triangle Inequality Theorem, for any triangle with sides a, b, and c: a + b > c, a + c > b, and b + c > a.

Testing each option based on the sum of the two smaller sides: ⇒ Option 1: 10, 8, 2 → 2 + 8 = 10. Since 10 is not greater than 10, it forms a straight line (degenerate triangle). ⇒ Option 2: 8, 9, 4 → 4 + 8 = 12. Since 12 > 9, these sides can form a triangle. ⇒ Option 3: 9, 12, 22 → 9 + 12 = 21. Since 21 is less than 22, the sides cannot meet to form a closed figure. ⇒ Option 4: 5, 6, 11 → 5 + 6 = 11. Since 11 is not greater than 11, it forms a straight line. ∴ The correct answer is {8 cm, 9 cm, 4 cm}.

Additional Information Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is always strictly greater than the length of the third side. Difference of Sides Property The absolute difference between any two sides of a triangle is always less than the length of the third side: |a − b| < c. Types of Triangles by Sides Scalene: All sides different. Isosceles: Two sides equal. Equilateral: All three sides equal.

Shortcut Trick

Use the Successive Multiplier method:

Let Manufacturing Cost = C

C × (1.20) × (1.25) × (1.24) = 372

C × 6/5 × 5/4 × 31/25 = 372

C × 3/2 × 31/25 = 372 ⇒ C × 93/50 = 372 ⇒ C = 4 × 50 = 200

∴ The correct answer is ₹200.

Alternate Method

Given:

Manufacturer Profit = 20% | Wholesaler Profit = 25% | Retailer Profit = 24%

Retailer Selling Price = ₹372

Let Manufacturing Cost be x.

⇒ Price for Wholesaler = x × (120/100) = 1.2x

⇒ Price for Retailer = 1.2x × (125/100) = 1.2x × 1.25 = 1.5x

⇒ Price for Customer (Retailer SP) = 1.5x × (124/100) = 372

⇒ 1.5x × 1.24 = 372

⇒ 1.86x = 372

⇒ x = 372 ÷ 1.86

⇒ x = 200

∴ The correct answer is ₹200.

Additional Information

Successive Percentage Formula

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

Profit and Loss Relation

Selling Price (SP) = Cost Price (CP) × (100 + Profit%)/100. For a chain of sales, final SP = Initial CP × MF₁ × MF₂ × ...

Base Change Rule

If the price increases by x%, to return to the original price, the value must decrease by [x/(100+x)] × 100%.

Shortcut Trick

Discount % = [Free Items ÷ Total Items (Paid + Free)] × 100

Discount % = [9 ÷ (47 + 9)] × 100

Discount % = (9 ÷ 56) × 100 ≈ 16.07%

∴ The correct answer is 16.07%.

Alternate Method

Given:

Number of Paid Items = 47

Number of Free Items = 9

Formula Used:

Discount % = (Free Items ÷ Total Items) × 100

Calculations:

⇒ Total items given to customer = 47 + 9 = 56 items

⇒ Let Marked Price (MP) of each item = ₹1

⇒ Total Marked Price of 56 items = ₹56

⇒ Selling Price (SP) = Price of 47 items = ₹47

⇒ Discount Amount = MP − SP = 56 − 47 = ₹9

⇒ Discount % = (9 ÷ 56) × 100

⇒ Discount % = 900 ÷ 56 = 16.0714...

∴ The correct answer is 16.07%.

Additional Information

Buy X Get Y Free

The effective discount is calculated on the total number of items (X+Y) carried home by the customer, not just the paid ones.

Successive Discounts

If two discounts a% and b% are given, total discount = [a + b − (ab/100)]%.

Marked Price (MP)

MP = CP × [100 + Profit% / 100 − Discount%], relating Cost Price and Marked Price directly.

Shortcut Trick

Average earnings per worker = 400 ÷ 5 = ₹80

Using Alligation: (Skilled ₹100) vs (Unskilled ₹50) with Mean ₹80

Ratio of workers = (80 − 50) : (100 − 80) = 30 : 20 = 3 : 2

Total units (3 + 2) = 5 workers. Since total workers are 5, 1 unit = 1 worker.

∴ The correct answer is 3.

Alternate Method

Given: Daily wage of skilled worker = ₹100, unskilled worker = ₹50.

Formula: Total Earnings = (Skilled × 100) + (Unskilled × 50)

⇒ Let the number of skilled workers be x.

⇒ Then unskilled workers = (5 − x).

⇒ (x × 100) + ((5 − x) × 50) = 400

⇒ 100x + 250 − 50x = 400

⇒ 50x = 400 − 250

⇒ 50x = 150

⇒ x = 150 ÷ 50 = 3

∴ The correct answer is 3.

Additional Information

Alligation Rule

The ratio of the weights of two components is the inverse ratio of the differences of their values from the mean value: (Value2 − Mean) : (Mean − Value1).

Weighted Average

The average value of a group where different items have different weights: A = (n₁w₁ + n₂w₂) ÷ (n₁ + n₂).

Shortcut Trick

In Percentage, any value X is considered 100% of itself.

Subtracting 1% from 100% gives 99%.

In decimal form, 99% is equal to 0.99.

∴ The correct answer is 0.99.

Alternate Method

Given: Original value is X. Percentage to subtract is 1% of X.

⇒ 1% of X = (1/100) × X = 0.01X

⇒ Required subtraction = X − 0.01X

⇒ Result = X (1 − 0.01)

⇒ Result = 0.99X

⇒ Multiplying factor = 0.99

∴ The correct answer is 0.99.

Additional Information

Percentage Change

To find the new value after a percentage decrease, multiply the original value by (1 − r/100).

Multiplying Factor (MF)

For a 10% increase, MF = 1.10. For a 10% decrease, MF = 0.90.

Base Change

Always assume the initial base as 100 to simplify manual calculations in percentage problems.

Shortcut Trick

Total Capacity (LCM of 360, 540, 720) = 2160 units.

Efficiencies: X = 6, Y = 4, Z = 3 units/min.

Work done by Y and Z in 30 mins = (4 + 3) × 30 = 210 units.

Remaining work = 2160 − 210 = 1950 units.

Time for remaining part = 1950 ÷ (6 + 4 + 3) = 1950 ÷ 13 = 150 minutes.

∴ The correct answer is 150 min.

Alternate Method

⇒ Total capacity of the tanker = LCM(360, 540, 720) = 2160 units.

⇒ Work done by Pipe Y and Pipe Z in 30 minutes = (4 + 3) × 30

⇒ 7 × 30 = 210 units.

⇒ Remaining work = Total Capacity − Work completed

⇒ 2160 − 210 = 1950 units.

⇒ Combined efficiency of X, Y, and Z = 6 + 4 + 3 = 13 units/min.

⇒ Time taken to fill remaining part = 1950 ÷ 13

⇒ Time = 150 minutes.

∴ The correct answer is 150 min.

Additional Information

Total Work Formula

Total Work = Time Taken × Efficiency. For pipes and cisterns, efficiency of an inlet pipe is positive, and an outlet pipe is negative.

Pipes and Cisterns Concept

If pipe A fills in x hours and pipe B fills in y hours, both together fill in (x × y) ÷ (x + y) hours.

Efficiency Ratio

The efficiency of a pipe is inversely proportional to the time taken to fill the tank (E ∝ 1/T).

Shortcut Trick

Using the deviation method by taking 13 as the baseline average.

Deviations: (6.5 − 13) = −6.5, (10.6 − 13) = −2.4, (19.7 − 13) = +6.7.

Sum of known deviations = −6.5 − 2.4 + 6.7 = −2.2.

For the total average to be 13, the deviation of a must be +2.2.

Value of a = 13 + 2.2 = 15.2.

∴ The correct answer is 15.2.

Alternate Method

Given: The observations are 6.5, 10.6, 19.7, and a. The Average of these 4 numbers is 13.

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

⇒ 13 = (6.5 + 10.6 + 19.7 + a) ÷ 4

⇒ 13 × 4 = 36.8 + a

⇒ 52 = 36.8 + a

⇒ a = 52 − 36.8

⇒ a = 15.2

∴ The correct answer is 15.2.

Additional Information

Sum of Observations

The total sum of a data set is calculated as: Sum = Average × Count. This is the primary step in missing value problems.

Average of Consecutive Numbers

For any Arithmetic Progression (AP), the average is simply: (First Term + Last Term) ÷ 2.

Net Deviation Principle

The sum of deviations of all observations from their arithmetic mean is always zero. This principle is used for faster calculations in large data sets.

Shortcut Trick

List the first 5 prime numbers: 2, 3, 5, 7, 11.

Sum = 2 + 3 + 5 + 7 + 11 = 28.

∴ The correct answer is 28.

Alternate Method

Given: Need to find the sum of the first 5 prime numbers.

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

⇒ First prime number = 2

⇒ Second prime number = 3

⇒ Third prime number = 5

⇒ Fourth prime number = 7

⇒ Fifth prime number = 11

⇒ Total Sum = 2 + 3 + 5 + 7 + 11

⇒ Total Sum = 28

∴ The correct answer is 28.

Additional Information

Definition of Prime Numbers

Numbers that have exactly two factors: 1 and the number itself. 1 is neither prime nor composite.

Even Prime Number

2 is the only even prime number; all other prime numbers are odd.

Prime Numbers up to 50

There are 15 prime numbers between 1 and 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Shortcut Trick

Total distance to cover = Sum of lengths = 410 + 440 = 850 m.

Relative speed (same direction) = 140 − 130 = 10 km/hr.

Relative speed in m/s = 10 × (5/18) = 25/9 m/s.

Time (seconds) = 850 ÷ (25/9) = 306 seconds.

Time (minutes) = 306 ÷ 60 = 5.1 minutes.

∴ The correct answer is 5.1 minutes.

Alternate Method

Given: Length of Train 1 (L₁) = 410 m, Length of Train 2 (L₂) = 440 m, Speed of Train 1 (S₁) = 130 km/hr, Speed of Train 2 (S₂) = 140 km/hr.

Formula Used: Time = (Total Distance) ÷ (Relative Speed)

⇒ Relative Speed = S₂ − S₁ = 140 − 130 = 10 km/hr

⇒ Conversion to m/s: 10 × (5/18) = 50/18 = 25/9 m/s

⇒ Total Distance = L₁ + L₂ = 410 + 440 = 850 m

⇒ Time (in sec) = 850 ÷ (25/9) = (850 × 9) ÷ 25 = 34 × 9 = 306 seconds

⇒ Time (in min) = 306 ÷ 60 = 5.1 minutes

∴ The correct answer is 5.1 minutes.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, the relative speed is the difference of their individual speeds (S₁ − S₂).

Relative Speed in Opposite Direction

When two objects move in opposite directions, the relative speed is the sum of their individual speeds (S₁ + S₂).

Unit Conversion

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

Shortcut Trick

Teacher's Age = New Average + (Original count × Increase in Average)

New Average = 38 years

Original number of students = 26

Increase in average = 38 − 37 = 1 year

Teacher's Age = 38 + (26 × 1) = 38 + 26 = 64 years

∴ The correct answer is 64 years.

Alternate Method

Given:

Initial number of students = 26

Initial average age = 37 years

New average age (including teacher) = 38 years

Formula Used: Total Age = Average Age × Number of Persons

⇒ Total age of 26 students = 26 × 37 = 962 years

⇒ Total number of persons including teacher = 26 + 1 = 27

⇒ Total age of 27 persons = 27 × 38 = 1026 years

⇒ Teacher's Age = Total age of group − Total age of students

⇒ Teacher's Age = 1026 − 962 = 64 years

∴ The correct answer is 64 years.

Additional Information

Net Deviation Concept

The sum of deviations from the average is always zero. If the new person increases the average, their value = New Average + (Previous count × Increase).

Average Property

If every value in a set is increased by a constant 'k', the average also increases by 'k'.

Weighted Average

Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where 'n' is the quantity and 'A' is the average.

Shortcut Trick

Ratio of Volume parts = 6 × 5 × 3 = 90 units³.

Given Volume = 720 m³ ⇒ 1 unit³ = 720 ÷ 90 = 8 ⇒ 1 unit = 2 m.

Ratio of Surface Area parts = 2 × (6×5 + 5×3 + 3×6) = 2 × (30 + 15 + 18) = 126 units².

Actual Surface Area = 126 × (1 unit)² = 126 × 2² = 126 × 4 = 504 m².

∴ The correct answer is 504 m².

Alternate Method

Given:

Ratio of length (l), breadth (b), and height (h) = 6 : 5 : 3

Volume of the cuboid = 720 m³

Formula Used:

Volume (V) = l × b × h

Total Surface Area (TSA) = 2(lb + bh + hl)

Calculations:

Let l = 6x, b = 5x, and h = 3x.

⇒ Volume = 6x × 5x × 3x = 90x³

⇒ 90x³ = 720

⇒ x³ = 720 ÷ 90 = 8

⇒ x = √8 (cube root) = 2

⇒ l = 6 × 2 = 12 m, b = 5 × 2 = 10 m, h = 3 × 2 = 6 m

⇒ TSA = 2(12 × 10 + 10 × 6 + 6 × 12)

⇒ TSA = 2(120 + 60 + 72)

⇒ TSA = 2 × 252 = 504 m²

∴ The correct answer is 504 m².

Additional Information

Lateral Surface Area (LSA) of Cuboid

The area of the four side faces of the cuboid is LSA = 2h(l + b).

Diagonal of a Cuboid

The longest rod that can be placed in a cuboid has length d = √(l² + b² + h²).

Properties of a Cube

For a cube with side 'a', Volume = a³ and total Surface Area = 6a².

Shortcut Trick

Profit Ratio = Investment Ratio (since time is equal)

Ratio of Vinod : Kajal : Arpit = 1040 : 1390 : 1890 = 104 : 139 : 189

Total Ratio Parts = 104 + 139 + 189 = 432

Arpit's share = (189 ÷ 432) × 1440 = 189 × (1440 ÷ 432) = 189 × 3.333... = 189 × (10 ÷ 3) = 63 × 10 = 630

∴ The correct answer is ₹630.

Alternate Method

Given:

Investment of Vinod (C_v) = ₹1,040

Investment of Kajal (C_k) = ₹1,390

Investment of Arpit (C_a) = ₹1,890

Total Profit (P_total) = ₹1,440

Time period (T) = 1 year (same for all)

Calculations:

⇒ Ratio of Profit = Ratio of (Investment × Time)

⇒ P_v : P_k : P_a = (1040 × 1) : (1390 × 1) : (1890 × 1)

⇒ Ratio = 104 : 139 : 189

⇒ Sum of ratio = 104 + 139 + 189 = 432

⇒ Arpit's share = (189 ÷ 432) × 1440

⇒ Arpit's share = 189 × (1440 ÷ 432)

⇒ Arpit's share = 189 × 3.3333

⇒ Arpit's share = 189 × (10 ÷ 3) = 63 × 10

⇒ Arpit's share = ₹630

∴ The correct answer is ₹630.

Additional Information

Partnership Ratio

Profit of Partner 1 : Profit of Partner 2 = (Capital₁ × Time₁) : (Capital₂ × Time₂). If time is constant, profit is proportional to capital.

Active vs Sleeping Partner

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

Compound Partnership

When partners invest money for different time periods, it is called a compound partnership, requiring the product of capital and time for calculation.

Shortcut Trick

Effective Simple Interest rate for Case 1 = 5% × 3 = 15%

Effective Simple Interest rate for Case 2 = 4% × 4 = 16%

Difference in effective rates = 16% − 15% = 1%

Required difference = 1% of Principal (₹5,000) = ₹50

∴ The correct answer is ₹50.

Alternate Method

Given:

Principal (P) = ₹5,000

Case 1: Rate (R₁) = 5%, Time (T₁) = 3 years

Case 2: Rate (R₂) = 4%, Time (T₂) = 4 years

Formula Used:

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

Calculations:

⇒ SI₁ = (5000 × 5 × 3) ÷ 100

⇒ SI₁ = 50 × 15 = ₹750

⇒ SI₂ = (5000 × 4 × 4) ÷ 100

⇒ SI₂ = 50 × 16 = ₹800

⇒ Difference = SI₂ − SI₁

⇒ Difference = 800 − 750 = ₹50

∴ The correct answer is ₹50.

Additional Information

Simple Interest Formula

The basic formula is SI = (P × R × T) ÷ 100, where P is Principal, R is Rate per annum, and T is Time in years.

Total Amount

The total amount (A) at the end of the period is the sum of the Principal and the Interest: A = P + SI.

Rate and Time Relationship

For a constant Principal, the Simple Interest is directly proportional to the product of Rate and Time (R × T).

Shortcut Trick

Simplify the ratio of the triangle sides: 120 : 126 : 174 → 20 : 21 : 29 (divided by 6).

Recognize (20, 21, 29) as a Pythagorean triplet because 20² + 21² = 400 + 441 = 841 = 29².

The triangle is right-angled with base 120 cm and height 126 cm.

Area = 1/2 × 120 × 126 = 60 × 126 = 7560 cm².

∴ The correct answer is 7560 cm².

Alternate Method

Given: Sides of the triangle are a = 120 cm, b = 174 cm, and c = 126 cm.

Formula Used: Heron's Formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2.

⇒ Semi-perimeter (s) = (120 + 174 + 126) ÷ 2

⇒ s = 420 ÷ 2 = 210 cm

⇒ Area = √[210 × (210 − 120) × (210 − 174) × (210 − 126)]

⇒ Area = √[210 × 90 × 36 × 84]

⇒ Area = √[18900 × 3024]

⇒ Area = √[57153600]

⇒ Area = 7560 cm²

∴ The correct answer is 7560 cm².

Additional Information

Heron's Formula

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

Pythagorean Triplets

A set of three integers satisfying a² + b² = c². Common triplets include (3,4,5), (5,12,13), (8,15,17), and (20,21,29).

Area of Right-Angled Triangle

If a triangle is right-angled, the area is simply 1/2 × product of the two sides containing the right angle.

Shortcut Trick

Loss of 12% means SP₁ is 88% of CP.

Gain of 12% means SP₂ should be 112% of CP.

Required SP = (528 ÷ 88) × 112

Required SP = 6 × 112 = 672

∴ The correct answer is ₹672.

Alternate Method

Given:

Selling Price (SP₁) = ₹528

Loss Percentage = 12%

Desired Gain Percentage = 12%

Formula Used:

Cost Price (CP) = [SP × 100] ÷ [100 − Loss%]

New Selling Price (SP₂) = [CP × (100 + Gain%)] ÷ 100

Calculations:

⇒ CP = (528 × 100) ÷ (100 − 12)

⇒ CP = (528 × 100) ÷ 88

⇒ CP = 6 × 100 = ₹600

⇒ SP₂ = [600 × (100 + 12)] ÷ 100

⇒ SP₂ = (600 × 112) ÷ 100

⇒ SP₂ = 6 × 112

⇒ SP₂ = ₹672

∴ The correct answer is ₹672.

Additional Information

Marked Price & Discount

Marked Price (MP) is the price printed on the label. Discount is always calculated on MP. SP = MP − Discount.

Successive Percentage Change

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

Profit and Loss Percentage

Profit% = (Profit ÷ CP) × 100 and Loss% = (Loss ÷ CP) × 100. Both are calculated on Cost Price unless specified otherwise.

Shortcut Trick

List the first 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Group them for faster addition: (2+3+5) + (7+13) + (11+19) + (17+23) + 29

⇒ 10 + 20 + 30 + 40 + 29 = 129

∴ The correct answer is 129.

Alternate Method

Given: First 10 prime numbers.

Formula Used: Sum = p₁ + p₂ + ... + p₁₀

⇒ The first 10 prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 5 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 10 + 7 + 11 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 17 + 11 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 28 + 13 + 17 + 19 + 23 + 29

⇒ Sum = 41 + 17 + 19 + 23 + 29

⇒ Sum = 58 + 19 + 23 + 29

⇒ Sum = 77 + 23 + 29

⇒ Sum = 100 + 29

⇒ Sum = 129

∴ The correct answer is 129.

Additional Information

Prime Number definition

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. 2 is the only even prime number.

Sum of Primes up to 50

There are 15 prime numbers up to 50 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), and their total sum is 328.

Composite Numbers

Numbers with more than two factors are composite numbers. Note that 1 is neither prime nor composite.

Shortcut Trick

Required percentage = (Value ÷ Base Value) × 100

Required percentage = (1/20 ÷ 10/20) × 100

Required percentage = (1/10) × 100 = 10%

∴ The correct answer is 10%.

Alternate Method

Given:

First fraction = 1/20

Second fraction (Base) = 10/20

Formula Used:

Percentage = (A / B) × 100

Calculations:

⇒ Required Percentage = [ (1/20) / (10/20) ] × 100

⇒ Since denominators are same, it simplifies to: (1 / 10) × 100

⇒ Required Percentage = 0.1 × 100

⇒ Required Percentage = 10%

∴ The correct answer is 10%.

Additional Information

Percentage Change

Percentage change = [(New Value − Old Value) ÷ Old Value] × 100.

Fraction to Percentage

To convert any fraction x/y into percentage, multiply the fraction by 100.

Base Value Identification

The value following the word 'of' is usually treated as the base (denominator) in percentage calculations.

7 persons: A, B, C, D, W, X, Y sitting in a straight line facing North

1. No one sits to the left of D

⇒ D at extreme left

2. Four people sit between D and C

3. Three people sit to the right of Y

4. X sits immediately left of A

5. W is not an immediate neighbour of Y

Remaining arrangement:

∴ Y sits third to the left of W.

Hence the correct answer is Option 1.

Expression:

44 B 30 A 10 D 46 C 3

Replace symbols and apply BODMAS rule

44 × 30 ÷ 10 − 46 + 3

⇒ 44 × 30 = 1320

⇒ 1320 ÷ 10 = 132

⇒ 132 − 46 = 86

⇒ 86 + 3 = 89

∴ Required value = 89.

Hence the correct answer is  OPTION 4.

Following the logic

Hence the correct answer is Option 4.

Series: 2, 8, 3, 27, 4, 64, 5, 125, 6, 216, 7, ?

Alternate pattern

Number and its cube

2 → 2³ = 8

3 → 3³ = 27

4 → 4³ = 64

5 → 5³ = 125

6 → 6³ = 216

7 → 7³ = 343

∴ Missing number = 343.

Hence the correct answer is Option 1.

Given:

Series:

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

Perfect squares (single digit) = 1, 4, 9

Odd numbers = 1, 3, 5, 7, 9

Count digits where:

Previous digit = Perfect square

Next digit = Odd

Check triplets:

4 5 7 → 4 square, 7 odd ✔

9 6 7 → 9 square, 7 odd ✔

1 7 3 → 1 square, 3 odd ✔

4 9 7 → 4 square, 7 odd ✔

Total valid digits = 4

∴ Required number of digits = 4.

Hence the correct answer is Option 1.

A + B → A is the son of B

A − B → A is the brother of B

A × B → A is the wife of B

A ÷ B → A is the father of B

Relation: D ÷ F × G − H + J

Decode relation from left → right

∴ D is the father-in-law of J’s son.

Hence the correct answer is Option 3.

Series: GEL, OMT, WUB, ECJ, ?

Alphabet positions:

G = 7, E = 5, L = 12

O = 15, M = 13, T = 20

W = 23, U = 21, B = 2

E = 5, C = 3, J = 10

Each step +8 in alphabetical order

First letters:

7 → 15 → 23 → 5

⇒ 5 + 8 = 13 → M

Second letters:

5 → 13 → 21 → 3

⇒ 3 + 8 = 11 → K

Third letters:

12 → 20 → 2 → 10

⇒ 10 + 8 = 18 → R

Required term = MKR

∴ The missing term is MKR. 

Hence the correct answer is Option 2.

kit son pit → ai lm go

boss pit poem → lm jn yt

Common word → Common code

kit son pit → ai lm go

boss pit poem → lm jn yt

Common word = pit

Common code = lm

⇒ pit = lm

∴ The code for pit is lm.

Hence the correct answer is Option 2.

PL 13 → SQ 16

IE 18 → LJ 21

OK 26 → ?

∴ Required relation = RP 29.

Hence the correct answer is Option 2.

Boxes: A, B, C, D, E, F, G (7 boxes)

Only three boxes are kept above G.

Only one box is kept between B and G.

Only three boxes are kept between B and C.

C is kept at some place above G.

F is kept immediately below C.

E is kept at one of the places above D.

A is not kept immediately above or below B.

Final Arrangement:

So, 2 boxes are kept below E.

Hence the correct answer is Option (4).

Persons: B, C, D, I, J, K, L (7 persons)

All sitting around a circular table facing centre

Left = clockwise

Right = anticlockwise 

3 persons between D and L from the right of L

3 persons between K and I from the left of K

B immediate right of D

J neighbour of L and I

Final arrangement 

Find second left of C:

Left of C → I

Second left → J

∴ J sits second to the left of C.

OH → QK → TV

LE → NH → QS

Options:

RK-TN-WZ

SK-TN-WZ

RK-TN-WY

SK-TM-WZ

1^st letter: +3 then +3

2^nd letter: +3 then +11

OH → QK

O(15) + 2 = Q(17)

H(8) + 3 = K(11)

QK → TV

Q(17) + 3 = T(20)

K(11) + 11 = V(22)

LE → NH

L(12) + 2 = N(14)

E(5) + 3 = H(8)

NH → QS

N(14) + 3 = Q(17)

H(8) + 11 = S(19)

Check option:

RK → TN

R(18) + 2 = T(20)

K(11) + 3 = N(14)

TN → WZ

T(20) + 3 = W(23)

N(14) + 12 ≈ Z(26) (largest jump like pattern)

∴ Correct triad = RK-TN-WZ.

Persons: A, B, C, D, E, F, G (7 persons)

All facing North

Only A sits to the left of D

4 persons between A and C

Only B between G and F

G not immediate neighbour of C

 Remaining positions

A D G B F C E

People right of G:

B, F, C, E = 4

∴ 4 people sit to the right of G.

Hence the correct answer is Option 2.

Boxes: M, N, O, P, Q, R, S (7 boxes)

Only 3 boxes above S ⇒ S at position 4

1 box between R and S ⇒ R is at position 2 or 6

3 boxes between R and Q

Q is above S

If R is 2, Q must be 6 (violates Q above S)

If R is 6, Q must be 2 (satisfies Q above S) ⇒ Q at position 2

N immediately below Q ⇒ N at position 3

Remaining positions: 1, 5, 7

P is not adjacent to R (position 6), so P cannot be 5 or 7 ⇒ P at position 1

M above O ⇒ M at position 5, O at position 7

Final order (Top → Bottom):

∴ Box at the lowest position = O.

Hence the correct answer is Option 1.

Hence the correct answer is Option 3.

Series: 501, 500, 497, 492, 485, ?

Increasing subtraction pattern

∴ Missing number = 476.

Hence the correct answer is Option 1.

Expression:

24 A 4 B 6 C 2 D 10

Replace symbols and apply BODMAS rule

24 ÷ 4 × 6 + 2 − 10

⇒ 24 ÷ 4 = 6

⇒ 6 × 6 = 36

⇒ 36 + 2 = 38

⇒ 38 − 10 = 28

∴ Required value = 28.

Hence the correct answer is Option 1.

Given:

Series:

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

Odd numbers: 1, 3, 5, 7, 9

Count numbers where:

Previous number = Odd

Current number = Odd

Next number = Odd

Check triplets:

1 1 7 → odd, odd, odd ✔

3 9 1 → odd, odd, odd ✔

Total valid numbers = 2

∴ Required count = 2.

Hence the correct answer is Option 1.

TOUR → 8571

DOUT → 5478

Common letters → Common digits

Common letters in both words:

T, O, U

Common digits in both codes:

8, 5, 7

⇒ T, O, U correspond to 8, 5, 7 (order may vary)

Remaining letter in second word = D

Remaining digit in second code = 4

⇒ D = 4

∴ Code for D = 4.

Hence the correct answer is Option 4.

423 + 9 × 19 − 6 ÷ 775

Interchange operations:

• ↔ ÷

− ↔ ×

Replace operators and apply BODMAS rule

423 + 9 × 19 − 6 ÷ 775

After interchange:

423 ÷ 9 − 19 × 6 + 775

⇒ 423 ÷ 9 = 47

⇒ 19 × 6 = 114

⇒ 47 − 114 + 775

⇒ −67 + 775

⇒ 708

∴ Required value = 708.

Hence the correct answer is option 2.

Start at A

East = 11 km

Right turn → South = 5 km

Right turn → West = 14 km

Right turn → North = 16 km

Right turn → East = 3 km

Stop at P

⇒ P is 11 km North of A

To reach A from P:

⇒ Drive 11 km South

∴ Shortest distance = 11 km towards South.