RRB Group D Official Paper (Held On: 11 Dec, 2025 Shift 3) (Previous Year Paper)

The correct answer is Molluscs.

Key Points

• Molluscs are classified as shellfish, which include organisms with soft bodies often protected by a hard shell. Examples include clams, oysters, mussels, and scallops.
• Shellfish are broadly categorized into molluscs and crustaceans. Molluscs are distinct due to their calcium carbonate shells and lack of jointed appendages.
• Unlike finned fish such as Silver carp, Bombay duck, and Tuna, molluscs are invertebrates and are harvested for their edible meat and sometimes their shells.
• Molluscs are primarily found in marine environments, although certain species like freshwater mussels exist in rivers and lakes.
• Shellfish, including molluscs, are a rich source of protein, omega-3 fatty acids, vitamins, and minerals, making them nutritionally valuable and widely consumed globally.

Additional Information

• Silver carp:
• Silver carp is a freshwater fish species commonly found in rivers, lakes, and reservoirs.
• It is widely used in aquaculture due to its rapid growth rate and ability to filter feed on plankton.
• Silver carp is not a shellfish; it belongs to the category of finned fish.
• Bombay duck:
• Bombay duck is a type of lizardfish and is predominantly found along the Arabian Sea, particularly in India.
• Despite the name, Bombay duck is not a shellfish but a finned fish consumed mainly in dried or fresh form.
• Tuna:
• Tuna is a marine finfish and one of the most commercially important fish species globally.
• It is known for its high nutritional value, especially its rich content of omega-3 fatty acids.
• Tuna does not fall under the shellfish category.
• Molluscs:
• Molluscs are invertebrates with soft bodies often enclosed in a calcium carbonate shell.
• They include species like clams, oysters, snails, squids, and octopuses.
• Molluscs are highly valued in the seafood industry for their nutritional benefits and culinary uses.
• They play a crucial role in maintaining marine ecosystems by filtering water and serving as prey for other marine organisms.

Important Points

• Shellfish Classification:
• Shellfish are divided into two major categories: molluscs (e.g., clams, oysters, scallops) and crustaceans (e.g., crabs, lobsters, shrimp).
• Molluscs are characterized by their soft bodies and hard shells, while crustaceans have jointed appendages and exoskeletons.
• Global Consumption:
• Shellfish, especially molluscs, are a key part of diets in coastal regions across the world.
• They are harvested for food, and their shells are used in various industries, including jewelry and construction.
• Environmental Importance:
• Molluscs like oysters and mussels are vital for water filtration in marine ecosystems.
• They help maintain ecological balance and improve water quality by removing pollutants.

The correct answer is the same direction.

Key Points

• When a current flows through a circular wire, it creates a magnetic field around it due to the motion of electric charges. This behavior is explained by Ampere's Circuital Law and Biot-Savart Law.
• Every small segment of the circular loop contributes to the magnetic field at the center of the loop in the same direction. This is due to the symmetry of the circular loop and the nature of the magnetic field lines.
• The direction of the magnetic field at the center of the loop is determined using the Right-Hand Thumb Rule: If the thumb of the right hand points in the direction of the current, the curled fingers around the loop indicate the direction of the magnetic field.
• The magnetic field created by each segment of the wire at the center of the loop adds up constructively because all the contributions are in the same direction.
• This collective effect results in a strong and uniform magnetic field at the center of the circular loop, while the field decreases as we move away from the center.

Additional Information

• Biot-Savart Law:
• The Biot-Savart Law provides the mathematical expression to calculate the magnetic field produced by a small segment of a current-carrying conductor.
• The law states that the magnetic field dB due to a small current element is proportional to the current (I), the length of the element (dl), and inversely proportional to the square of the distance (r²) from the element.
• Mathematically, dB = (μ₀ / 4π) × (I × dl × sinθ) / r², where μ₀ is the permeability of free space.
• Right-Hand Thumb Rule:
• This rule helps determine the direction of the magnetic field surrounding a current-carrying conductor.
• If you curl the fingers of your right hand around a conductor in the direction of current flow, your thumb points in the direction of the resulting magnetic field lines.
• Magnetic Field in Circular Loops:
• The magnetic field at the center of a current-carrying circular loop is stronger compared to other points because all the magnetic field lines from different sections of the loop converge constructively at the center.
• The magnetic field at the center of the loop is given by B = (μ₀ × I) / (2 × R), where I is the current, R is the radius of the loop, and μ₀ is the permeability of free space.
• Applications of Magnetic Fields in Circular Loops:
• Circular loops are used in electromagnets and magnetic coils to create strong and uniform magnetic fields.
• They are also used in inductors in electronic circuits and devices like MRI machines, where controlled magnetic fields are essential.

Important Points

• Option 1 (Opposite Direction): Incorrect. If sections of the wire contributed in the opposite direction, the magnetic field at the center would cancel out, which does not happen in a circular loop.
• Option 2 (Random Direction): Incorrect. The contributions are not random; they follow a specific pattern determined by the Right-Hand Thumb Rule.
• Option 3 (Perpendicular Direction): Incorrect. The magnetic field contributions are not perpendicular but are aligned in the same direction at the center.
• Option 4 (Same Direction): Correct. All sections of the current-carrying circular wire contribute to the magnetic field at the center in the same direction.

The correct answer is Apical meristem.

Key Points

• Apical meristem is a region of actively dividing cells found at the tips of roots and shoots in plants. It is responsible for the primary growth of the plant, which includes elongation of stems and roots.
• The apical meristem contains undifferentiated or embryonic cells that continuously divide to produce new cells. These cells later differentiate to form various tissues and organs of the plant.
• This meristematic tissue plays a key role in increasing the height of the plant and allows it to grow in specific directions (e.g., upward for shoots and downward for roots).
• The apical meristem is protected by the root cap in roots and by young leaves in shoots, ensuring it remains functional during growth.
• In contrast, other plant tissues such as sieve tubes, sclerenchyma, and vessels are differentiated tissues that primarily perform specific functions like transport, support, or conduction, and do not contribute directly to growth through cell division.

Additional Information

• Meristematic Tissues:
• Meristematic tissues are regions in plants where active cell division occurs. These tissues are responsible for the growth and development of plants.
• They can be classified into apical meristem (at tips of roots and shoots), intercalary meristem (found at internodes or base of leaves), and lateral meristem (responsible for secondary growth, e.g., cambium).
• Sieve Tubes:
• Sieve tubes are specialized cells of the phloem tissue responsible for transporting nutrients, particularly sucrose, throughout the plant.
• These tubes lack nuclei and rely on companion cells for their functions. They do not contribute to growth by cell division.
• Sclerenchyma:
• Sclerenchyma is a type of permanent tissue that provides mechanical support to plants.
• It consists of dead cells with thick, lignified walls. These cells lose their ability to divide and hence do not contribute to growth.
• Vessels:
• Vessels are tube-like structures found in the xylem tissue that help in the conduction of water and minerals from roots to other parts of the plant.
• They are composed of dead cells and are specialized for conduction, not for growth through cell division.
• Primary vs Secondary Growth:
• Primary growth is the growth in length of roots and shoots and is facilitated by the apical meristem.
• Secondary growth, on the other hand, involves an increase in the girth or thickness of the plant and is caused by lateral meristems like vascular cambium and cork cambium.

Important Points

• Examples of Meristematic Tissue:
• Apical meristem: Found at the tips of roots and shoots, promotes vertical growth.
• Intercalary meristem: Found at the base of leaves or internodes, helps in regrowth and elongation.
• Lateral meristem: Includes vascular cambium and cork cambium, responsible for secondary growth (girth).
• Characteristics of Meristematic Cells:
• They are small, densely cytoplasmic cells with prominent nuclei.
• They have thin primary cell walls and lack vacuoles to allow rapid division.
• Real-life Application:
• Understanding meristematic tissue is essential for horticulture, as it helps in techniques like grafting and tissue culture.
• Apical meristems are studied for improving crop growth and yield by regulating hormone activity.

The correct answer is Carbon-12.

Key Points

• Carbon-12 is chosen as the standard reference for determining atomic masses due to its stability and wide occurrence in nature.
• The atomic mass unit (amu), also known as the unified mass unit, is defined as one-twelfth the mass of a single carbon-12 atom.
• The decision to use Carbon-12 as the standard was made by the International Union of Pure and Applied Chemistry (IUPAC) in 1961.
• Carbon-12 is a stable isotope of carbon that contains 6 protons and 6 neutrons, making its atomic mass exactly 12 atomic mass units.
• This standardization ensures consistency and accuracy in the measurement of atomic masses across all elements.

Additional Information

• Atomic Mass Unit (amu):
• The atomic mass unit is used to express the masses of atoms and molecules.
• 1 amu is approximately equal to 1.660539066 x 10^-27 kilograms.
• It is a standard unit of measurement in chemistry, physics, and molecular biology.
• Carbon Isotopes:
• Carbon has three naturally occurring isotopes: Carbon-12, Carbon-13, and Carbon-14.
• Carbon-12 is the most abundant isotope, making up about 98.93% of all carbon found in nature.
• Carbon-13 is stable and used in chemical research and spectroscopy.
• Carbon-14 is radioactive and used in radiocarbon dating to determine the age of ancient objects.
• Why Carbon-12 is Preferred:
• Carbon-12 is chemically inert and does not react easily, making it ideal for a stable reference.
• Its atomic structure is simple and well-understood, ensuring precision in calculations.
• The abundance of Carbon-12 in nature provides practical advantages for standardization.
• Applications of Atomic Mass Standardization:
• Atomic masses are used in chemical equations, stoichiometry, and molecular mass calculations.
• They are crucial in fields like materials science, biochemistry, and pharmacology for accurate molecular composition analysis.
• Atomic mass plays a significant role in determining isotopic distributions and studying nuclear reactions.

Important Points

• Historical Context:
• Before Carbon-12, oxygen isotopes were used as a reference for atomic mass calculations.
• The shift to Carbon-12 eliminated discrepancies between the physicists' and chemists' scales.
• Standardization by IUPAC:
• IUPAC standardized the atomic mass scale globally, making scientific data uniform across disciplines.
• This move facilitated international collaboration in research and development.
• Significance in Research:
• Atomic mass standardization allows scientists to predict chemical reactions and study the properties of elements.
• It also supports advancements in fields like nanotechnology and quantum mechanics.

The correct answer is 4 kg.

Key Points

• The formula for kinetic energy is given as:
  

K.E. = (1/2) × m × v², where:

• K.E. = Kinetic Energy (in joules)
• m = Mass of the object (in kilograms)
• v = Velocity of the object (in m/s)
• In the question, the kinetic energy is 450 J, and the velocity is 15 m/s.
• Rearranging the formula to calculate the mass:
  

m = (2 × K.E.) / v²

• Substituting the given values:
  

m = (2 × 450) / (15 × 15) = 900 / 225 = 4 kg.

• The calculated mass of the body is 4 kg, which matches the correct answer.

Additional Information

• Kinetic Energy:
• Kinetic Energy is the energy possessed by an object due to its motion.
• It depends on two factors: the mass of the object and the square of its velocity.
• K.E. is always a scalar quantity and is measured in joules (J).
• Examples of kinetic energy include a moving car, a flowing river, and a flying bird.
• Velocity:
• Velocity is the rate of change of an object's position with respect to time in a specific direction.
• It is a vector quantity, meaning it has both magnitude and direction.
• Units of velocity are measured in meters per second (m/s).
• Examples of velocity include the speed of a train moving north or a plane flying west at a specific speed.
• Mass:
• Mass is the amount of matter contained in an object.
• It is a scalar quantity and is measured in kilograms (kg) in the SI system.
• Mass is different from weight, which is the force of gravity acting on an object.
• In physics problems, mass plays a significant role in determining an object's inertia and energy.
• Applications of Kinetic Energy Formula:
• It is used in calculating the energy of moving objects, such as vehicles, projectiles, and machinery.
• Helps in designing safety measures such as airbags and helmets by understanding the impact of kinetic energy.
• Widely used in sports science to analyze the motion of athletes and equipment.
• Used in industries like aviation and space exploration for trajectory predictions.

Important Points

• Conservation of Energy: Kinetic energy is a part of the law of conservation of energy, which states that energy can neither be created nor destroyed but can only transform from one form to another.
• Work-Energy Theorem: It states that the work done on an object is equal to the change in its kinetic energy.
• SI Units:
• Kinetic Energy: Joules (J)
• Mass: Kilograms (kg)
• Velocity: Meters per second (m/s)
• Understanding these concepts is essential for fields like physics, engineering, mechanics, and energy analysis.

The correct answer is It is used in fire extinguishers to produce foam.

Key Points

• Sodium hydrogen carbonate (baking soda) is a crucial component in certain fire extinguishers. It reacts with acid to release carbon dioxide (CO₂) gas, which helps in extinguishing fire by displacing oxygen and creating a foam.
• This foam acts as a barrier to prevent the fire from spreading further and helps in cooling the burning material.
• When baking soda is used in fire extinguishers, it decomposes upon heating to release carbon dioxide and water vapor, which are effective in putting out flames.
• The reaction can be represented as:
  

2NaHCO₃ → Na₂CO₃ + H₂O + CO₂.

• These extinguishers are particularly effective in dealing with Class B and Class C fires, which involve flammable liquids and gases.

Additional Information

• Uses of Baking Soda:
• Option 1: It is used to make soaps non-greasy. This is incorrect because baking soda is not commonly used in soap manufacturing to remove grease.
• Option 2: It is used to neutralize strong acids in lead-acid batteries. This is incorrect because, while baking soda can neutralize acids, it is not specifically used for lead-acid batteries.
• Option 3: It is used to prepare edible salts. This is incorrect because baking soda itself is an edible salt but is not primarily used to prepare other salts.
• Option 4: It is used in fire extinguishers to produce foam. This is correct because baking soda reacts to release carbon dioxide, forming foam to extinguish fires.
• Properties of Baking Soda:
• Its chemical formula is NaHCO₃, and it is a mild alkali.
• It is commonly used as a leavening agent in baking due to the release of carbon dioxide gas during decomposition.
• Baking soda is also used as a cleaning agent, deodorizer, and in medical applications for neutralizing stomach acid.
• Types of Fire Extinguishers:
• Fire extinguishers are categorized based on the types of fires they are designed to combat, such as Class A, B, C, D, and electrical fires.
• Baking soda-based extinguishers are effective against Class B (flammable liquids) and Class C (electrical) fires.
• Environmental Impact:
• The use of baking soda in fire extinguishers is environmentally friendly because it produces non-toxic byproducts like sodium carbonate, water, and carbon dioxide.

Important Points

• Reaction in Fire Extinguishers: Baking soda reacts with acids like sulfuric acid to produce carbon dioxide, which extinguishes fire.
• Handling Fires: Always use the appropriate type of fire extinguisher depending on the material involved in the fire (e.g., wood, liquid, gas, or electrical equipment).
• Storage and Safety: Baking soda-based fire extinguishers should be stored in accessible locations and inspected regularly for proper functionality.

The correct answer is 3 oxygen atoms.

Key Points

• A molecule of ozone (O₃) consists of three oxygen atoms bonded together.
• Ozone is a triatomic molecule and is an allotrope of oxygen, differing from the more common diatomic oxygen molecule (O₂).
• The ozone molecule has a bent structure with an angle of approximately 116.8° between the oxygen atoms.
• It exists naturally in the Earth's atmosphere, particularly in the stratosphere, where it forms the ozone layer that protects life by absorbing harmful ultraviolet (UV) radiation from the Sun.
• Ozone can also be found near the Earth's surface (troposphere) as a result of pollution, where it acts as a harmful air pollutant and a key component of smog.

Additional Information

• Properties of Ozone:
• Ozone is a pale blue gas at standard temperature and pressure, with a strong, pungent odor.
• It is a powerful oxidizing agent and is used in various industrial and medical applications, such as water purification and sterilization.
• Formation of Ozone:
• Ozone is formed naturally in the stratosphere when ultraviolet (UV) light from the Sun splits an oxygen molecule (O₂) into two individual oxygen atoms, which then bond with another oxygen molecule to form ozone (O₃).
• Ozone Layer and Its Importance:
• The ozone layer, located in the stratosphere, absorbs the majority of the Sun's harmful ultraviolet (UV-B and UV-C) radiation, protecting living organisms from DNA damage and other harmful effects.
• The thickness of the ozone layer varies by location and time, and its depletion due to human-made chemicals like chlorofluorocarbons (CFCs) has been a major environmental concern.
• Industrial and Environmental Relevance:
• In the troposphere, ozone is a pollutant that contributes to respiratory problems and environmental damage.
• Ozone is used in industries for applications like water treatment, deodorization, and bleaching.

Important Points

• Ozone Depletion:
• Human-made chemicals such as chlorofluorocarbons (CFCs), halons, and other ozone-depleting substances (ODS) release chlorine and bromine atoms in the stratosphere, which destroy ozone molecules.
• Efforts to protect the ozone layer include the Montreal Protocol (1987), an international treaty aimed at phasing out the production of ODS.
• Health Effects of Ozone:
• While ozone in the stratosphere is essential for life, ozone at ground level can cause respiratory issues, throat irritation, and exacerbate conditions like asthma.
• Exposure to high levels of ozone can damage crops, trees, and other vegetation, impacting agriculture and ecosystems.

The correct answer is About 3°C.

Key Points

• The temperature of the scrotum is maintained at approximately 3°C lower than the normal body temperature to ensure optimal conditions for spermatogenesis (the production of sperm).
• The normal human body temperature is about 37°C, while the scrotal temperature remains around 34°C to support the proper functioning of the testes.
• This temperature difference is crucial because sperm production and maturation require a cooler environment than the core body temperature.
• The cremaster muscle and dartos muscle in the scrotum regulate temperature by contracting or relaxing, thus moving the testes closer to or farther away from the body.
• High temperatures can impair sperm quality and lead to issues such as reduced fertility, emphasizing the importance of maintaining this cooler scrotal environment.

Additional Information

• Function of Scrotum:
• The scrotum houses the testes and plays a vital role in thermoregulation to ensure proper sperm production.
• The scrotal skin contains sweat glands that help in cooling through evaporation.
• Role of Muscles:
• The cremaster muscle adjusts the position of the testes relative to the body, moving them closer in cold conditions and away in warm conditions.
• The dartos muscle causes the scrotal skin to wrinkle or relax to aid in temperature regulation.
• Impact of Temperature on Fertility:
• Exposure to higher temperatures for prolonged periods can lead to a temporary decline in sperm count.
• Occupational hazards, such as working in hot environments, can affect male fertility over time.
• Scientific Studies:
• Studies indicate that even a slight increase in scrotal temperature (1-2°C) can negatively affect sperm motility and morphology.
• Wearing tight clothing or prolonged sitting can elevate scrotal temperature and impact sperm quality.
• Evolutionary Advantage:
• The external location of the scrotum in mammals is an evolutionary adaptation to optimize sperm production by maintaining a cooler environment.

Important Points

• Normal Body Temperature: The average human body temperature is 37°C, though it can vary slightly depending on the individual and time of day.
• Testes in Other Mammals: In certain mammals like whales and elephants, the testes are located inside the body, and their sperm production mechanisms have adapted to function at body temperature.
• Medical Conditions: Conditions like cryptorchidism (undescended testes) can lead to infertility due to higher temperatures inside the abdomen.
• Preventive Measures: Avoiding prolonged exposure to high temperatures, wearing loose clothing, and maintaining a healthy lifestyle are recommended to support male fertility.

The correct answer is osmosis.

Key Points

• Osmosis is the process by which water molecules move from a region of their higher concentration to a region of their lower concentration through a selectively permeable membrane.
• The selectively permeable membrane allows only certain molecules, such as water, to pass through while blocking others, ensuring the balance of substances between the two sides of the membrane.
• Osmosis plays a vital role in biological systems, especially in regulating the water content of cells, maintaining cell turgor, and balancing fluids in organisms.
• It is a passive transport mechanism, meaning it does not require energy (ATP) to occur, as it relies on the natural concentration gradient of water molecules.
• Examples of osmosis include water absorption by plant roots and the regulation of water balance in animal cells.

Additional Information

• Transpiration:
• Transpiration refers to the process by which water vapor is lost from the aerial parts of plants, primarily through the stomata in leaves.
• It is essential for maintaining the water cycle in plants and aiding in nutrient transport from the roots to other parts.
• This process is different from osmosis as it involves the evaporation of water rather than diffusion through a membrane.
• Active Transport:
• This is a biological process in which substances move against their concentration gradient (from lower to higher concentration) across a cell membrane.
• It requires energy in the form of ATP, unlike osmosis, which is passive.
• Examples include the sodium-potassium pump in animal cells and the uptake of minerals by plant roots.
• Diffusion:
• Diffusion is the movement of molecules (e.g., gases like oxygen and carbon dioxide) from a region of higher concentration to a region of lower concentration.
• It can occur in solids, liquids, and gases and does not necessarily involve a selectively permeable membrane.
• Unlike osmosis, diffusion applies to any substance and not just water.
• Importance of Osmosis in Daily Life:
• In plants, osmosis helps maintain turgidity in cells, which supports the plant structure.
• In humans, osmosis regulates the balance of fluids in cells and tissues, such as in kidneys during the filtration of blood.
• Industrial applications include water purification through reverse osmosis systems.

The correct answer is Uterus.

Key Points

• The uterus is a key reproductive organ in the human female body that prepares itself every month for the possibility of receiving a fertilized egg.
• This preparation is part of the menstrual cycle, which is regulated by hormones such as estrogen and progesterone.
• The inner lining of the uterus, called the endometrium, thickens each month to create a suitable environment for the implantation of a fertilized egg.
• If fertilization does not occur, the endometrial lining sheds, leading to menstruation.
• The uterus plays a critical role in supporting and nourishing a developing fetus during pregnancy and is also involved in childbirth through contractions.

Additional Information

• Menstrual Cycle:
• The menstrual cycle typically lasts 28 days, though it can range from 21 to 35 days.
• The cycle has four main phases: menstrual phase, follicular phase, ovulation, and luteal phase.
• Hormones such as follicle-stimulating hormone (FSH) and luteinizing hormone (LH) play key roles in regulating the cycle.
• Role of Hormones:
• Estrogen: Promotes the thickening of the endometrium.
• Progesterone: Maintains the endometrial lining in preparation for implantation.
• If pregnancy does not occur, the levels of these hormones drop, triggering menstruation.
• Structure of the Uterus:
• The uterus is a hollow, muscular organ located in the pelvis.
• It has three main layers: the perimetrium (outer layer), myometrium (muscular middle layer), and endometrium (inner lining).
• The uterus is connected to the fallopian tubes and the vagina.
• Functions of the Uterus:
• Supports implantation of the fertilized egg.
• Provides nourishment to the developing fetus through the placenta.
• Facilitates childbirth through muscular contractions.

Important Points

• Kidney:
• The kidney is part of the urinary system and is responsible for filtering blood, removing waste products, and maintaining fluid and electrolyte balance.
• It is not involved in the reproductive process.
• Pancreas:
• The pancreas is a gland that plays a dual role in the digestive system and endocrine system.
• It produces insulin and digestive enzymes, but it does not contribute to reproductive functions.
• Pharynx:
• The pharynx is part of the respiratory and digestive systems, serving as a pathway for air and food.
• It has no role in reproduction.

The correct answer is Displacement.

Key Points

• Displacement is defined as the shortest distance between the initial and final positions of an object, along with its direction.
• When an object moves with a uniform velocity, the product of velocity and time gives the displacement of the object.
• Formula: Displacement (s) = Velocity (v) × Time (t). This is derived from the basic equation of motion when acceleration is zero.
• The SI unit of displacement is the same as distance, which is the meter (m).
• In this case, since the velocity is uniform and there is no change in direction, the displacement will equal the distance covered by the object.

Additional Information

• Acceleration:
• Acceleration is the rate of change of velocity of an object with respect to time.
• It is calculated using the formula: Acceleration (a) = Change in velocity (Δv) / Time (t).
• Its SI unit is meters per second squared (m/s²).
• It is not obtained by multiplying velocity with time.
• Momentum:
• Momentum is the product of an object's mass (m) and its velocity (v).
• Formula: Momentum (p) = Mass (m) × Velocity (v).
• Its SI unit is kilogram meter per second (kg·m/s).
• It is unrelated to the product of velocity and time.
• Force:
• Force is the interaction that, when unopposed, changes the motion of an object.
• It is calculated using Newton's second law: Force (F) = Mass (m) × Acceleration (a).
• Its SI unit is the Newton (N).
• Force is not derived from the product of velocity and time.
• Displacement Features:
• Displacement is a vector quantity, meaning it has both magnitude and direction.
• For example, if an object moves 5 meters east, its displacement is 5 meters in the east direction.
• Displacement can be zero if the object returns to its starting point.
• It is different from distance, which is a scalar quantity and only measures the path traveled.

Important Points

• Uniform Velocity:
• When an object moves with a constant speed in a straight line, its velocity is uniform.
• Under uniform velocity, displacement is directly proportional to time.
• Difference Between Distance and Displacement:
• Distance: Total path traveled, scalar quantity, always positive.
• Displacement: Shortest path between two points, vector quantity, can be positive, negative, or zero.
• Applications:
• Displacement is crucial in physics for motion analysis in both linear and rotational mechanics.
• It is used in navigation, robotics, and engineering to determine exact positioning.

The correct answer is esters.

Key Points

• Reaction between ethanol and ethanoic acid: When ethanol (an alcohol) reacts with ethanoic acid (a carboxylic acid) in the presence of concentrated sulfuric acid (H₂SO₄), an ester is formed. This process is known as esterification.
• Role of concentrated H₂SO₄: Concentrated sulfuric acid acts as a dehydrating agent in the reaction, removing water (H₂O) formed during the reaction and driving the equilibrium towards ester formation.
• Product of the reaction: The ester formed in this reaction is ethyl ethanoate, which has a characteristic fruity smell.
• Chemical equation: The reaction can be represented as:
  

CH₃COOH + C₂H₅OH → CH₃COOC₂H₅ + H₂O

• Applications of esters: Esters are widely used in perfumes, flavoring agents, and as solvents in various industrial processes.

Additional Information

• What is esterification?
• Esterification is a chemical reaction in which a carboxylic acid reacts with an alcohol in the presence of an acid catalyst (commonly concentrated H₂SO₄) to form an ester and water.
• This is an example of a condensation reaction, as two molecules combine to form a larger molecule with the elimination of a smaller molecule (water).
• Characteristics of esters:
• Esters are organic compounds with the general formula RCOOR', where R and R' are alkyl or aryl groups.
• They are known for their pleasant fruity odors, making them useful in the perfume and food industries.
• Role of concentrated H₂SO₄:
• Concentrated sulfuric acid is a strong acid that catalyzes the esterification reaction by donating protons (H⁺) to enhance the reactivity of the carboxylic acid and alcohol.
• It also acts as a dehydrating agent, removing water from the reaction mixture and shifting the equilibrium toward ester formation.
• Other reactions involving ethanol and ethanoic acid:
• In the absence of a dehydrating agent, ethanol and ethanoic acid do not readily form esters, highlighting the importance of concentrated H₂SO₄.
• Both ethanol and ethanoic acid are also involved in various other reactions, such as oxidation and neutralization, but esterification specifically leads to the formation of esters.

Important Points

• Ester usage: Esters like ethyl acetate (ethyl ethanoate) are widely used as solvents in the production of paints, varnishes, and adhesives.
• Reverse reaction: The reverse of esterification is known as ester hydrolysis, where esters react with water to form carboxylic acids and alcohols in the presence of an acid or base catalyst.
• Fischer–Speier esterification: The esterification reaction described is also referred to as Fischer–Speier esterification, named after the scientists Hermann Emil Fischer and Arthur Speier.
• Industrial relevance: Understanding esterification is crucial for industries like pharmaceuticals, cosmetics, and food flavoring.

The correct answer is Only C.

Key Points

• Statement A: Baking soda (sodium hydrogen carbonate) is prepared industrially by reacting sodium chloride, ammonia, carbon dioxide, and water. This reaction is part of the Solvay process, which is widely used in chemical industries. Hence, this statement is correct.
• Statement B: On heating, baking soda decomposes to release carbon dioxide gas (CO₂), water, and sodium carbonate. This property makes it useful in baking as it helps dough rise. The decomposition reaction is as follows:
  

2NaHCO₃ → Na₂CO₃ + H₂O + CO₂. Hence, this statement is correct.

• Statement C: Baking soda is used in antacids to neutralize excess stomach acid; however, it does not produce only salt and water. The reaction also releases carbon dioxide gas, which may lead to belching. The reaction is as follows:
  

NaHCO₃ + HCl → NaCl + H₂O + CO₂. Hence, this statement is NOT correct.

• Conclusion: Since statement C is incorrect, the correct answer is Only C.

Additional Information

• Properties of Baking Soda (Sodium Hydrogen Carbonate):
• Its chemical formula is NaHCO₃. It is a white crystalline solid, often appearing as a fine powder.
• Baking soda is mildly alkaline in nature and has a pH of around 8-9.
• It is soluble in water and decomposes upon heating to produce sodium carbonate, water, and carbon dioxide.
• Uses of Baking Soda:
• It is used in baking as a leavening agent, where it releases carbon dioxide to make dough rise.
• It is a component of some antacids, as it neutralizes stomach acid.
• Baking soda is used in cleaning products as a mild abrasive and deodorizer.
• It finds application in fire extinguishers, especially for grease and electrical fires.
• Solvay Process:
• This is an industrial process used to manufacture sodium carbonate and sodium bicarbonate.
• It involves the reaction of ammonia, carbon dioxide, and sodium chloride in water.
• The process is efficient and widely used in the chemical industry.
• Health Effects of Baking Soda:
• While it is safe in small amounts (e.g., in baking or as an antacid), excessive consumption may lead to alkalosis or an electrolyte imbalance.
• It should not be used in large doses without medical supervision.

Important Points

• Decomposition Reaction: Heating baking soda leads to its decomposition into sodium carbonate, water, and carbon dioxide. This reaction is crucial for its use in baking.
• Neutralization Reaction: Baking soda reacts with acids like vinegar or lemon juice to produce carbon dioxide, which is useful in cooking and cleaning applications.
• Storage: Baking soda should be stored in a dry, airtight container to prevent it from reacting with moisture and losing its effectiveness.

The correct answer is The direction of magnetic field around a straight current-carrying conductor can be given by the right-hand thumb rule..

Key Points

• The right-hand thumb rule is used to determine the direction of the magnetic field generated around a current-carrying conductor. According to this rule, if you hold the conductor with your right hand such that your thumb points in the direction of the current, then the curl of your fingers around the conductor indicates the direction of the magnetic field lines.
• Magnetic field lines are not straight lines. They are closed continuous curves that emerge from the north pole of a magnet, loop around, and re-enter at the south pole. Hence, the statement that "magnetic field lines are always straight lines" is incorrect.
• Magnetic field lines do not start from the south pole and end at the north pole outside the magnet. They originate from the north pole and terminate at the south pole outside the magnet. Inside the magnet, they travel from the south pole to the north pole, completing a closed loop. Hence, the second statement is incorrect.
• Magnetic field is not zero at the surface of a current-carrying wire. In fact, the magnetic field is strongest at the surface of the conductor and decreases as the distance from the conductor increases. Hence, the third statement is also incorrect.
• The fourth statement, regarding the right-hand thumb rule, is correct as it accurately describes the method to determine the direction of the magnetic field produced by a straight current-carrying conductor.

Additional Information

• Right-hand thumb rule:
• This rule is specifically applicable to straight current-carrying conductors.
• It helps visualize the magnetic field lines in a circular loop around the conductor.
• It is an essential concept in electromagnetism, widely used in physics and engineering applications.
• Magnetic field lines:
• These are imaginary lines used to represent the direction and strength of the magnetic field.
• The density of the lines indicates the strength of the field; closer lines represent a stronger magnetic field.
• They never intersect each other because a magnetic field cannot have two directions at the same point.
• Magnetic field near a conductor:
• The strength of the magnetic field near a current-carrying conductor depends on the current's magnitude and the distance from the conductor.
• The magnetic field is proportional to the current and inversely proportional to the distance (as per Ampere's circuital law).
• Applications of the right-hand thumb rule:
• Used in designing electric motors, generators, and transformers.
• Helpful in understanding electromagnetic induction and the behavior of magnetic fields in devices like solenoids and toroids.

Important Points

• Biot-Savart Law: This law quantifies the magnetic field produced at a point due to a small segment of current-carrying wire.
• Ampere's Circuital Law: This law relates the integrated magnetic field around a closed loop to the electric current passing through the loop.
• Electromagnetic Induction: The process of generating an electromotive force (EMF) in a conductor due to a changing magnetic field.
• Practical Uses: The understanding of magnetic fields and their direction is pivotal in developing technologies like MRI machines, electric power transmission systems, and maglev trains.

The correct answer is At the focus of the lens.

Key Points

• Convex lens: A convex lens is a type of lens that is thicker at the center and thinner at the edges. It is also known as a converging lens because it bends light rays inward.
• Parallel rays: When light rays that are parallel to the principal axis of a convex lens pass through the lens, they converge at a specific point on the principal axis known as the focus.
• The focus of a convex lens is the point where rays of light parallel to the principal axis converge after refraction through the lens. This property is a fundamental characteristic of convex lenses.
• When three rays of light are incident parallel and close to the principal axis, they will always converge at the focus of the lens, irrespective of their distance from the principal axis as long as they are within the paraxial region.
• The correct option reflects this fundamental behavior of a convex lens, making “At the focus of the lens” the accurate answer.

Additional Information

• Principal Axis: The principal axis is an imaginary straight line that passes through the optical center and the center of curvature of the lens. It acts as a reference line for analyzing the behavior of light rays.
• Focal Length: The focal length of a convex lens is the distance between the optical center and the focus of the lens. It determines the lens’s converging power. A shorter focal length indicates stronger converging power.
• Optical Center: The optical center of a lens is the point inside the lens through which light rays pass without any deviation. It lies on the principal axis.
• Refraction in Convex Lens: Refraction occurs when light rays pass from one medium to another with different optical densities (e.g., from air to glass). In a convex lens, this refraction causes the light rays to bend inward and converge.
• Applications: Convex lenses are widely used in various optical devices such as cameras, microscopes, telescopes, magnifying glasses, and corrective lenses for farsightedness.

Important Points

• Other behaviors of light rays in a convex lens:
• If a light ray passes through the optical center of the lens, it remains undeviated.
• If a light ray passes through the focus of the lens before incidence, it emerges parallel to the principal axis after refraction.
• Light rays that are not parallel to the principal axis converge at points on the focal plane, not the focus.
• Other types of lenses:
• Concave Lens: A concave lens diverges light rays that are parallel to the principal axis. It is also known as a diverging lens.
• Concave lenses are primarily used in devices like peepholes, eyeglasses for nearsightedness, and lasers.
• Real vs. Virtual Images: Convex lenses usually form real and inverted images when the object is placed beyond the focal length. However, when the object is within the focal length, the lens forms a virtual, erect, and magnified image.

The correct answer is C₂H₆ (Ethane).

Key Points

• Organic compounds are chemical compounds primarily composed of carbon and hydrogen, often with other elements like oxygen, nitrogen, sulfur, or phosphorus.
• C₂H₆ (Ethane) is a simple organic compound that belongs to the category of hydrocarbons, specifically an alkane, which contains only carbon (C) and hydrogen (H) atoms. Hence, C₂H₆ is an organic compound.
• In contrast, compounds like NaCl, CaCO₃, and Na₂CO₃ are considered inorganic compounds because they do not fit the typical structure of organic molecules with carbon-hydrogen bonds.
• Ethane is a gaseous compound under standard conditions and is a component of natural gas. It is used as a fuel and a feedstock in the chemical industry.
• Organic compounds like ethane are essential in biological processes and industrial applications, making them a key focus in chemistry.

Additional Information

• NaCl (Sodium Chloride):
• NaCl is an inorganic compound commonly known as table salt.
• It consists of sodium (Na) and chlorine (Cl) ions arranged in a crystalline lattice.
• It is widely used in cooking, food preservation, and industrial applications.
• CaCO₃ (Calcium Carbonate):
• CaCO₃ is an inorganic compound found naturally as limestone, chalk, and marble.
• It is used in construction materials, cement, and as a dietary calcium supplement.
• While it contains carbon, it lacks the C-H bonds required to classify it as an organic compound.
• Na₂CO₃ (Sodium Carbonate):
• Na₂CO₃, also known as soda ash or washing soda, is an inorganic compound.
• It is widely used in the glass manufacturing industry, water treatment, and as a cleaning agent.
• It contains carbonate ions (CO₃²⁻) but does not belong to organic compounds because of the absence of C-H bonds.
• Organic Compounds:
• Organic chemistry focuses on compounds containing carbon-hydrogen (C-H) bonds.
• Examples of organic compounds include methane (CH₄), ethane (C₂H₆), glucose (C₆H₁₂O₆), and ethanol (C₂H₅OH).
• They play an essential role in life processes, fuels, medicines, and industrial products.
• Inorganic Compounds:
• These compounds include salts, oxides, and other substances that do not have C-H bonds, such as water (H₂O), carbon dioxide (CO₂), and NaCl.
• They are crucial for various applications like construction, agriculture, and chemical synthesis.

The correct answer is 56.8 km/hr.

Key Points

• To calculate the average speed, we use the formula: Average Speed = Total Distance / Total Time.
• The car travels at a speed of 62 km/hr for 2.5 hours. The distance covered during this time is calculated as: Distance = Speed × Time = 62 × 2.5 = 155 km.
• The car then travels at a speed of 48 km/hr for 1.5 hours. The distance covered during this time is calculated as: Distance = Speed × Time = 48 × 1.5 = 72 km.
• The Total Distance covered by the car is: Total Distance = 155 km + 72 km = 227 km.
• The Total Time taken by the car is: Total Time = 2.5 hours + 1.5 hours = 4 hours.
• The Average Speed is therefore: Average Speed = Total Distance / Total Time = 227 / 4 = 56.75 km/hr. This value is approximately 56.8 km/hr.
• Hence, the correct answer is 56.8 km/hr.

Additional Information

• Average Speed Calculation:
• The formula for average speed is applicable only when the distances traveled at different speeds are known.
• If the time intervals are the same for different speeds, then the harmonic mean can be used to calculate the average speed.
• Average speed is a scalar quantity and does not take direction into account.
• Units of Speed:
• Speed is measured in units such as kilometers per hour (km/hr), meters per second (m/s), or miles per hour (mph).
• To convert from km/hr to m/s, divide the speed by 3.6. For example, 62 km/hr = 62 ÷ 3.6 = 17.22 m/s.
• Applications of Average Speed:
• Average speed is used in transportation planning, vehicle performance analysis, and time-distance calculations.
• It is a crucial metric in industries such as logistics and aviation for optimizing routes and fuel usage.
• :
• The distance-time graph provides a visual representation of speed. A steeper slope indicates higher speed.
• Uniform speed results in a straight line, while variable speed creates a curve.

Important Points

• Key Considerations in Calculations:
• Ensure all units are consistent before performing calculations (e.g., time in hours and speed in km/hr).
• For accurate results, avoid rounding off intermediate values during calculations.
• Real-Life Example:
• If a vehicle travels at different speeds due to traffic conditions, the average speed helps in estimating the total journey time.
• For example, a bus traveling at varying speeds between cities uses average speed to calculate arrival time.

The correct answer is Replication.

Key Points

• Replication is the process by which DNA duplicates into two identical molecules, ensuring that genetic information is passed on accurately during cell division.
• It is a semiconservative process, meaning that each of the two new DNA molecules consists of one original strand and one newly synthesized strand.
• DNA replication begins at specific sites called origins of replication, where the double helix unwinds, and replication forks are formed.
• The enzyme DNA helicase unwinds the DNA strands, while DNA polymerase synthesizes the new DNA by adding complementary nucleotides to the template strand.
• Replication ensures that each daughter cell has an exact copy of the parent cell's DNA, which is critical for maintaining genetic stability in organisms.

Additional Information

• Mutation:
• A mutation is a change in the DNA sequence that can occur due to errors during replication, exposure to mutagens, or other factors.
• Mutations are not responsible for duplicating DNA but can lead to variations in genetic material.
• Examples include point mutations, deletions, insertions, or chromosomal rearrangements.
• Translation:
• Translation is the process of synthesizing proteins from messenger RNA (mRNA) in the ribosome.
• This process involves decoding the genetic code carried by mRNA into a sequence of amino acids, forming a polypeptide chain.
• It is not related to DNA duplication.
• Transcription:
• Transcription is the process of synthesizing RNA from a DNA template by the enzyme RNA polymerase.
• It involves converting genetic information stored in DNA into a complementary RNA sequence.
• Transcription does not duplicate DNA but plays a role in gene expression.
• Semiconservative replication:
• This model was confirmed by the Meselson and Stahl experiment in 1958.
• It ensures that each new DNA molecule retains one strand from the parent DNA and one newly synthesized strand.
• This mechanism is crucial for minimizing errors during replication.
• Important enzymes in DNA replication:
• DNA helicase: Unwinds the DNA double helix.
• DNA polymerase: Adds complementary nucleotides to the growing DNA strand.
• Primase: Synthesizes RNA primers to initiate replication.
• Ligase: Joins fragments of DNA together, especially on the lagging strand.

The correct answer is They synthesise energy as ATP.

Key Points

• Mitochondria are often referred to as the "powerhouse of the cell" because they are the primary site for ATP (Adenosine Triphosphate) production, which is the main energy currency of the cell.
• ATP is synthesised during the process of cellular respiration, specifically in the electron transport chain, which occurs in the inner mitochondrial membrane.
• The mitochondria are equipped with a double membrane structure: an outer membrane and a highly folded inner membrane, which contains proteins essential for ATP production.
• Through the breakdown of glucose and other molecules in processes like the Krebs cycle (in the mitochondrial matrix) and oxidative phosphorylation (on the inner membrane), mitochondria efficiently produce ATP.
• This function is crucial for maintaining cellular activities such as muscle contraction, nerve impulse propagation, and synthesis of biomolecules. Hence, mitochondria are indispensable for energy metabolism in the cell.

Additional Information

• Structure and Function of Mitochondria:
• The mitochondria have a double membrane structure. The outer membrane is permeable to ions and small molecules, while the inner membrane is impermeable and contains folds called cristae, which increase surface area for ATP production.
• The mitochondrial matrix contains enzymes for the Krebs cycle, mitochondrial DNA, and ribosomes.
• Mitochondria have their own DNA and ribosomes, allowing them to synthesise some of their own proteins and enzymes independently of the cell nucleus.
• Other Cellular Functions of Mitochondria:
• In addition to ATP synthesis, mitochondria are involved in calcium ion storage, which is critical for cell signalling, muscle contraction, and other cellular processes.
• They play a role in the regulation of apoptosis, or programmed cell death, which is essential for removing damaged or unwanted cells.
• Mitochondria are also involved in heat production in brown adipose tissue through a process known as non-shivering thermogenesis.
• Explanation of Incorrect Options:
• Option 1: "They contain digestive enzymes" – Incorrect. Digestive enzymes are found in lysosomes, not mitochondria.
• Option 3: "They control protein synthesis" – Incorrect. Protein synthesis is primarily controlled by ribosomes and the cell nucleus.
• Option 4: "They store calcium ions" – Partially correct. While mitochondria do store calcium ions, this is not their primary function and does not justify their title as the "powerhouse of the cell."
• Key Processes Related to Mitochondria:
• Glycolysis: Occurs in the cytoplasm and provides pyruvate for the Krebs cycle in the mitochondria.
• Krebs Cycle: Takes place in the mitochondrial matrix and produces electron carriers like NADH and FADH2.
• Oxidative Phosphorylation: Occurs on the inner mitochondrial membrane, where ATP is generated from ADP and inorganic phosphate.

The correct answer is Image formation by a convex mirror.

Key Points

• Refraction of light occurs when light travels from one medium to another and bends due to a change in speed. This phenomenon is responsible for effects like image formation by lenses, the twinkling of stars, and the delayed sunset.
• Image formation by a convex lens is due to refraction. Light bends when passing through the curved lens surface, focusing at a particular point to create an image. Hence, this is correct.
• Delayed sunset happens due to atmospheric refraction. Light bends as it passes through the Earth's atmosphere, making the Sun visible even after it has set below the horizon. Hence, this is correct.
• Twinkling of stars is caused by refraction of starlight through layers of the Earth's atmosphere with varying densities. This bending of light causes stars to appear as if they are twinkling. Hence, this is correct.
• Image formation by a convex mirror is not due to refraction, but rather due to reflection. Convex mirrors reflect light rays outward, forming virtual and diminished images. Hence, this is the correct answer.

Additional Information

• Refraction:
• Definition: The bending of light when it passes from one transparent medium to another due to a change in its speed.
• Applications: Refraction is used in lenses (e.g., eyeglasses, microscopes, telescopes) to focus light and form images.
• Examples: Phenomena like the bending of a straw in water, mirages in deserts, and rainbows are due to refraction.
• Convex Lens:
• Function: Convex lenses are thicker at the center and converge light rays to a focal point.
• Uses: Found in cameras, magnifying glasses, and optical instruments.
• Image Formation: Can form both real and virtual images depending on the object’s position relative to the focal length.
• Atmospheric Refraction:
• Definition: Refraction caused by the Earth’s atmosphere due to varying air densities.
• Effects: Causes phenomena such as the apparent position of celestial bodies and delayed sunrise/sunset.
• Example: Stars twinkle because their light is refracted multiple times through atmospheric layers.
• Convex Mirror:
• Function: Convex mirrors reflect light rays outward, spreading them apart.
• Applications: Commonly used as rear-view mirrors in vehicles due to their wide field of view.
• Image Formation: Always forms virtual, upright, and diminished images.

Important Points

• Key Differences between Refraction and Reflection:
• Refraction: Light bends due to a change in medium (e.g., glass to air).
• Reflection: Light bounces back into the same medium from a surface (e.g., mirrors).
• Convex mirrors: Rely entirely on reflection, not refraction, for image formation.
• Applications of Refraction: Essential for designing optical devices like lenses, prisms, and fiber optics.
• Applications of Reflection: Used in mirrors, solar panels, and optical instruments that rely on light bouncing back.

The correct answer is To detect cracks and flaws inside the block.

Key Points

• Ultrasonic waves are widely used in industries for non-destructive testing (NDT) of materials, particularly metals, to detect cracks and flaws inside the structures.
• This technique works by transmitting high-frequency sound waves (typically above 20 kHz) into the material and analyzing the reflected signals to identify irregularities.
• The reflected waves, also called echoes, indicate the presence of flaws, such as cracks, voids, or inclusions, within the metallic block.
• Ultrasonic testing is crucial for ensuring the structural integrity and safety of components like machine parts, bridges, pipelines, and aircraft structures.
• It is a highly effective method because it is non-invasive, meaning it does not damage the component being tested, and can detect flaws even at considerable depths.

Additional Information

• Applications of Ultrasonic Testing:
• Used extensively in the aerospace industry for inspecting aircraft components to detect fatigue cracks and corrosion.
• In the automotive industry, it is utilized to check the quality of welds and the integrity of engine components.
• In civil engineering, ultrasonic waves are used to assess the condition of concrete structures, bridges, and tunnels.
• The oil and gas industry employs ultrasonic testing to detect defects in pipelines and storage tanks.
• Used in the manufacturing sector to ensure the quality of castings, forgings, and other metal products.
• Principle of Ultrasonic Testing:
• Ultrasonic testing is based on the principle of sound wave reflection. When sound waves encounter a boundary between two different materials or a defect, part of the wave is reflected back.
• The time taken for the reflected wave to return helps determine the location and size of the defect.
• The equipment typically includes a transducer (to generate and receive waves), a pulser/receiver, and a display unit for data interpretation.
• Advantages of Ultrasonic Testing:
• It provides precise results and can detect even small defects deep inside a material.
• It is a non-destructive technique, meaning the tested material remains unharmed.
• Ultrasonic testing can be automated and integrated into production lines for real-time quality control.
• Limitations of Ultrasonic Testing:
• Requires skilled operators and proper calibration of equipment for accurate results.
• Materials with coarse grains, such as cast iron, can scatter sound waves and reduce the accuracy of the test.
• Surface preparation is often necessary to ensure proper coupling of the transducer.
• Other Uses of Ultrasonic Waves:
• Medical Imaging: Used in ultrasound scans to visualize internal organs, monitor pregnancies, and diagnose conditions.
• Cleaning: Used in ultrasonic cleaning devices to remove contaminants from delicate objects like jewelry, surgical instruments, and electronic components.
• Welding: Ultrasonic waves are used in ultrasonic welding to join materials, particularly plastics, without using adhesives or fasteners.
• Flow Measurement: Used to measure the flow rate of liquids and gases in pipelines.

The correct answer is Gravitational force is strongest among small bodies.

Key Points

• The value of G is constant everywhere:
• The gravitational constant (G) is a universal constant with a value of approximately 6.674 × 10^-11 N·m²/kg². It remains constant throughout the universe.
• This constant is fundamental to Newton's law of gravitation, which governs the attractive force between two masses. Hence, this statement is correct.
• Gravitational force is always attractive:
• Gravitational force acts as an attractive force between two masses, pulling them towards each other.
• Unlike other forces such as electromagnetic forces, gravity does not have a repulsive component. Hence, this statement is correct.
• Gravitational force is strongest among small bodies:
• This statement is incorrect as gravitational force depends on the masses of the objects and the distance between them.
• Gravitational force is stronger between large celestial bodies, such as planets, stars, or moons, compared to smaller objects due to their larger masses.
• For small objects, the gravitational force is negligible. Hence, this statement is incorrect.
• Every object attracts every other object in the universe:
• According to Newton's Universal Law of Gravitation, every object in the universe exerts a gravitational pull on every other object.
• This mutual attraction is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Hence, this statement is correct.

Additional Information

• Newton's Universal Law of Gravitation:
• The law states that every mass in the universe attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The formula is given by: F = G × (m₁ × m₂) / r², where F is the gravitational force, m₁ and m₂ are the masses of the two objects, r is the distance between their centers, and G is the gravitational constant.
• Gravitational Force Characteristics:
• It is a long-range force that acts over astronomical distances.
• It is the weakest of the fundamental forces but dominates at large scales due to its cumulative effect over massive bodies.
• It is always attractive and never repulsive.
• Applications of Gravity:
• Gravity is responsible for holding planets in orbit around the sun and moons in orbit around planets.
• It governs the motion of celestial bodies, ocean tides, and the structure of galaxies.
• It plays a crucial role in phenomena such as free fall and weight measurement.
• Role of Mass and Distance:
• Greater the mass of the objects, stronger is the gravitational force.
• Greater the distance between the objects, weaker is the gravitational force due to the inverse square law.
• Gravity on Different Celestial Bodies:
• The acceleration due to gravity varies on different celestial bodies due to differences in their mass and radius.
• For example, the gravity on the Moon is about 1/6th of Earth's gravity, making objects weigh less on the Moon.

Important Points

• Gravitational Constant:
• Its value is 6.674 × 10^-11 N·m²/kg².
• G is a universal constant and remains the same irrespective of location.
• Inverse Square Law:
• The gravitational force decreases as the square of the distance between two objects increases.
• This is why celestial bodies far apart exert negligible gravitational force on each other.
• Gravitational Force vs. Other Forces:
• Gravity is weaker compared to electromagnetic and nuclear forces but dominates over large distances.
• It plays a fundamental role in shaping the universe's structure.
• Weight vs. Mass:
• Mass is a measure of the amount of matter in an object and remains constant everywhere.
• Weight is the gravitational force acting on an object and depends on the location due to variations in gravity.

The correct answer is Brass.

Key Points

• Brass is an alloy, which means it is a mixture of two or more metals. Its composition can vary depending on the desired properties, making it a material with variable composition.
• The primary components of brass are copper (Cu) and zinc (Zn). The ratio of these two metals can be adjusted to achieve different levels of strength, ductility, corrosion resistance, and color.
• Unlike pure substances like hydrogen chloride, water, or methane, brass does not have a fixed chemical formula. Instead, the proportion of its constituents can be tailored according to specific applications.
• Brass is widely used in various industries, including construction, musical instruments, plumbing, and decorative items, due to its malleability and attractive appearance.
• Its variable composition is an essential characteristic that makes brass suitable for diverse purposes, as different ratios of copper and zinc produce variations in physical and chemical properties.

Additional Information

• Hydrogen chloride (HCl):
• It is a pure compound with a fixed composition of one hydrogen atom and one chlorine atom.
• Hydrogen chloride is used in industries for chemical synthesis, cleaning agents, and the production of hydrochloric acid.
• It has a specific molecular formula that does not vary, making it a compound with a fixed composition.
• Water (H₂O):
• Water is a pure compound composed of two hydrogen atoms bonded to one oxygen atom.
• Its composition remains constant, and it is vital for life processes, serving purposes such as drinking, sanitation, and industrial cooling.
• Water is not a mixture or alloy, and it does not exhibit variable composition.
• Methane (CH₄):
• Methane is a simple hydrocarbon with a fixed molecular composition of one carbon atom and four hydrogen atoms.
• It is used as a fuel in homes and industries and is a major component of natural gas.
• Since methane has a fixed chemical formula, it does not exhibit variable composition.
• Brass:
• As an alloy, brass has a variable composition depending on the percentage of copper and zinc.
• For example, brass with higher copper content tends to be more corrosion-resistant, while higher zinc content increases strength.
• Brass is used for applications like electrical connectors, jewelry, mechanical parts, and marine hardware.
• Its adjustable properties make it versatile and widely applicable in various fields.

Important Points

• Pure substances: Compounds like hydrogen chloride, water, and methane have fixed compositions, meaning their molecular structures and chemical formulas remain constant.
• Alloys: Mixtures like brass are composed of two or more elements, with their ratios adjusted to meet specific requirements. This makes alloys materials with variable compositions.
• Brass is an example of an alloy, as its copper and zinc proportions can be modified for different mechanical, electrical, and aesthetic purposes.
• Understanding the difference between pure substances and alloys is crucial in materials science and industrial applications.

The correct answer is 2.

Key Points

• Valency is the combining capacity of an element, which determines how many atoms of an element can combine with atoms of another element to form a compound.
• In water (H₂O), oxygen combines with two hydrogen atoms. Each hydrogen atom has a valency of 1, meaning it can form one bond.
• Oxygen forms two bonds with hydrogen atoms, indicating that its valency is 2. These bonds are covalent in nature, meaning they involve sharing of electrons.
• The concept of valency is based on the number of valence electrons in the outermost shell of an element. Oxygen, with an atomic number of 8, has 6 electrons in its outer shell and needs 2 more electrons to achieve a stable octet configuration.
• Hence, the valency of oxygen in H₂O is 2.

Additional Information

• Definition of Valency:
• Valency is defined as the number of electrons an atom can lose, gain, or share to achieve a full outer shell of electrons.
• For elements in Groups 1-4 of the periodic table, valency is equal to the group number, while for Groups 5-8, it is 8 minus the group number.
• Oxygen's Electronic Configuration:
• The atomic number of oxygen is 8, and its electronic configuration is 2, 6.
• Oxygen requires 2 electrons to complete its octet (8 electrons in the outermost shell), giving it a valency of 2.
• Water Molecule Structure:
• Water is formed by the covalent bonding of two hydrogen atoms and one oxygen atom.
• The molecular structure is bent, with a bond angle of approximately 104.5° due to the lone pairs of electrons on the oxygen atom.
• Characteristics of Oxygen in Water:
• Oxygen is more electronegative than hydrogen, causing a partial negative charge on oxygen and a partial positive charge on hydrogen, making water a polar molecule.
• This polarity leads to properties like high surface tension, excellent solvent capabilities, and hydrogen bonding in water.

Important Points

• Valency vs. Oxidation State:
• Valency refers to the number of bonds an atom can form, while the oxidation state indicates the charge of an atom in a compound.
• For oxygen in most compounds, the oxidation state is -2, but it can vary in certain compounds like peroxides (-1) and superoxides (-1/2).
• Other Elements' Valencies in Water:
• Hydrogen has a valency of 1, as it can form a single bond with oxygen in the water molecule.
• The ratio of elements in H₂O is 2:1 (two hydrogens for one oxygen), maintaining the valency balance.

The correct answer is Plant health is affected.

Key Points

• Insects sucking sap from leaves: When insects such as aphids, whiteflies, or mealybugs suck the sap from crop plants, they deplete the plant of essential nutrients. This disrupts the plant's physiological processes, leading to weakened growth and reduced productivity.
• Impact on plant health: The extraction of sap causes significant damage to plant tissues, making the plants more susceptible to diseases and environmental stressors like drought and temperature fluctuations.
• Transmission of diseases: Many sap-sucking insects are vectors for plant diseases, such as viruses and bacteria. For example, aphids are known to transmit over 150 plant viruses.
• Reduction in photosynthesis: Damage to leaves caused by sap-sucking insects reduces the plant's ability to photosynthesize effectively. This impacts energy production and overall growth.
• Visible symptoms: Infestation by sap-sucking insects can cause visible symptoms such as yellowing of leaves, curling, stunted growth, and even premature leaf drop, all of which signal poor plant health.

Additional Information

• Function of sap in plants: Sap contains vital nutrients, including sugars, minerals, and water, which are essential for plant growth and development. Loss of sap weakens the plant's overall vitality.
• Examples of sap-sucking insects: Common sap-sucking pests include aphids, whiteflies, mealybugs, scale insects, and leafhoppers. These insects damage crops like cotton, tomatoes, and citrus plants.
• Prevention and control: Farmers often use integrated pest management (IPM) techniques such as introducing natural predators (e.g., ladybugs), using insecticidal soaps, or applying organic and chemical pesticides to control sap-sucking pests.
• Economic impact: Sap-sucking insects can cause significant economic losses in agriculture. For instance, in crops like cotton, infestations by pests like whiteflies have been known to reduce yield drastically, impacting farmers' incomes.
• Plant diseases spread by sap-sucking insects: Examples include Tomato Yellow Leaf Curl Virus transmitted by whiteflies and Barley Yellow Dwarf Virus spread by aphids. These diseases can devastate crops if not managed effectively.

Important Points

• Misconception about plant responses: Options such as "Plant gets pollinated" or "Plant grows stronger" are incorrect because sap-sucking insects harm plants rather than benefiting them. Pollination is typically carried out by pollinators like bees, not sap-sucking insects.
• Not all insects are harmful: While sap-sucking insects harm plants, other insects like bees and butterflies are beneficial as they aid in pollination, which is crucial for fruit and seed production.
• Environmental factors: Factors such as climate, humidity, and the presence of natural predators can influence the severity of sap-sucking insect infestations.
• Crop-specific damage: Some crops are more vulnerable to sap-sucking insects. For example, cotton is particularly susceptible to whiteflies, while citrus plants are often attacked by aphids and scale insects.

Shortcut Trick

Profit = 5% of Cost Price

Profit = 5% × 27,650

Profit = 27,650 ÷ 20 = 1,382.50

Rounding off 1,382.50 to the nearest rupee → 1,383

∴ The correct answer is ₹1,383.

Alternate Method

Given:

Cost Price (CP) = ₹27,650

Profit Percentage = 5%

Formula Used:

Profit = (Profit% ÷ 100) × Cost Price

⇒ Profit = (5 ÷ 100) × 27,650

⇒ Profit = 0.05 × 27,650

⇒ Profit = 1,382.5

⇒ Rounding off to the nearest rupee ≈ 1,383

∴ The correct answer is ₹1,383.

Additional Information

Selling Price Formula

Selling Price (SP) = CP × (100 + Profit%)/100. In case of loss, SP = CP × (100 − Loss%)/100.

Profit Percentage

Profit % = (Profit ÷ CP) × 100. Always calculate profit and loss percentages on the Cost Price unless specified otherwise.

Successive Discounts

The effective discount for two successive discounts a% and b% is given by (a + b − ab/100)%.

Shortcut Trick

Assume all 25 items are pencils: Cost = 25 × 4 = ₹100.

Difference from actual cost = 180 − 100 = ₹80.

Difference in cost between a pen and a pencil = 12 − 4 = ₹8.

Number of pens = 80 ÷ 8 = 10.

∴ The correct answer is 10.

Alternate Method

Cost of a pen = ₹12

Cost of a pencil = ₹4

Total items = 25

Total cost = ₹180

⇒ Average price per item = 180 ÷ 25 = ₹7.2

⇒ By Alligation, Ratio of pens to pencils = (7.2 − 4) : (12 − 7.2)

⇒ Ratio = 3.2 : 4.8 = 2 : 3

⇒ Total ratio parts = 2 + 3 = 5 units

⇒ 5 units = 25 items

⇒ 1 unit = 5 items

⇒ Number of pens = 2 × 5 = 10

∴ The correct answer is 10.

Additional Information

Alligation Method

It is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price.

Simultaneous Equations

Algebraic method where x + y = Total Quantity and Ax + By = Total Value are solved by substitution or elimination.

Weighted Average Formula

Average = (n₁w₁ + n₂w₂) ÷ (n₁ + n₂), where n is the quantity and w is the weight (cost) of each type.

Shortcut Trick

Efficiency ratio (A+B) : B = (1/6) : (1/7) = 7 : 6

Efficiency of A = (Efficiency of A+B) − (Efficiency of B) = 7 − 6 = 1 unit

Total Work = LCM(6, 7) = 42 units

Double the work = 42 × 2 = 84 units

Time taken by A alone = 84 ÷ 1 = 84 days

∴ The correct answer is 84.

Alternate Method

Given:

Time taken by A and B together = 6 days

Time taken by B alone = 7 days

Formula Used:

Total Work = Efficiency × Time

Calculations:

⇒ Let total work be LCM of 6 and 7 = 42 units.

⇒ Efficiency of (A + B) = 42 ÷ 6 = 7 units/day.

⇒ Efficiency of B = 42 ÷ 7 = 6 units/day.

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

⇒ Efficiency of A = 7 − 6 = 1 unit/day.

⇒ Double the work = 2 × 42 = 84 units.

⇒ Time taken by A to complete double work = 84 ÷ 1 = 84 days.

∴ The correct answer is 84.

Additional Information

Total Work Concept

Total work is always assumed to be the Least Common Multiple (LCM) of the individual time taken by different persons to simplify efficiency calculations.

Pipes and Cisterns

Similar to time and work, where an inlet pipe has positive efficiency and an outlet/leakage pipe has negative efficiency.

Work and Wages

Wages are always distributed in the ratio of the work done by individuals, which is often the same as the ratio of their efficiencies if they work for the same duration.

Shortcut Trick

Semi-perimeter (s) = (39 + 62 + 85) ÷ 2 = 93 cm

Area = √[93 × (93 − 39) × (93 − 62) × (93 − 85)]

Area = √[93 × 54 × 31 × 8] = 1116 cm²

Altitude to 62 cm side = (2 × Area) ÷ Base = (2 × 1116) ÷ 62

Altitude = 1116 ÷ 31 = 36 cm

∴ The correct answer is 36 cm.

Alternate Method

Given: Sides a = 39 cm, b = 62 cm, c = 85 cm.

Formula Used: Area = √[s(s − a)(s − b)(s − c)]; Area = (1/2) × base × altitude

⇒ Semi-perimeter s = (39 + 62 + 85) ÷ 2 = 93 cm

⇒ Area = √[93 × (93 − 39) × (93 − 62) × (93 − 85)]

⇒ Area = √[93 × 54 × 31 × 8]

⇒ Area = √[(31 × 3) × (2 × 27) × 31 × (2 × 4)]

⇒ Area = √[31² × 3 × 2 × 3³ × 2 × 2²]

⇒ Area = √[31² × 3⁴ × 2⁴] = 31 × 9 × 4 = 1116 cm²

⇒ Area = (1/2) × base × altitude

⇒ 1116 = (1/2) × 62 × h

⇒ 1116 = 31 × h

⇒ h = 1116 ÷ 31 = 36 cm

∴ The correct answer is 36 cm.

Additional Information

Heron's Formula

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

Altitude and Area Relationship

For any triangle, the altitude (h) to a specific base (b) is given by h = (2 × Area) ÷ b.

Obtuse Triangle Altitude

In an obtuse-angled triangle, the altitudes to the sides containing the obtuse angle fall outside the triangle on the extensions of the bases.

Shortcut Trick

Increase in average = (New Score − Old Average) ÷ Total New Innings

Increase = (195 − 57) ÷ 11

Increase = 138 ÷ 11 ≈ 12.54

Rounding 12.54 to the nearest whole number → 13

∴ The correct answer is 13.

Alternate Method

Given:

Average of 10 innings = 57 runs

Score in 11^th inning = 195 runs

Formula Used:

Average = Total Sum of Observations ÷ Number of Observations

⇒ Total runs in 10 innings = Average × Number of innings

⇒ Total runs = 57 × 10 = 570

⇒ Total runs after 11 innings = 570 + 195 = 765

⇒ New Average = 765 ÷ 11

⇒ New Average ≈ 69.545

⇒ Increase in average = New Average − Old Average

⇒ Increase = 69.545 − 57 = 12.545

⇒ Nearest whole number ≈ 13

∴ The correct answer is 13.

Additional Information

Net Deviation Method

The sum of deviations of all observations from the average is always zero. Surplus score beyond the current average is divided equally among all entities to find the increase.

Average of n Numbers

Average = (x₁ + x₂ + ... + x_n) ÷ n. If a new value x is added, the new average A′ = (n × A + x) ÷ (n + 1).

Rounding Rules

If the decimal part is 0.5 or greater, round up to the next whole number; if less than 0.5, round down.

Shortcut Trick

Rate of typing = 40 minutes ÷ 2 pages = 20 minutes per page.

Total time for 101 pages = 101 × 20 = 2020 minutes.

∴ The correct answer is 2020.

Alternate Method

Given:

Time taken for 2 pages = 40 minutes

Total pages to be typed = 101

Formula Used:

In direct proportion: (Work₁ / Time₁) = (Work₂ / Time₂)

By unitary method:

⇒ Time for 2 pages = 40 minutes

⇒ Time for 1 page = 40 ÷ 2

⇒ Time for 1 page = 20 minutes

⇒ Time for 101 pages = 101 × 20

⇒ Time for 101 pages = 2020 minutes

∴ The correct answer is 2020.

Additional Information

Direct Proportion

Two quantities are in direct proportion if an increase in one leads to a proportional increase in the other. Formula: x/y = Constant.

Work and Time

Total Work = Rate × Time. If the rate of work is constant, the time taken is directly proportional to the amount of work.

Unitary Method

A technique where you first find the value of a single unit (1 page) and then multiply to find the value of the required units (101 pages).

Shortcut Trick

Equate total work: (11M + 17W) × 6 = (18M + 15W) × 4

Divide both sides by 2: 3(11M + 17W) = 2(18M + 15W)

Expand terms: 33M + 51W = 36M + 30W

Rearrange like terms: 36M − 33M = 51W − 30W

3M = 21W ⇒ 1M = 7W

∴ The correct answer is 7.

Alternate Method

Given: 11 men and 17 women in 6 days | 18 men and 15 women in 4 days

Formula Used: Total Work = Efficiency × Time

Let efficiency of 1 man be M and 1 woman be W.

⇒ Total Work = (11M + 17W) × 6

⇒ Total Work = (18M + 15W) × 4

⇒ 6(11M + 17W) = 4(18M + 15W)

⇒ 66M + 102W = 72M + 60W

⇒ 72M − 66M = 102W − 60W

⇒ 6M = 42W

⇒ 1M = 7W

∴ The correct answer is 7.

Additional Information

Man-Day Formula

For group work, M₁D₁H₁/W₁ = M₂D₂H₂/W₂, where M is number of people, D is days, H is hours, and W is work done.

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete the work (E ∝ 1/T).

Work Equivalence

When different groups complete the same work, their total man-days must be equated to find individual efficiency ratios.

Shortcut Trick

Direct Formula: Ratio of CP/MP = (100 − Discount%)/(100 + Profit%)

Given: Discount = 32% and Profit = 57%

⇒ CP : MP = (100 − 32) : (100 + 57)

⇒ CP : MP = 68 : 157

∴ The correct answer is 68 : 157.

Alternate Method

Given:

Discount allowed = 32%

Profit earned = 57%

Formula Used:

Selling Price (SP) = CP × (100 + Profit%)/100

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

Calculations:

⇒ Let Cost Price be CP and Listed Price (Marked Price) be MP.

⇒ SP = CP × (100 + 57)/100 = 157CP/100

⇒ SP = MP × (100 − 32)/100 = 68MP/100

⇒ Equating the Selling Price: 157CP/100 = 68MP/100

⇒ 157 × CP = 68 × MP

⇒ CP/MP = 68 / 157

⇒ CP : MP = 68 : 157

∴ The correct answer is 68 : 157.

Additional Information

CP-MP Relation

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

Successive Discounts

If two discounts d₁% and d₂% are given, Net Discount = d₁ + d₂ − (d₁ × d₂)/100.

Profit and Loss Percentage

Profit % = (Profit/CP) × 100. Always calculate profit or loss percentage on the Cost Price unless specified.

Shortcut Trick

Use Algebraic Identity: (a + b)² + (a − b)² = 2(a² + b²)

Here, a = 5x and b = 11y

⇒ Result = 2[(5x)² + (11y)²]

⇒ 2[25x² + 121y²] = 50x² + 242y²

∴ The correct answer is 50x² + 242y².

Alternate Method

Given expression: (5x + 11y)² + (5x − 11y)²

Formula used: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

⇒ (5x + 11y)² = (5x)² + 2(5x)(11y) + (11y)²

⇒ (5x + 11y)² = 25x² + 110xy + 121y²

⇒ (5x − 11y)² = (5x)² − 2(5x)(11y) + (11y)²

⇒ (5x − 11y)² = 25x² − 110xy + 121y²

⇒ Adding both results: (25x² + 110xy + 121y²) + (25x² − 110xy + 121y²)

⇒ (25x² + 25x²) + (110xy − 110xy) + (121y² + 121y²)

⇒ 50x² + 0 + 242y²

⇒ 50x² + 242y²

∴ The correct answer is 50x² + 242y².

Additional Information

Standard Algebraic Identities

(a + b)² − (a − b)² = 4ab

Sum and Difference Identity

(a + b)(a − b) = a² − b²

Perfect Square Trinomial

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

Shortcut Trick

Area of rectangular strip (outside) = 2w(L + B + 2w) = 2 × 10(240 + 180 + 20) = 8800 m²

Volume of earth spread = Area of land × Rise in level = 240 × 180 × 0.25 = 10800 m³

Volume dug = Volume spread ⇒ Area of strip × Depth = 10800

Depth = 10800 ÷ 8800 = 1.227...

∴ The correct answer is 1.23 m.

Alternate Method

Given:

Dimensions of land = 240 m × 180 m

Width of strip (w) = 10 m

Rise in land level (h) = 25 cm = 0.25 m

Formula Used:

Volume of earth dug = Area of strip × Depth

Area of rectangular path (outside) = (L + 2w)(B + 2w) − (L × B)

⇒ Area of land = 240 × 180 = 43200 m²

⇒ Volume of earth spread = 43200 × 0.25 = 10800 m³

⇒ Outer Length = 240 + 10 + 10 = 260 m

⇒ Outer Width = 180 + 10 + 10 = 200 m

⇒ Area of strip = (260 × 200) − (240 × 180)

⇒ Area of strip = 52000 − 43200 = 8800 m²

⇒ Depth of strip × 8800 = 10800

⇒ Depth = 10800 ÷ 8800

⇒ Depth = 27 ÷ 22 ≈ 1.2272 m

∴ The correct answer is 1.23 m.

Additional Information

Area of Path Around a Rectangle

For a path of width 'w' outside a rectangle of L × B, Area = 2w(L + B + 2w). For an inside path, Area = 2w(L + B − 2w).

Conservation of Volume in Earthwork

Volume of earth removed from a pit/trench is equal to the volume of the platform or layer formed by spreading that earth.

Unit Conversions

Always ensure all dimensions are in the same unit. 1 cm = 0.01 m, 1 m² = 10000 cm², 1 m³ = 1000 litres.

Shortcut Trick

Merge ratios: R : B = 7 : 5 and B : Y = 8 : 4 (or 2 : 1).

R : B : Y = (7 × 8) : (5 × 8) : (5 × 4) = 56 : 40 : 20 = 14 : 10 : 5.

Total units = 14 + 10 + 5 = 29 units.

29 units = 6409 ⇒ 1 unit = 221.

Yellow blocks = 5 × 221 = 1105.

∴ The correct answer is 1105.

Alternate Method

Given:

Total colour blocks = 6409

Ratio of Red : Blue = 7 : 5

Ratio of Blue : Yellow = 8 : 4

Formula Used:

If A : B = x : y and B : C = p : q, then A : B : C = (x × p) : (y × p) : (y × q)

⇒ Red : Blue = 7 : 5

⇒ Blue : Yellow = 8 : 4 = 2 : 1

⇒ To make Blue parts equal, multiply (7 : 5) by 2 and (2 : 1) by 5.

⇒ Red : Blue = 14 : 10

⇒ Blue : Yellow = 10 : 5

⇒ Combined Ratio R : B : Y = 14 : 10 : 5

⇒ Sum of ratio parts = 14 + 10 + 5 = 29 units

⇒ 29 units = 6409

⇒ 1 unit = 6409 ÷ 29 = 221

⇒ Number of yellow blocks = 5 × 221

⇒ Yellow blocks = 1105

∴ The correct answer is 1105.

Additional Information

Ratio Merging

To combine A:B and B:C into A:B:C, the common term (B) must have the same value in both ratios. Use the LCM of the B values to normalize.

Distribution based on Ratio

Value of a specific part = (Particular ratio part ÷ Sum of all ratio parts) × Total quantity.

Compound Ratio

The compound ratio of a:b and c:d is (a × c) : (b × d). It is the product of the individual ratios.

Shortcut Trick

Speed of train = 1768 ÷ 68 = 26 m/s.

Total distance to cross platform = Train length + Platform length = 1768 + 338 = 2106 m.

Time taken = Total Distance ÷ Speed = 2106 ÷ 26 = 81 seconds.

∴ The correct answer is 81.

Alternate Method

Given:

Length of train (L_T) = 1768 m

Time taken to cross a pole (t₁) = 68 s

Length of platform (L_P) = 338 m

Formula Used:

Speed = Distance ÷ Time

When crossing a platform: Total Distance = Length of Train + Length of Platform

Calculations:

⇒ Speed of the train = L_T ÷ t₁

⇒ Speed = 1768 ÷ 68 = 26 m/s

⇒ Total distance to cross platform = L_T + L_P

⇒ Total distance = 1768 + 338 = 2106 m

⇒ Time taken to cross platform (t₂) = Total Distance ÷ Speed

⇒ t₂ = 2106 ÷ 26

⇒ t₂ = 81 seconds

∴ The correct answer is 81.

Additional Information

Relative Speed

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

Unit Conversion

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

Train Crossing Objects

When a train crosses a stationary point object (pole, man), distance = length of train. When crossing an object with length (platform, bridge), distance = train length + object length.

Shortcut Trick

Assume Income = 100 units, Savings = 18.5 units, Expenditure = 81.5 units.

Change in Income = 17% of 100 = +17 units.

Change in Expenditure = 25% of 81.5 = +20.375 units.

Net Change in Savings = Change in Income − Change in Expenditure = 17 − 20.375 = −3.375 units.

Given 100 units = ₹68,800 ⇒ 1 unit = ₹688.

Change in Savings = −3.375 × 688 = −₹2,322 (Decrease).

∴ The correct answer is decrease by ₹2,322.

Alternate Method

Given:

Initial Income = ₹68,800

Initial Savings % = 18.5%

Income Increase = 17% | Expenditure Increase = 25%

Formula Used:

Income = Expenditure + Savings

⇒ Initial Savings = 18.5% of 68,800 = 0.185 × 68,800 = ₹12,728

⇒ Initial Expenditure = 68,800 − 12,728 = ₹56,072

⇒ New Income = 68,800 × (100 + 17)% = 68,800 × 1.17 = ₹80,496

⇒ New Expenditure = 56,072 × (100 + 25)% = 56,072 × 1.25 = ₹70,090

⇒ New Savings = New Income − New Expenditure = 80,496 − 70,090 = ₹10,406

⇒ Net Change in Savings = 10,406 − 12,728 = −₹2,322

∴ The correct answer is decrease by ₹2,322.

Additional Information

Income-Expenditure Fundamental

The basic relation is Income = Expenditure + Savings. Any percentage change in one or two components directly affects the third component based on their relative weights.

Deviation Method

Total Change = (Exp Weight × Exp % Change) + (Sav Weight × Sav % Change). This can be used to find the missing percentage change quickly.

Percentage Change Rule

Percentage Change = (New Value − Old Value) ÷ Old Value × 100. A positive result indicates an increase, while a negative result indicates a decrease.

Shortcut Trick

Net percentage change = a + b + (ab/100)

Where a = markup % = +75 and b = discount % = −52

Net % = 75 − 52 + (75 × −52)/100

Net % = 23 − (3 × 13) = 23 − 39

Net % = −16% (Negative sign denotes loss)

∴ The correct answer is 16% loss.

Alternate Method

Given:

Markup % = 75%

Discount % = 52%

Formula Used:

Marked Price (MP) = CP × (100 + Markup%)/100

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

⇒ Let Cost Price (CP) = 100 units

⇒ Marked Price (MP) = 100 + 75% of 100 = 175 units

⇒ Discount = 52% of MP = (52/100) × 175

⇒ Discount = 52 × 1.75 = 91 units

⇒ Selling Price (SP) = MP − Discount = 175 − 91 = 84 units

⇒ Loss = CP − SP = 100 − 84 = 16 units

⇒ Loss % = (Loss/CP) × 100 = (16/100) × 100 = 16%

∴ The correct answer is 16% loss.

Additional Information

Relation between CP and MP

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

Markup Percentage

Markup % = [(MP − CP)/CP] × 100. It is the percentage by which the cost price is increased to set the marked price.

Successive Discount Formula

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

Shortcut Trick

To find the percentage of one fraction relative to another, use the ratio of their decimal values or cross-multiply.

1/20 = 0.05 and 11/50 = 0.22

Required % = (0.05 ÷ 0.22) × 100

Required % = 5/22 × 100 = 500/22

Required % = 22.7272... ≈ 22.73%

∴ The correct answer is 22.73%.

Alternate Method

Given:

Number 1 = 1/20

Number 2 (Base) = 11/50

Formula:

Percentage = (Value ÷ Base) × 100

⇒ Percentage = [(1/20) ÷ (11/50)] × 100

⇒ Percentage = (1/20) × (50/11) × 100

⇒ Percentage = (1 × 50) ÷ (20 × 11) × 100

⇒ Percentage = 50/220 × 100

⇒ Percentage = 5/22 × 100

⇒ Percentage = 500/22

⇒ Percentage = 22.7272...

⇒ Rounding to two decimal places, we get 22.73%.

∴ The correct answer is 22.73%.

Additional Information

Percentage of Fractions

To calculate what percentage (a/b) is of (c/d), use: [(a × d) ÷ (b × c)] × 100.

Decimal Conversion

Fractions can be converted to decimals first to simplify calculation: 1/20 = 0.05 and 11/50 = 0.22.

Rounding Rules

When rounding to two decimal places, if the third decimal digit is 5 or greater, increase the second decimal digit by 1.

Shortcut Trick

Number of cubes = (Volume of Cuboid) ÷ (Volume of one Cube)

⇒ (24 × 16 × 10) ÷ (4 × 4 × 4)

⇒ (24 ÷ 4) × (16 ÷ 4) × (10 ÷ 4)

⇒ 6 × 4 × 2.5 = 60

∴ The correct answer is 60.

Alternate Method

Given:

Dimensions of cuboid: L = 24 cm, B = 16 cm, H = 10 cm

Side of small cube (a) = 4 cm

Formula Used:

Volume of Cuboid = Length × Breadth × Height

Volume of Cube = (Side)³

⇒ Total volume of the cuboid = 24 × 16 × 10

⇒ Total volume = 3840 cm³

⇒ Volume of one small cube = 4 × 4 × 4

⇒ Volume of one small cube = 64 cm³

⇒ Number of cubes = Total volume of cuboid ÷ Volume of one cube

⇒ Number of cubes = 3840 ÷ 64

⇒ Number of cubes = 60

∴ The correct answer is 60.

Additional Information

Total Surface Area of Cuboid

The total area of all six faces is given by TSA = 2 × (LB + BH + HL).

Diagonal of a Cuboid

The longest rod that can be placed inside is D = √(L² + B² + H²).

Surface Area of a Cube

The total surface area of a cube with side a is 6a².

Shortcut Trick

Relative speed in same direction = 150 − 120 = 30 km/hr.

Total distance to be covered = 210 + 390 = 600 m = 0.6 km.

Time taken (in hours) = Distance ÷ Relative Speed = 0.6 ÷ 30 = 0.02 hours.

Time (in minutes) = 0.02 × 60 = 1.2 minutes.

∴ The correct answer is 1.2.

Alternate Method

Given: L₁ = 210 m, L₂ = 390 m, S₁ = 120 km/hr, S₂ = 150 km/hr (same direction).

Formula: Time = (Length₁ + Length₂) ÷ (Speed₂ − Speed₁)

⇒ Relative Speed = 150 − 120 = 30 km/hr

⇒ Speed in m/s = 30 × (5/18) = 25/3 m/s

⇒ Total distance = 210 + 390 = 600 m

⇒ Time taken (seconds) = 600 ÷ (25/3) = 600 × (3/25)

⇒ Time taken = 24 × 3 = 72 seconds

⇒ Time (minutes) = 72 ÷ 60 = 1.2 minutes

∴ The correct answer is 1.2.

Additional Information

Relative Speed in Opposite Direction

When two objects move toward each other, their relative speed is the sum of their individual speeds (S₁ + S₂).

Speed Conversion

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

Train Crossing Stationary Object

When a train crosses a pole or a person, the distance covered equals the length of the train (L). If it crosses a platform, distance = L + Length of platform.

Shortcut Trick

Let total voters = x

Successive percentage equation: x × 80% × 98% × 75% = 8,820

x × (4/5) × (49/50) × (3/4) = 8,820

x × 147/250 = 8,820

x = 60 × 250 = 15,000

∴ The correct answer is 15,000.

Alternate Method

Given:

Voters cast = 80% | Invalid votes = 2% (Valid = 98%) | Candidate share = 75% of valid votes | Candidate votes = 8,820

⇒ Let total registered voters = x

⇒ Number of cast votes = 0.80 × x

⇒ Number of valid votes = 98% of (0.80 × x) = 0.98 × 0.80 × x

⇒ Winning candidate's votes = 75% of valid votes

⇒ 0.75 × (0.98 × 0.80 × x) = 8,820

⇒ 0.75 × 0.784x = 8,820

⇒ 0.588x = 8,820

⇒ x = 8,820 ÷ 0.588

⇒ x = 15,000

∴ The correct answer is 15,000.

Additional Information

Successive Percentage Application

When multiple percentages are applied to a base value in sequence, the final value = Base × (P₁/100) × (P₂/100) × ... × (P_n/100).

Election Terminology

Valid Votes = Total Cast Votes − Invalid Votes. Winning Margin = (Winner% − Loser%) × Valid Votes.

Fractional Conversion Shortcuts

Using fractions simplifies calculation: 80% = 4/5, 75% = 3/4, 25% = 1/4, 20% = 1/5.

Shortcut Trick

Product of extremes = Product of means

18 × 16 = a × 8a

288 = 8a²

a² = 36

a = 6

∴ The correct answer is 6.

Alternate Method

Given: The terms are 18, a, 8a, and the fourth proportional is 16.

Formula Used: If x, y, z, and w are in proportion, then x : y = z : w or x/y = z/w.

⇒ 18/a = 8a/16

⇒ 18 × 16 = a × 8a

⇒ 288 = 8a²

⇒ a² = 288 ÷ 8

⇒ a² = 36

⇒ a = √36

⇒ a = 6 (as a is positive)

∴ The correct answer is 6.

Additional Information

Fourth Proportional

If four quantities a, b, c, and d are in proportion, then d is the fourth proportional to a, b, and c such that a/b = c/d.

Mean Proportional

The mean proportional between two numbers a and b is given by x = √(a × b).

Third Proportional

If a, b, and c are in continued proportion such that a/b = b/c, then c is the third proportional given by c = b²/a.

Shortcut Trick

Assume a base mean value of 60.

Find deviations from 60: +36, +10, +25, −8, −20, +28, +3, −20, 0.

Sum of deviations = 36 + 10 + 25 + 28 + 3 − 8 − 20 − 20 = 54.

Average deviation = 54 ÷ 9 = 6.

Arithmetic Mean = Base + Average Deviation = 60 + 6 = 66.

∴ The correct answer is 66.

Alternate Method

Given observations: 96, 70, 85, 52, 40, 88, 63, 40, 60

Formula: Arithmetic Mean = (Sum of observations) ÷ (Number of observations)

⇒ Total number of observations (n) = 9

⇒ Sum = 96 + 70 + 85 + 52 + 40 + 88 + 63 + 40 + 60

⇒ Sum = 594

⇒ Arithmetic Mean = 594 ÷ 9

⇒ Arithmetic Mean = 66

∴ The correct answer is 66.

Additional Information

Sum of Deviations

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

Effect of Linear Change

If every observation is increased or decreased by a constant k, the arithmetic mean is also increased or decreased by k.

Combined Mean

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

Shortcut Trick

Rate difference = 7.5% − 3.5% = 4% per annum.

Total interest percentage difference for 3 years = 4% × 3 = 12%.

Required difference = 12% of Principal = 12/100 × 2,20,000.

Required difference = 12 × 2,200 = 26,400.

∴ The correct answer is 26,400.

Alternate Method

Given:

Principal (P) = ₹2,20,000

Rate 1 (R₁) = 3.5% | Rate 2 (R₂) = 7.5%

Time (T) = 3 years

Formula Used:

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

⇒ SI paid to Bank A = (2,20,000 × 3.5 × 3) ÷ 100

⇒ SI paid to Bank A = 2,200 × 10.5 = 23,100

⇒ SI paid to Bank B = (2,20,000 × 7.5 × 3) ÷ 100

⇒ SI paid to Bank B = 2,200 × 22.5 = 49,500

⇒ Positive Difference = SI paid to Bank B − SI paid to Bank A

⇒ Positive Difference = 49,500 − 23,100

⇒ Positive Difference = 26,400

∴ The correct answer is 26,400.

Additional Information

Simple Interest Concept

Simple interest is calculated only on the initial principal amount. Formula: SI = (P × R × T) ÷ 100.

Interest Difference Rule

When Principal and Time are constant, the difference in SI is proportional to the difference in rates: ΔSI = P × ΔR × T ÷ 100.

Amount Calculation

The total amount (A) payable after time T is the sum of Principal and Simple Interest: A = P + SI.

Shortcut Trick

Unitary Method: Time taken per page = 40 / 4 = 10 minutes.

For 112 pages, time required = 112 × 10 = 1120 minutes.

∴ The correct answer is 1120.

Alternate Method

Given:

Work₁ = 4 pages | Time₁ = 40 minutes

Work₂ = 112 pages | Time₂ = ?

Formula used: Work₁ / Time₁ = Work₂ / Time₂ (Direct Proportion)

⇒ Let the time taken for 112 pages be x.

⇒ 4 / 40 = 112 / x

⇒ 1 / 10 = 112 / x

⇒ x = 112 × 10

⇒ x = 1120 minutes

∴ The correct answer is 1120.

Additional Information

Direct Proportion

When one quantity increases as the other increases, the ratio x/y remains constant. Formula: x₁/y₁ = x₂/y₂.

Work and Time Relationship

For a constant efficiency, the amount of work done is directly proportional to the time taken (W ∝ T).

Rate of Work

Rate is defined as Work done divided by Time taken. It represents the efficiency of the person or machine.

Shortcut Trick

Age difference between two people is always constant = 4 years.

20 years ago, ratio (Elder : Younger) = 1.25 : 1 = 5 : 4.

Ratio difference = 5 − 4 = 1 unit → 4 years.

Elder age (20 years ago) = 5 × 4 = 20 years.

Current age of Elder = 20 + 20 = 40 years.

∴ The correct answer is 40.

Alternate Method

Given:

Current difference (E − Y) = 4 years.

20 years ago, Elder age = 1.25 × Younger age.

Let current age of Elder brother be E and Younger be Y.

Formula: Age_past = Age_current − Time_interval

⇒ E − Y = 4

⇒ Y = E − 4

20 years ago:

⇒ (E − 20) = 1.25 × (Y − 20)

⇒ E − 20 = 1.25 × (E − 4 − 20)

⇒ E − 20 = 1.25 × (E − 24)

⇒ E − 20 = 1.25E − 30

⇒ 30 − 20 = 1.25E − E

⇒ 10 = 0.25E

⇒ E = 10 / 0.25

⇒ E = 40

∴ The correct answer is 40.

Additional Information

Constant Age Difference

The difference between the ages of two individuals remains the same throughout their lives, regardless of how many years pass.

Decimal to Ratio Conversion

Converting multipliers to fractions (e.g., 1.25 = 5/4, 1.5 = 3/2, 1.75 = 7/4) simplifies the Unitary Method for solving age problems.

Future/Past Age Calculation

If current age is X, age after n years is X + n, and age n years ago was X − n.

Shortcut Trick

Ratio of Height : Distance from foot = 5 : 3

Given, distance (3 units) = 36 m

1 unit = 36 / 3 = 12 m

Height (5 units) = 5 × 12 = 60 m

∴ The correct answer is 60.

Alternate Method

Given: Distance of person from tower foot (Base) = 36 m

Angle of elevation = θ, where tan θ = 5/3

Formula: tan θ = Perpendicular (Height) / Base (Distance)

⇒ tan θ = Height / 36

⇒ 5 / 3 = Height / 36

⇒ Height = (5 × 36) / 3

⇒ Height = 5 × 12

⇒ Height = 60 m

∴ The correct answer is 60.

Additional Information

Trigonometric Ratios

In a right-angled triangle, tan θ = Perpendicular/Base, sin θ = Perpendicular/Hypotenuse, and cos θ = Base/Hypotenuse.

Angle of Elevation

It is the angle between the horizontal line and the line of sight when an observer looks at an object above the horizontal level.

Pythagoras Theorem

For a right triangle with height h and base b, the hypotenuse l is given by l = √(h² + b²).

Shortcut Trick

Convert all bases to power 2: 16 = 2⁴, 8 = 2³, 4 = 2².

Numerator: 2⁻⁴ × 2⁶ = 2²

Denominator: 2⁻⁶ × 2⁵ = 2⁻¹

Expression simplifies to: (2² ÷ 2⁻¹)^1/3 = (2²⁻⁽⁻¹⁾)^1/3 = (2³)^1/3

Result = 2^{3 × 1/3} = 2

∴ The correct answer is 2.

Alternate Method

Given:

((16⁻¹ × 8²) ÷ (4⁻³ × 2⁵))^1/3

Formula Used:

1. a^m × aⁿ = a^m+n
2. a^m ÷ aⁿ = a^m−n
3. (a^m)ⁿ = a^m×n

Calculations:

⇒ [((2⁴)⁻¹ × (2³)²) ÷ ((2²)⁻³ × 2⁵)]^1/3

⇒ [(2⁻⁴ × 2⁶) ÷ (2⁻⁶ × 2⁵)]^1/3

⇒ [2^{(−4 + 6)} ÷ 2^{(−6 + 5)}]^1/3

⇒ [2² ÷ 2⁻¹]^1/3

⇒ [2^{2 − (−1)}]^1/3

⇒ [2³]^1/3

⇒ 2^{3 × 1/3}

⇒ 2¹ = 2

∴ The correct answer is 2.

Additional Information

Negative Exponent Rule

Any base raised to a negative power is the reciprocal of the base raised to the positive power: a⁻ⁿ = 1/aⁿ.

BODMAS in Indices

Always simplify the innermost brackets and powers before applying the outer roots or exponents.

Fractional Exponents

An exponent of 1/n represents the nth root of the number: a^1/n = ⁿ√a.

Given: In a certain code language, ‘keep turning pages’ is coded as ‘kd vt fg’ and ‘count the pages’ is coded as ‘kl vt za’.

Thus, 'pages' is coded as 'vt' in the given language.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: PI 13 is related to SM 17 in a certain way. In the same way, HA 24 is related to KE 28. 

The logic followed here is:

Similarly,

Thus, RK35 is related to UO39, following the same logic.

Hence, the correct answer is "Option 3".

Given: In a certain code language, ‘lab bird set’ is coded as 'ef xo ej' and ‘set web hall’ is coded as ‘ot ef hs’.

Thus, ‘set’ is coded as 'ef' in that language.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: UDB VEC WFD XGE ?

The logic followed here is:

Thus, YHF should come in place of the question mark (?) in the given series based on the English alphabetical order.

Hence, the correct answer is "Option 4".

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

Now,

Even numbers, each of which is immediately preceded by an odd number and also immediately followed by an even number.

So, the needed pattern is: Odd — Even — Even

Thus, there are 2 even numbers, each of which is immediately preceded by an odd number and also immediately followed by an even number.

Hence, the correct answer is "Option 4".

Given: A, B, C, E, F, P and Q are sitting around a circular table facing the centre.

1. C sits third to the left of P.
2. Q is an immediate neighbour of both E and P.

3. Only four people sit between C and E when counted from the right of C.

The above statement cannot be satisfied in case I, so it is eliminated.

4. B sits third to the left of F.

A, the only person left will occupy the vacant position.

Thus, two people sit between A and Q when counted from the right of A.

Hence, the correct answer is "Option 4".

Given:

Mr. Zzz starts from Point A and drives 11 km East.

Turns left and drives 19 km (North), then turns left and drives 21 km (West).

Turns left and drives 24 km (South), then takes a final left turn and drives 10 km (East) to reach Point P.

  North   South West   East    Point A  11 km  19 km  21 km  24 km  10 km  Point P  (5 km distance)

Since Point P is 5 km South of Point A, to reach Point A again, he must drive 5 km towards the North.

Hence, the correct answer is "Option 4".

The logic followed here is:

Position of Saurabh from the top = 17th

Position of Saurabh from the bottom = 28th

So,

Total number of students = (Position from the top + Position from the bottom) - 1

= 17 + 28 - 1

= 45 - 1 = 44.

Hence, the correct answer is "Option 1".

Given: A, B, C, D, W, X and Y are sitting around a circular table facing the centre of the table.

1. C sits to the immediate left of B. 2. D sits third to the left of B.

2. W sits to the immediate left of A.

3. A sits second to the left of X.

The above statement cannot be satisfied in case II, so it is eliminated.

Y, the only person left will occupy the vacant position.

Thus, D sits third to the right of Y.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

The pattern followed here is:

Each letter-cluster follows the pattern -1, +2.

Let's check each option:

1. AZB

This letter-cluster follows the pattern -1, +2.

2. CBD

This letter-cluster follows the pattern -1, +2.

3. VUW

This letter-cluster follows the pattern -1, +2.

4. TTU

This letter-cluster does not follow the pattern -1, +2.

Thus, TTU is the odd one out.

Hence, the correct answer is "Option 4".

BODMAS Rule:

Given equation: 26 × 2 − 8 ÷ 4 + 9 = ?

Now, if ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged, then the equation becomes:

⇒ 26 ÷ 2 × 8 − 4 + 9 = ?

⇒ 13 × 8 − 4 + 9 = ?

⇒ 104 − 4 + 9 = ?

⇒ 113 − 4 = ?

⇒ 109 = ?

Hence, the correct answer is "Option 3".

Statements:

All golds are hives.

No hive is a cow.

The least possible Venn diagram for the given statement is:

Conclusions:

(I): No gold is a cow. - Follow (Because, All golds are hives and No hive is a cow, which means all gold being hives are not cow.)

(II): Some cows are hives. - Does not follow (Because, No hive is a cow, which means no cows are hives.)

Thus, Only conclusion (I) follows.

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, the missing term in the series is 80.

Hence, the correct answer is "Option 1".

The logic followed here is:

OWES - WOES - OESW

LITE - ILTE - LTEI

Now, let's check each option:

1. JUTE - UJTE - EUTJ

This group of triads does not follow the same logic.

2. VAST - VSAT - TSAV

This group of triads does not follow the same logic.

3. EARN - AERN - ERNA

This group of triads follows the same logic.

4. PAIN - APIN - AINP

This group of triads does not follow the same logic.

Thus, EARN - AERN - ERNA from the given options is the one which follows the same logic.

Hence, the correct answer is "Option 3".

Given:

74, 43

98, 55

The logic followed here is:

Logic: (a, b) → b = (a ÷ 2) + 6

74 ÷ 2 + 6 = 37 + 6 = 43

98 ÷ 2 + 6 = 49 + 6 = 55

Let's check each option:

1. (122, 66) → 122 ÷ 2 + 6 = 61 + 6 = 67 ≠ 66
2. (142, 76) → 142 ÷ 2 + 6 = 71 + 6 = 77 ≠ 76
3. (46, 29) → 46 ÷ 2 + 6 = 23 + 6 = 29
4. (64, 39) → 64 ÷ 2 + 6 = 32 + 6 = 38 ≠ 39

Thus, (46, 29) is the number-pair in which the numbers are related in the same way as the given pairs.

Hence, the correct answer is "Option 3".

The position and positional values or English Alphabets are:

Given: HR 15 is related to KU 12 in a certain way. In the same way, SK 18 is related to VN 15. 

The logic followed here is:

Similarly,

Thus, EW 21 is related to HZ18, following the same logic.

Hence, the correct answer is "Option 2".

The logic followed here is:

Thus, the missing term in the series is 12.

Hence, the correct answer is "Option 3".

The position and positional values or English Alphabets are:

Given: IHT KJS MLR ONQ ?

The logic followed here is:

Thus, QPP will replace the question mark (?) to complete the given series.

Hence, the correct answer is "Option 2".

Given: Seven boxes, A, B, C, D, E, F and G, are kept one over the other but not necessarily in the same order.

1. Only E is kept above B.
2. Only two boxes are kept between B and A.
3. Only C is kept below G.
4. F is not kept immediately above A.

G, the only box left will occupy the vacant position.

Thus, 2 boxes are kept below A.

Hence, the correct answer is "Option 2".

Given: (Left) 1 © % £ 8 8 @ % 4 # 1 © 3 7 6 3 % * & € 9 7 (Right)

Now,

Symbols, each of which is immediately preceded by a number and also immediately followed by a symbol.

So, the needed pattern is: Number — Symbol — Symbol

Thus, there are 3 such symbols, each of which is immediately preceded by a number and also immediately followed by a symbol.

Hence, the correct answer is "Option 1".

The logic followed here is:

Thus, the missing term in the series is 562.

Hence, the correct answer is "Option 1".

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

Now,

Even numbers, each of which is immediately preceded by an even number and also immediately followed by an odd number.

So, the needed pattern is: Even — Even — Odd

Thus, there is one even number, which is immediately preceded by an even number and also immediately followed by an odd number.

Hence, the correct answer is "Option 2".

BODMAS Rule:

Decoding the given information:

Given equation: 36 B 8 C 528 A 22 D 55 = ?

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 36 × 8 + 528 ÷ 22 − 55 = ?

⇒ 36 × 8 + 24 − 55 = ?

⇒ 288 + 24 − 55 = ?

⇒ 312 − 55 = ?

⇒ 257 = ?

Hence, the correct answer is "Option 2".

Statements:

All phones are cars.

All phones are lights.

The least possible Venn diagram for the given statement is:

Conclusions:

(I) Some cars are lights. - Follow (Because, All phones are cars and All phones are lights, which means some cars being phones are lights.)

(II) All lights are phones. - Does not follow (Because, All phones are lights, which means some lights are phones.)

Thus, Only conclusion (I) follows.

Hence, the correct answer is "Option 4".

Given: M, N, O, P, W, X and Y are sitting around a circular table facing the centre of the table.

1. Only one person sits between P and O when counted from the right of P.
2. Only one person sits between O and X when counted from the right of O.
3. Only one person sits between P and W when counted from the right of W.
4. Only two people sit between N and X when counted from the right of N.
5. Only one person sits between M and W when counted from the right of M.

Y, the only person left will occupy the vacant position.

Thus, O sits third to the right of Y.

Hence, the correct answer is "Option 4".