RRB Group D Official Paper (Held On: 17 Dec, 2025 Shift 2) (Previous Year Paper)

Shortcut Trick

Total time = 2 minutes 30 seconds = 150 seconds.

Number of rounds = Total time ÷ Time per round = 150 ÷ 50 = 3 rounds.

Distance of 1 round = π × Diameter = 3.14 × 300 = 942 m.

Total distance = 3 × 942 = 2826 m.

∴ The correct answer is 2826 m.

Alternate Method

Given:

Diameter (d) = 300 m

Time for one round = 50 seconds

Total running time = 2 minutes 30 seconds

Value of π = 3.14

Formula Used:

Circumference of circular track = π × d

Total distance = Number of rounds × Circumference

⇒ Total running time = (2 × 60) + 30 = 150 seconds

⇒ Circumference (Distance of 1 round) = 3.14 × 300

⇒ Circumference = 942 m

⇒ Number of rounds = 150 / 50

⇒ Number of rounds = 3

⇒ Total Distance = 3 × 942

⇒ Total Distance = 2826 m

∴ The correct answer is 2826 m.

Additional Information

Displacement on Circular Path

If an athlete completes an integer number of rounds, the net displacement is 0 as the starting and ending points are the same.

Area of Circular Track

The total area enclosed by the circular track is calculated using Area = πr², where r is the radius (diameter/2).

Average Speed

Average Speed = Total Distance ÷ Total Time. In this case, Speed = 2826 ÷ 150 = 18.84 m/s.

The correct answer is The ratio of the sine of the angle of incidence to the sine of the angle of refraction remains constant.

Key Points

• Snell’s Law of Refraction is a fundamental principle in optics that describes how light bends when passing from one medium to another.
• It states that the ratio of the sine of the angle of incidence (represented as sin i) to the sine of the angle of refraction (represented as sin r) is a constant for a given pair of media and for light of a specific color or wavelength. Hence, statement 2 is correct.
• The mathematical expression for this law is sin i / sin r = μ (or n), where μ is the Refractive Index of the second medium relative to the first.
• Snell’s Law is also known as the second law of refraction. It was discovered by the Dutch astronomer and mathematician Willebrord Snellius in the year 1621.
• The angle of incidence is the angle between the incoming ray and the normal (perpendicular) to the interface, while the angle of refraction is the angle between the refracted ray and the normal.
• This law applies specifically to monochromatic light, as different colors of light have different refractive indices in the same medium, a phenomenon known as dispersion.

Additional Information

• Refraction of Light:
• Refraction is the change in direction of a wave passing from one medium to another caused by its change in speed.
• When light travels from a rarer medium (like air) to a denser medium (like glass or water), it slows down and bends towards the normal.
• Conversely, when light travels from a denser medium to a rarer medium, it speeds up and bends away from the normal.
• Refractive Index (n):
• The refractive index is a dimensionless number that indicates the light-bending ability of a medium.
• It is defined as the ratio of the speed of light in a vacuum (approximately 3 x 10^8 m/s) to the speed of light in the medium.
• Absolute Refractive Index is calculated relative to a vacuum, while Relative Refractive Index is calculated between two specific media.
• Behavior of Other Options:
• The statement that "Both angles are always equal" refers to the Law of Reflection, not refraction. Hence, statement 1 is incorrect.
• The statement "Their difference is always constant" does not correspond to any standard physical law of optics. Hence, statement 3 is incorrect.
• The statement "Their sum is always 90°" is only true in the specific case of Brewster's Law, where the reflected and refracted rays are perpendicular to each other. Hence, statement 4 is incorrect.

Important Points

• Change in Properties: During the process of refraction, the velocity and wavelength of light change, but the frequency remains constant. The frequency is a characteristic of the source of light.
• Normal Incidence: If a ray of light strikes the interface of two media at an angle of 0 degrees (along the normal), the angle of refraction is also 0 degrees. In this specific case, the ray passes through without bending, although its speed still changes.
• Total Internal Reflection (TIR): This occurs when light travels from a denser medium to a rarer medium at an angle of incidence greater than the critical angle. In this scenario, Snell’s Law implies that the sine of the angle of refraction would be greater than 1, which is mathematically impossible, resulting in the light reflecting back entirely into the denser medium.
• Applications: Snell’s Law is essential in the design of optical lenses, spectacles, microscopes, telescopes, and fiber optic cables used in telecommunications.

Shortcut Trick

Total distance covered = 6 km + 6 km = 12 km = 12,000 m

Total time taken = 20 min + 30 min = 50 min = 3,000 s

Average speed = Total Distance ÷ Total Time

Average speed = 12,000 ÷ 3,000 = 4 m/s

∴ The correct answer is 4.

Alternate Method

Given:

Distance from A to B = 6 km

Time from A to B = 20 minutes

Time from B to A = 30 minutes

Formula Used:

Average Speed = Total Distance ÷ Total Time

1 km = 1,000 m | 1 minute = 60 seconds

⇒ Total Distance = 6 km + 6 km = 12 km

⇒ Total Distance in meters = 12 × 1,000 = 12,000 m

⇒ Total Time = 20 min + 30 min = 50 min

⇒ Total Time in seconds = 50 × 60 = 3,000 s

⇒ Average Speed = 12,000 ÷ 3,000

⇒ Average Speed = 4 m/s

∴ The correct answer is 4.

Additional Information

Average Speed for Equal Distances

When an object travels equal distances at speeds x and y, the average speed is given by the harmonic mean: 2xy ÷ (x + y).

Unit Conversion Shortcut

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

Uniform Speed vs Average Speed

Uniform speed is constant throughout the journey, whereas average speed is the ratio of total path length to the total time interval taken.

The correct answer is Pressure increases when area decreases.

Key Points

• Pressure is defined as the thrust (force acting perpendicular to a surface) acting per unit area of that surface.
• The mathematical relationship between these physical quantities is expressed by the formula: Pressure = Thrust / Area.
• When the thrust remains constant, pressure is inversely proportional to the area of contact. This means that if the area increases, the pressure decreases, and if the area decreases, the pressure increases.
• Evaluating the given options based on the inverse relationship (P ∝ 1/A):
• Option 1: Pressure decreases when area decreases. This statement is incorrect because pressure should increase.
• Option 2: Pressure increases when area increases. This statement is incorrect because pressure should decrease.
• Option 3: Pressure increases when area decreases. This statement is correct as it aligns with the inverse proportionality.
• Option 4: Pressure does not depend on area. This statement is incorrect because area is a primary factor in calculating pressure.
• Hence, the correct answer is Pressure increases when area decreases.

Additional Information

• Thrust:
• It is the total force acting perpendicularly on a surface.
• The S.I. unit of thrust is the Newton (N), the same as force.
• Thrust is a vector quantity, meaning it has both magnitude and direction.
• Pressure Units:
• The S.I. unit of pressure is Newton per square metre (N/m²).
• In honour of the scientist Blaise Pascal, the unit is also called the Pascal (Pa). 1 Pa = 1 N/m².
• Other common units include the bar, torr, and atmosphere (atm). 1 atm is approximately 101,325 Pa.
• Practical Applications of Pressure-Area Relationship:
• Sharp Knives: A sharp knife has a very small area at its cutting edge. For the same thrust applied by the hand, it produces a high pressure that easily cuts through objects.
• School Bags: These are provided with broad straps so that the weight (thrust) of the bag is distributed over a larger area of the shoulder, resulting in less pressure and more comfort for the student.
• Foundations of Buildings: High-rise buildings have wide foundations to ensure the massive weight is spread over a large surface area, preventing the structure from sinking into the ground due to excessive pressure.
• Tractors: Tractors have broad tires to reduce the pressure on the soft ground of agricultural fields, preventing them from sinking.

Important Points

• While force and thrust are vector quantities, pressure is a scalar quantity because it does not have a specific direction associated with it at a point in a fluid or on a surface; it acts in all directions.
• In fluids (liquids and gases), pressure increases with depth because the weight of the fluid column above (which acts as the thrust) increases as you go deeper.
• Atmospheric Pressure: The weight of the Earth's atmosphere exerts pressure on everything on the surface. At sea level, the standard atmospheric pressure is 1.013 x 10⁵ Pa.

The correct answer is Each striated muscle fibre contains a single nucleus.

Key Points

• Striated muscles are primarily those muscles that are under the conscious control of the somatic nervous system, which is why they are commonly referred to as voluntary muscles. Hence, statement 1 is correct.
• A defining structural feature of striated muscle fibers is that they are multinucleated. During development, several myoblasts fuse together to form a single muscle fiber, resulting in a syncytium where multiple nuclei are present in a single cell, typically located at the periphery. Hence, statement 2 is incorrect.
• These muscles are found extensively throughout the body, particularly in the limbs (arms and legs), body wall, face, and neck, where they facilitate complex movements. Hence, statement 3 is correct.
• Striated muscles are mostly attached to bones via tendons. Their primary function is to contract and relax to produce force and motion, helping in body movement and maintaining posture. Hence, statement 4 is correct.
• Microscopically, these muscles appear to have alternating light and dark bands (striations) when viewed under a microscope, which is caused by the specific arrangement of actin and myosin filaments.

Additional Information

• Types of Muscles:
• Smooth Muscles: These are involuntary muscles found in the walls of internal organs like the stomach, intestines, and blood vessels. They are spindle-shaped and, unlike striated muscles, contain only a single nucleus (uninucleated) and lack striations.
• Cardiac Muscles: Located exclusively in the heart, these muscles are striated but involuntary. They are characterized by branched fibers and intercalated discs that allow for synchronized contractions. They usually contain one or two nuclei.
• Structural Components:
• The functional unit of a striated muscle is the sarcomere. It is the segment between two Z-lines and contains the machinery required for muscle contraction.
• The Sarcolemma is the specialized cell membrane that surrounds the striated muscle fiber, while the Sarcoplasm is the cytoplasm containing high amounts of glycosomes and myoglobin.
• Functions of Skeletal Muscle:
• Beyond locomotion, these muscles play a critical role in heat production (thermogenesis) to maintain body temperature.
• They also serve as a storage site for amino acids and glucose (in the form of glycogen), which can be utilized by the body during metabolic stress.

Important Points

• Muscle Contraction Mechanism: The process is explained by the Sliding Filament Theory, proposed in 1954. It states that muscle contraction occurs when thin filaments (actin) slide over thick filaments (myosin), shortening the sarcomere without the filaments themselves changing length.
• Fast-twitch vs. Slow-twitch: Striated muscles can be categorized into Type I (slow-twitch), which are dense with capillaries and mitochondria for endurance, and Type II (fast-twitch), which are used for short bursts of strength or speed.
• Energy Source: These muscles utilize Adenosine Triphosphate (ATP) as the primary energy source for contraction, which is replenished through aerobic respiration or anaerobic glycolysis depending on the intensity of the physical activity.

The correct answer is By using protection such as condoms during intercourse.

Key Points

• Sexually Transmitted Infections (STIs) are infections primarily passed from one person to another through vaginal, anal, or oral sexual contact, caused by more than 30 different pathogens including bacteria, viruses, and parasites.
• Condoms are highly effective barrier methods that prevent the transmission of infections by blocking the direct exchange of bodily fluids like semen, pre-ejaculate, and vaginal secretions.
• According to scientific data, consistent and correct use of latex condoms reduces the risk of Human Immunodeficiency Virus (HIV) transmission by approximately 80% to 90%.
• Condoms also significantly lower the transmission rates of other fluid-borne bacterial infections such as Gonorrhea (caused by Neisseria gonorrhoeae) and Chlamydia (caused by Chlamydia trachomatis) by preventing the pathogens from reaching the mucous membranes.
• While they are most effective against fluid-borne diseases, they also provide a degree of protection against infections spread through skin-to-skin contact, such as Syphilis and Genital Herpes, provided the infected area is covered by the barrier.

 Additional Information

• Pathogen Categories:
• Bacterial STIs: Infections like Syphilis, Gonorrhea, and Chlamydia can be cured with a full course of antibiotics prescribed by medical professionals.
• Viral STIs: Infections like HIV, Human Papillomavirus (HPV), and Hepatitis B are generally incurable but manageable with antiviral therapy and vaccines.
• Preventive Misconceptions:
• Antibiotic Prophylaxis: Taking antibiotics before sexual activity without a medical diagnosis is ineffective against viruses and contributes to the global threat of antimicrobial resistance.
• Environmental Factors: The time of day or the setting of sexual activity has zero biological impact on the viability or transmission of STI-causing pathogens.

The correct answer is four times.

Key Points

• Kinetic Energy (KE) is the energy possessed by an object due to its motion. It is mathematically expressed by the formula: KE = 1/2 mv^2, where m represents the mass of the body and v represents its velocity.
• From the formula, it is evident that the kinetic energy of a body is directly proportional to the square of its velocity (KE ∝ v^2), assuming the mass remains constant.
• If the velocity of a body is doubled (v becomes 2v), the new kinetic energy (KE') can be calculated as: KE' = 1/2 m(2v)^2 = 1/2 m(4v^2) = 4 (1/2 mv^2).
• Therefore, when the velocity is increased by a factor of 2, the kinetic energy increases by a factor of 2 squared, which is 4. Hence, the kinetic energy becomes four times its original value.
• This relationship highlights that kinetic energy is highly sensitive to changes in speed compared to changes in mass. For instance, if the velocity were tripled, the kinetic energy would become nine times the original.

Additional Information

• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy. If positive work is done on a body, its velocity increases, leading to a rise in kinetic energy.
• Relationship with Momentum: Kinetic energy is also related to linear momentum (p) by the formula KE = p^2 / 2m. This shows that if the momentum of a body is doubled, its kinetic energy also becomes four times, provided the mass is constant.
• Units and Nature: The SI unit of kinetic energy is the Joule (J), which is equivalent to 1 kg m^2/s^2. Kinetic energy is a scalar quantity, meaning it has only magnitude and no direction, and it can never be negative because mass is positive and the square of velocity is always non-negative.
• Practical Applications: This principle is critical in road safety. Since KE ∝ v^2, doubling the speed of a car quadruples the energy involved in a collision and significantly increases the braking distance required to stop the vehicle.

Important Points

• Mass vs. Velocity: While kinetic energy is linearly proportional to mass, it is quadratically proportional to velocity. Doubling the mass only doubles the kinetic energy, but doubling the velocity quadruples it.
• Frame of Reference: The value of kinetic energy depends on the observer's frame of reference. A person sitting in a moving train has zero kinetic energy relative to the train but significant kinetic energy relative to an observer standing on the platform.
• Types of Kinetic Energy: Besides translational kinetic energy (1/2 mv^2), objects can also possess rotational kinetic energy (1/2 Iω^2) and vibrational kinetic energy, depending on their type of motion.

The correct answer is Sodium.

Key Points

• Sodium is a soft, silvery-white, highly reactive alkali metal belonging to Group 1 of the periodic table with atomic number 11.
• It reacts most vigorously with cold water among the given options, producing sodium hydroxide and hydrogen gas. This reaction is highly exothermic, often generating enough heat to ignite the hydrogen gas produced. Hence, the correct answer is Sodium.
• The chemical equation for the reaction is 2Na + 2H2O → 2NaOH + H2 + heat energy.
• Due to its extreme reactivity with oxygen and moisture in the air, sodium must be stored under kerosene or mineral oil to prevent accidental combustion.
• In the Reactivity Series, Sodium is placed near the top, just below Potassium, indicating its high tendency to lose its valence electron and form positive ions.

Additional Information

• Copper (Option 1):
• Copper is a transition metal located in Group 11 of the periodic table.
• It is highly unreactive with water. It does not react with cold water, hot water, or even steam.
• Because of its low reactivity and high thermal conductivity, it is used for making hot water tanks and electrical wires. Hence, statement 1 is incorrect.
• Iron (Option 3):
• Iron is a transition metal with atomic number 26.
• It does not react with cold water or hot water. It only reacts with steam to form iron(III) oxide and hydrogen gas.
• The reaction with steam is 3Fe + 4H2O → Fe3O4 + 4H2. Hence, statement 3 is incorrect.
• Cobalt (Option 4):
• Cobalt is a transition metal (atomic number 27) found in Group 9.
• Like iron, it is relatively stable and does not react vigorously with cold water under standard conditions.
• It is primarily used in batteries, magnets, and high-strength alloys. Hence, statement 4 is incorrect.

Important Points

• Potassium: It is even more reactive than sodium and reacts violently with cold water, catching fire instantly.
• Calcium: It reacts with cold water but the reaction is less violent. The hydrogen bubbles produced stick to the surface of the metal, causing the calcium to float.
• Magnesium: It does not react with cold water at all. It reacts with hot water to form magnesium hydroxide and hydrogen.
• Noble Metals: Metals like Gold, Silver, and Platinum do not react with water in any form, which is why they are used in jewelry.

The correct answer is Their formation takes millions of years.

Key Points

• Coal and petroleum are categorized as fossil fuels, which are derived from the organic remains of prehistoric plants and organisms buried under layers of sediment.
• Fossil fuels are non-renewable resources of energy because they are available in limited quantities in nature and cannot be replenished at the same rate they are consumed. Hence, statement 1 is incorrect.
• The process of carbonization (for coal) and the transformation of marine organisms into petroleum occur under conditions of high temperature and high pressure over a period of millions of years. Hence, statement 2 is correct.
• The extraction of these resources is not inexpensive; it requires massive capital investment, deep-sea drilling, advanced machinery, and refining processes to make them usable. Hence, statement 3 is incorrect.
• The combustion of coal and petroleum products releases significant amounts of carbon dioxide (CO2), sulfur dioxide, and nitrogen oxides, which are major contributors to air pollution and global warming. Hence, statement 4 is incorrect.
• Hence, the correct answer is Their formation takes millions of years.

Additional Information

• Types of Coal:
• Coal is classified into different types based on its carbon content and age. Anthracite is the highest quality coal with over 80% carbon, followed by Bituminous, Lignite (brown coal), and Peat (the first stage of coal formation).
• In India, coal is the most abundantly available fossil fuel and meets a substantial part of the country's energy needs.
• Petroleum and its Products:
• Petroleum is often called "Black Gold" because of its high commercial value. Through fractional distillation in refineries, crude oil is separated into petrol, diesel, kerosene, wax, and lubricants.
• It is primarily found in sedimentary rocks and is often associated with deposits of natural gas.
• Conservation Strategies:
• Conservation is essential because these resources are exhaustible. Strategies include the 3Rs (Reduce, Reuse, Recycle) and shifting toward renewable energy sources like solar energy, wind energy, and tidal power.
• The Petroleum Conservation Research Association (PCRA) in India provides guidelines for saving fuel, such as maintaining constant speed and ensuring regular vehicle maintenance.

Important Points

• Environmental Impact: The excessive use of fossil fuels leads to the greenhouse effect, which results in a rise in the Earth's average temperature. Furthermore, the release of sulfur and nitrogen oxides leads to acid rain, which damages soil, vegetation, and aquatic life.
• Geological Significance: The Carboniferous Period, roughly 300 to 360 million years ago, was a major era for the formation of the world's coal beds, highlighting the vast geological timescale required to produce these energy sources.
• Energy Security: Many countries are now focusing on Sustainable Development to ensure that the energy needs of the present are met without compromising the ability of future generations to meet their own needs.

The correct answer is 60 W.

Key Points

• Electric Power is defined as the rate at which electrical energy is consumed or dissipated in an electric circuit. It represents the work done per unit time.
• In this problem, the Voltage (V) provided by the generator is 240 V and the Current (I) drawn by the electric bulb is 0.25 A.
• The standard mathematical formula to calculate electric power is P = V × I, where P is the power, V is the potential difference (voltage), and I is the current.
• Substituting the given numerical values into the formula: Power = 240 × 0.25.
• The calculation yields 60. Since the SI unit of power is the Watt (W), the power consumed by the bulb is 60 W. Hence, the correct answer is 60 W.

Additional Information

• Ohm's Law and Resistance:
• According to Ohm's Law, the relationship between voltage, current, and resistance is given by V = IR.
• If we calculate the resistance of this specific bulb, R = V / I, which is 240 / 0.25 = 960 Ohms. Note that 960 is the Resistance, not the power.
• Alternative Power Formulas:
• Power can also be expressed in terms of resistance: P = I²R or P = V² / R.
• Using P = V² / R for this bulb: (240 × 240) / 960 = 57600 / 960 = 60 W. These formulas are useful when the current or voltage is unknown but resistance is provided.
• Units of Measurement:
• The Watt is named after the Scottish inventor James Watt. 1 Watt is equal to 1 Joule per second (1 J/s).
• Larger units of power include the Kilowatt (kW), where 1 kW = 1,000 Watts, and the Horsepower (hp), where 1 hp is approximately 746 Watts.

Important Points

• Distinction between Power and Energy:
• Power (measured in Watts) is the rate of energy consumption, whereas Energy (measured in Joules) is the total amount of work done over a period.
• The options 60 J and 960 J are incorrect because the Joule is a unit of energy, not power.
• Commercial Unit of Electrical Energy:
• The commercial unit used for electricity bills is the Kilowatt-hour (kWh), which is often referred to as a "unit."
• 1 kWh is the energy consumed by a 1,000 W appliance running for one hour, which equals 3.6 × 10⁶ Joules.
• Practical Application:
• Most domestic appliances in India operate at a standard voltage of 220 V to 240 V. The current rating helps in determining the appropriate fuse or circuit breaker required for safety features.

The correct answer is AgNO3 + NaCl → AgCl + NaNO3.

Key Points

• A double displacement reaction is a type of chemical reaction where two compounds react, and the positive ions (cations) and the negative ions (anions) of the two reactants switch places, forming two new compounds.
• In the reaction AgNO3 + NaCl → AgCl + NaNO3, the silver ion (Ag+) from silver nitrate combines with the chloride ion (Cl-) from sodium chloride to form silver chloride (AgCl). Simultaneously, the sodium ion (Na+) combines with the nitrate ion (NO3-) to form sodium nitrate (NaNO3). Hence, statement 4 is correct.
• The reaction CaCO3 → CaO + CO2 involves a single reactant breaking down into two simpler products upon heating. This is classified as a thermal decomposition reaction. Hence, statement 1 is incorrect.
• The reaction 2Mg + O2 → 2MgO involves two reactants (magnesium and oxygen) combining to form a single product (magnesium oxide). This is known as a combination reaction or synthesis reaction. Hence, statement 2 is incorrect.
• The reaction Zn + H2SO4 → ZnSO4 + H2 involves Zinc displacing Hydrogen from sulfuric acid because zinc is more reactive. This is a single displacement reaction. Hence, statement 3 is incorrect.
• Hence, the correct answer is AgNO3 + NaCl → AgCl + NaNO3.

Additional Information

• Precipitation Reactions: Many double displacement reactions result in the formation of an insoluble solid that settles out of the solution, known as a precipitate. In the given correct option, AgCl is a white precipitate.
• Neutralization Reactions: This is another common type of double displacement reaction that occurs between an acid and a base to produce salt and water. For example, HCl + NaOH → NaCl + H2O.
• Reactivity Series: While single displacement depends heavily on the reactivity series (where a more active metal displaces a less active one), double displacement usually depends on the solubility rules or the formation of a stable product like water or a gas.
• Gas Evolution: Some double displacement reactions produce a gas. For instance, the reaction between sodium carbonate and hydrochloric acid eventually releases carbon dioxide (CO2) gas after an initial ion exchange.

Important Points

• Ionic State: These reactions typically occur in aqueous solutions where the compounds are dissociated into free ions, allowing them to move and collide to form new bonds.
• Balanced Equations: In all chemical reactions, including AgNO3 + NaCl → AgCl + NaNO3, the law of conservation of mass must be satisfied, meaning the number of atoms of each element must be the same on both the reactant and product sides.
• Oxidation State: Unlike redox reactions (like single displacement or combination), the oxidation states of the elements in a typical double displacement reaction usually remain unchanged.
• Example Compounds: AgNO3 (Silver Nitrate) is often used in photography and medical antiseptics, while NaCl is common table salt used in food preservation and industrial chemistry.

The correct answer is Joule.

Key Points

• The International System of Units (SI) is founded on seven base units, which are the fundamental building blocks for all other physical measurements.
• The Metre (m) is the SI base unit for length. It is defined by taking the fixed numerical value of the speed of light in vacuum. Hence, statement 2 is a base unit.
• The Kilogramme (kg) is the SI base unit for mass. Following the 2019 redefinition, it is defined in terms of the Planck constant (h). Hence, statement 3 is a base unit.
• The Second (s) is the SI base unit for time. It is determined by the fixed numerical value of the caesium frequency, which is the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom. Hence, statement 4 is a base unit.
• The Joule (J) is the unit of energy, work, or heat. It is classified as a derived unit rather than a base unit because it is expressed in terms of the base units as 1 kg m2/s2. Hence, statement 1 is NOT a base unit.

Additional Information

• The Seven SI Base Units:
• Metre (m): Measures length.
• Kilogramme (kg): Measures mass.
• Second (s): Measures time.
• Ampere (A): Measures electric current.
• Kelvin (K): Measures thermodynamic temperature.
• Mole (mol): Measures the amount of substance.
• Candela (cd): Measures luminous intensity.
• Derived Units:
• These units are formed by combining base units according to the algebraic relations linking the corresponding quantities.
• Examples include the Newton (force), Pascal (pressure), Watt (power), and Hertz (frequency).
• The Joule is named after the English physicist James Prescott Joule (1818–1889), who studied the nature of heat and its relationship to mechanical work.

Important Points

• The SI system was officially established in 1960 by the 11th General Conference on Weights and Measures (CGPM).
• In May 2019, a historic revision of the SI came into force, where all base units are now defined by constants of nature rather than physical artifacts like the "International Prototype of the Kilogram."
• The International Bureau of Weights and Measures (BIPM), located in France, ensures worldwide unification of measurements and their traceability to the International System of Units.
• Understanding the distinction between base units and derived units is essential in physics and engineering for maintaining dimensional consistency in equations.

The correct answer is Vitamin A, D and E.

Key Points

• Poultry nutrition requires a precise balance of nutrients to support rapid growth and high productivity in intensive farming systems.
• Vitamin A is crucial for maintaining the health of epithelial tissues, improving vision, and enhancing the overall growth rate of chickens.
• Vitamin D, particularly in the form of Vitamin D₃, is essential for the metabolism of calcium and phosphorus, which ensures strong bone development and eggshell quality.
• Vitamin E acts as a potent biological antioxidant, protecting cell membranes from oxidative damage and supporting the immune system and reproductive health.
• Maintaining high levels of these fat-soluble vitamins in the feed helps prevent specific deficiency diseases such as xerophthalmia, rickets, and encephalomalacia.

 Additional Information

• Vitamin A:
• Primary sources include yellow maize and green fodder rich in carotene.
• Deficiency leads to nutritional roup and poor resistance to coccidiosis.
• Vitamin D:
• Poultry utilize Cholecalciferol (D₃) much more efficiently than Ergocalciferol (D₂).
• It is critical for the prevention of osteomalacia in laying hens.
• Vitamin E:
• Its requirement is closely linked to the Selenium content in the diet.
• It prevents exudative diathesis and muscular dystrophy in growing chicks.
• B-complex Vitamins:
• While Vitamin B₂ (Riboflavin) prevents curled-toe paralysis and Vitamin B₁₂ is needed for growth, the core health-supportive trio typically emphasized at high levels for broad-spectrum growth is A, D, and E.

The correct answer is They are active cells with dense cytoplasm, thin walls, prominent nuclei and no vacuoles.

Key Points

• Meristematic tissues are a group of immature cells that are in a continuous state of division or retain the power of cell division.
• The cells of meristematic tissue are very active, have dense cytoplasm, thin cellulose walls, and prominent nuclei. Hence, statement 1 is correct.
• These cells lack vacuoles. Vacuoles are typically used for storage of food or waste and providing turgidity to the cell. Since meristematic cells are rapidly dividing, they do not need to store food and a large vacuole might obstruct the process of cell division.
• The cells are compactly arranged, meaning there are no intercellular spaces between them.
• They are generally small in size and can be spherical, oval, polygonal, or rectangular in shape.

Additional Information

• Classification of Meristematic Tissue based on Location:
• Apical Meristem: Located at the growing tips of stems and roots. It is responsible for the increase in the length of the plant, known as primary growth.
• Lateral Meristem (Cambium): Located on the lateral sides of stems and roots. It is responsible for increasing the girth or thickness of the plant, known as secondary growth.
• Intercalary Meristem: Located at the base of leaves or internodes (on twigs). It helps in the longitudinal growth of the plant parts.
• Permanent Tissues:
• When meristematic cells take up a specific role and lose the ability to divide, they form permanent tissue. This process is called differentiation.
• Parenchyma: These are live cells, usually loosely packed with large intercellular spaces. They provide support and store food. Hence, statement 3 is incorrect.
• Sclerenchyma: These are dead cells with thickened walls (due to lignin) and no internal space. They make the plant hard and stiff. Hence, statement 2 and 4 are incorrect.

Important Points

• The term Meristem was coined by C. Nageli in 1858.
• Metabolic Activity: Meristematic cells have a very high metabolic rate because they are constantly preparing for and undergoing mitosis.
• Ergastic Substances: These cells generally do not contain ergastic substances (non-living inclusions like starch grains, oil droplets) because they consume nutrients as fast as they are produced.
• Cell Wall Composition: The cell wall is primary in nature and made exclusively of cellulose to remain flexible for expansion and division.

The correct answer is Cellulose.

Key Points

• Cellulose is the primary structural component of the plant cell wall, providing the necessary mechanical strength, rigidity, and protection to the plant cell.
• Chemically, it is a polysaccharide (a complex carbohydrate) consisting of a long, linear chain of hundreds to thousands of glucose units joined by beta(1→4) glycosidic bonds.
• It is considered the most abundant organic polymer on Earth, making up a significant portion of the biomass in terrestrial ecosystems.
• In the cell wall, cellulose molecules are bundled together to form microfibrils, which create a high-tensile scaffold that helps the cell withstand turgor pressure and maintain its shape.
• Unlike starch, humans cannot digest cellulose because they lack the enzyme cellulase; however, it serves as essential dietary fiber in the human diet.

Additional Information

• Lignin:
• It is a complex organic polymer found primarily in the secondary cell walls of vascular plants.
• Lignin provides structural support, making the plant woody and rigid, and plays a crucial role in water transport by making cell walls hydrophobic. Hence, option 1 is incorrect.
• Suberin:
• This is a waxy, fatty substance found in the cell walls of cork tissues and the endodermis of roots.
• It acts as a protective barrier against water loss and pathogen entry, effectively waterproofing specific plant tissues. Hence, option 2 is incorrect.
• Cutin:
• Cutin is one of the main components of the plant cuticle, which covers the epidermis of leaves, fruits, and soft stems.
• It is a waxy polymer that prevents excessive transpiration (water loss) and protects the plant from ultraviolet radiation and external stress. Hence, option 3 is incorrect.
• Hemicellulose and Pectin:
• While cellulose provides the framework, hemicellulose acts as a cross-linking agent between cellulose microfibrils.
• Pectin is a gel-like polysaccharide that fills the spaces between the fibers, providing flexibility and helping in cell-to-cell adhesion through the middle lamella.

Important Points

• Composition: The plant cell wall is typically composed of approximately 35% to 50% cellulose, 20% to 35% hemicellulose, and 10% to 25% lignin.
• Turgor Pressure: The rigidity of the cellulose-based cell wall is vital for maintaining turgor pressure, which allows plants to stand upright without a skeletal system.
• Discovery: Cellulose was first discovered and isolated by the French chemist Anselme Payen in the year 1838.
• Industrial Uses: Beyond its biological role, cellulose is used extensively in the production of paper, textiles (like cotton and rayon), and biofuels.

The correct answer is rancid.

Key Points

• When fats and oils are exposed to air, they undergo a chemical reaction known as oxidation. This process results in the development of a foul smell and an unpleasant taste, a condition commonly referred to as rancidity.
• The primary cause of this phenomenon is the oxidation of unsaturated fatty acids present in the food. During this reaction, oxygen molecules attack the double bonds of the fatty acid chains.
• This chemical breakdown leads to the formation of highly volatile substances, including aldehydes, ketones, and short-chain carboxylic acids, which provide the characteristic "off" flavor and pungent odor.
• Rancidification is a form of food spoilage that not only makes the food unpalatable but can also lead to the loss of essential vitamins and the formation of potentially harmful free radicals.
• The rate of oxidation is significantly influenced by environmental factors such as temperature, light, and the presence of moisture or metal catalysts. Hence, the correct answer is rancid.

Additional Information

• Types of Rancidity:
• Oxidative Rancidity: This is the most common type, occurring when oxygen reacts with fats. It is often accelerated by exposure to ultraviolet (UV) light and heat.
• Hydrolytic Rancidity: This occurs when water breaks down the triglyceride bonds into free fatty acids and glycerol. This process is often catalyzed by enzymes known as lipases, which are produced by microorganisms.
• Ketonic Rancidity: This involves the action of molds (like Penicillium) on short-chain fatty acids, resulting in the production of ketones.
• Methods to Prevent Rancidity:
• Use of Antioxidants: Synthetic chemicals like BHA (Butylated Hydroxyanisole) and BHT (Butylated Hydroxytoluene), or natural ones like Vitamin E (Tocopherol) and Vitamin C, are added to foods to inhibit the oxidation process.
• Nitrogen Flushing: Commercial food manufacturers often pack snacks like potato chips in bags filled with Nitrogen gas. Since Nitrogen is an inert gas, it displaces Oxygen, preventing the oil in the chips from turning rancid.
• Vacuum Packaging: Removing air completely from the packaging ensures that there is no Oxygen available to react with the fats.
• Refrigeration: Storing food at low temperatures slows down the kinetic energy of the molecules, thereby reducing the rate of the chemical oxidation reaction.
• Storage in Dark Containers: Using opaque or amber-colored containers helps block light, which acts as a catalyst for photo-oxidation.
• Health and Industrial Impact:
• Consuming rancid fats can lead to oxidative stress in the human body, potentially contributing to digestive issues and long-term health complications.
• In the industrial sector, the Peroxide Value (PV) and Acid Value are standard tests used to measure the degree of rancidity in oils and fats to ensure quality control.

The correct answer is Resistor.

Key Points

• A resistor is a passive electrical component that provides resistance to the flow of electric current. In circuit diagrams, it is most commonly represented by a zig-zag line (standard in the United States and Japan) or a rectangular box (standard in International Electrotechnical Commission - IEC).
• The primary function of a resistor is to limit current, divide voltage, or generate heat. By controlling the flow of electrons, it protects sensitive components like transistors and Light Emitting Diodes (LEDs) from damage caused by excessive current.
• The value of resistance is measured in Ohms, denoted by the symbol Ω (Omega). This relationship is defined by Ohm’s Law, which states that Voltage (V) = Current (I) × Resistance (R).
• Resistors are essential in almost all electronic circuits, ranging from simple household appliances to complex industrial machinery and communication systems. Hence, the correct answer is Resistor.

Additional Information

• Battery:
• In a circuit diagram, a battery is represented by two or more parallel lines of unequal length.
• The longer line represents the positive terminal, while the shorter, thicker line represents the negative terminal.
• A battery provides the Potential Difference required to drive the flow of electric charge in a closed loop.
• Electric Bulb:
• An electric bulb is typically represented by a circle with a cross (X) inside it or a circle with a loop-like filament symbol.
• It converts electrical energy into light energy and thermal energy.
• In an ideal circuit, the brightness of the bulb depends on the power it consumes, which is a product of voltage and current.
• Switch:
• A switch is a device used to make or break an electrical circuit.
• In a diagram, an open switch is shown as a line with a gap, indicating that current cannot flow. A closed switch is shown as a continuous line, allowing current to pass.
• It is a fundamental control element in any electrical system.

Important Points

• Variable Resistor (Rheostat): This is a resistor whose value can be changed. It is represented by a zig-zag line with an arrow passing through it diagonally. It is widely used in volume controls for audio devices.
• Ammeter and Voltmeter:
• An Ammeter measures electric current in Amperes (A) and is always connected in series with the circuit.
• A Voltmeter measures electric potential difference in Volts (V) and is always connected in parallel to the component across which voltage is measured.
• Ohm’s Law: Discovered by Georg Simon Ohm in 1827, it remains the foundation of electrical engineering. It establishes that at a constant temperature, the current through a conductor is directly proportional to the voltage across it.
• Materials: Resistors are often made from carbon composition, metal films, or wire-wound coils, depending on the required precision and power rating.

The correct answer is An atom or group of atoms with a positive or negative charge..

Key Points

• An ion is a chemical species that carries a net electrical charge because the number of electrons is unequal to the number of protons in the species.
• Ions are formed through a process called ionization, where an atom or a molecule either gains or loses electrons to achieve a stable electronic state, often resembling the nearest noble gas.
• These species can be classified as monatomic ions, which consist of a single charged atom (like K⁺), or polyatomic ions, which consist of two or more atoms covalently bonded that act as a single charged unit (like SO₄^2-).
• The presence of a charge allows ions to conduct electricity in aqueous solutions or molten states, and they are the primary components of ionic compounds held together by electrostatic forces.

 Additional Information

• Classification of Ions:
• Cations: Positively charged ions formed when an atom loses electrons. Metals typically form cations (e.g., Magnesium Mg²⁺).
• Anions: Negatively charged ions formed when an atom gains electrons. Non-metals typically form anions (e.g., Fluoride F^-).
• Explanation of Other Terms:
• Neutral Atom: An atom where the number of protons equals the number of electrons, resulting in a net charge of zero.
• Molecule: An electrically neutral group of two or more atoms held together by chemical bonds.
• Proton: A subatomic particle located in the nucleus of an atom with a positive charge of +1; it defines the atomic number of an element.

The correct answer is Physical maturity.

Key Points

• Physical maturity is the state of full development of an organism's body, including its reproductive system, which is essential for the process of mating and reproduction.
• Without reaching a certain age or biological stage, animals do not possess the functional gametes (sperm or eggs) or the developed organs necessary to successfully participate in mating. Hence, statement 1 is correct.
• While attractive eyes or physical traits like being tall might be factors in sexual selection for specific species (like peacocks or giraffes), they are not universal biological requirements for the act of mating across the animal kingdom. Hence, statements 2 and 4 are incorrect.
• Sexual immaturity refers to the stage where an animal has not yet reached puberty. At this stage, the endocrine system has not triggered the release of sex hormones required for mating behaviors and fertility. Hence, statement 3 is incorrect.
• The process of reaching maturity involves significant physiological changes, governed by hormones such as testosterone in males and estrogen or progesterone in females.

Additional Information

• Puberty and Maturity:
• In most mammals, the onset of puberty marks the transition from a juvenile state to sexual maturity.
• The age at which maturity is reached varies greatly by species; for example, mice may reach maturity in 6 to 8 weeks, while elephants may take 10 to 15 years.
• Secondary Sexual Characteristics:
• Upon reaching physical maturity, many animals develop secondary sexual characteristics, such as manes in lions, antlers in deer, or colorful plumage in birds.
• These features serve as signals to other members of the species that the individual is physically mature and ready to mate.
• Role of the Endocrine System:
• The Hypothalamus-Pituitary-Gonadal (HPG) axis is the primary regulatory system that controls the timing of physical maturity.
• Environmental factors such as nutrition, climate, and daylight length (photoperiod) can influence the year or month an animal reaches maturity.
• Reproductive Success:
• Mating is not just about the physical act but also about the ability to produce viable offspring. Physical maturity ensures that the parent's body is capable of gametogenesis and, in females, supporting pregnancy or egg-laying.

Important Points

• Mating Rituals: Once physical maturity is attained, animals often engage in courtship rituals. These are behaviors like singing, dancing, or fighting to demonstrate fitness to a potential mate.
• Biological Fitness: In the context of evolution, reaching maturity is a key component of fitness, as it allows an individual to pass its genes to the next generation.
• Fertility Windows: Many mature animals are not fertile year-round but follow estrus cycles or breeding seasons, which are only activated once physical maturity has been established.

The correct answer is Both sugar and salt.

Key Points

• Matter is generally classified into three states: solids, liquids, and gases, based on the arrangement of their particles.
• Iron is a metallic solid where the atoms are arranged in a very tightly packed crystalline lattice, held together by strong metallic bonds. Because it is a continuous solid mass, there is negligible macroscopic space between its constituent parts. Hence, iron is not the correct choice for having sufficient spaces.
• Sugar and salt are considered granular solids. Although the individual crystals of sugar and salt are solids with tightly packed molecules, their bulk form consists of many tiny particles.
• When sugar or salt is placed in a container, they do not pack perfectly. There are sufficient spaces (air gaps) between the individual granules, allowing them to flow and take the shape of the container. Hence, sugar and salt are correct as they are not tightly packed in their common aggregate form.
• The inter-particle space in granular materials like salt and sugar is much higher compared to a solid block of iron, which has a very high density and low porosity.
• Hence, the correct answer is Both sugar and salt.

Additional Information

• Iron (Fe):
• It is a metal with the atomic number 26.
• In its solid state, iron has a density of approximately 7.87 g/cm³ at room temperature.
• It is used extensively in construction and manufacturing due to its high strength and rigidity, which results from its closely packed atomic structure.
• Sugar (Sucrose):
• Common table sugar has the chemical formula C12H22O11.
• It is a carbohydrate that forms monoclinic crystals.
• In a jar, the bulk density of sugar is much lower than its actual particle density (1.59 g/cm³) because of the air trapped in the interstitial spaces.
• Salt (Sodium Chloride):
• Commonly known as NaCl, it is an ionic compound.
• It forms cubic crystals where sodium and chloride ions are arranged in a lattice.
• Like sugar, it is handled as a granular material, and the void fraction between grains is significant.

Important Points

• Porosity: This is a measure of the "empty" spaces in a material. Granular substances like salt and sugar have high porosity, whereas a piece of iron has nearly zero macroscopic porosity.
• States of Matter Comparison:
• Solids: Particles are closely packed; have a fixed shape and volume.
• Liquids: Particles are less tightly packed than solids; have a fixed volume but no fixed shape.
• Gases: Particles have large spaces between them; have no fixed shape or volume.
• Granular Flow: Because of the spaces between them, granular solids like sand, salt, and sugar exhibit fluid-like behavior, which is a key feature used in industrial applications such as silo storage and food processing.

The correct answer is Parallel to the incident ray but shifted sideward.

Key Points

• When a ray of light enters a rectangular glass slab from air, it undergoes refraction twice: once at the air-glass interface and again at the glass-air interface.
• At the first surface, light travels from a rarer medium (air) to a denser medium (glass), causing the ray to bend towards the normal.
• At the second surface, light travels from the denser medium (glass) back into the rarer medium (air), causing the ray to bend away from the normal.
• Since the opposite faces of a rectangular glass slab are parallel to each other, the extent of bending of the ray of light at the first surface is equal and opposite to the extent of bending at the second surface.
• As a result, the emergent ray emerges in a direction parallel to the incident ray. However, there is a lateral shift or sideward displacement between the original path of the incident ray and the emergent ray. Hence, the statement describing it as parallel but shifted sideward is correct.

Additional Information

• Refractive Index: It is a dimensionless number that describes how fast light travels through a medium. The refractive index of glass is typically around 1.5, while for air, it is approximately 1.0003.
• Snell's Law: This law governs refraction and is expressed as n1 sin(i) = n2 sin(r), where n1 and n2 are refractive indices of the two media, i is the angle of incidence, and r is the angle of refraction.
• Lateral Displacement: The perpendicular distance between the direction of the incident ray and the emergent ray is known as lateral displacement.
• Factors Affecting Lateral Displacement:
• Thickness of the glass slab: Lateral displacement is directly proportional to the thickness of the slab. A thicker slab causes a larger sideward shift.
• Angle of Incidence: As the angle of incidence (i) increases, the lateral displacement also increases.
• Refractive Index: A higher refractive index of the material leads to greater lateral displacement.
• Wavelength of Light: Lateral displacement depends on the wavelength (color) of light. Since violet light bends more than red light, it experiences more lateral displacement.

Important Points

• If a light ray falls normally (perpendicularly) on the surface of the glass slab, the angle of incidence is 0 degrees. In this specific case, the ray passes through without any deviation or lateral displacement.
• The angle of incidence is always equal to the angle of emergence (i = e) for a rectangular glass slab when the medium on both sides of the slab is the same.
• This phenomenon is a practical application of optics and is used to demonstrate the rectilinear propagation of light and the laws of refraction in laboratory experiments.
• In 2024, understanding these principles remains fundamental in fields like lens design, fiber optics, and microscopy.

The correct answer is By transferring pollen from the anther to the stigma, enabling fertilisation.

Key Points

• Pollination is defined as the act of transferring pollen grains from the male anther of a flower to the female stigma. This process is the vital first step in the sexual reproduction of flowering plants.
• The primary contribution of pollination to seed formation is that it enables fertilisation. Without this transfer, the male gametes contained within the pollen would never reach the female gametes (ovules) located in the ovary. Hence, option 3 is correct.
• Once a compatible pollen grain lands on the stigma, it germinates and produces a pollen tube. This tube grows down through the style to deliver two male gametes into the embryo sac within the ovule.
• Upon reaching the ovule, one male gamete fuses with the egg cell to form a zygote. This fusion is the essence of fertilisation, which eventually triggers the transformation of the ovule into a seed.
• While nectar production (Option 4) attracts pollinators, it is a biological "bribe" and not the direct mechanism that forms a seed. Similarly, seed dispersal (Option 2) occurs only after the seed is fully formed.

Additional Information

• Types of Pollination:
• Self-pollination: Occurs when pollen from the anther is deposited on the stigma of the same flower or another flower on the same plant. This is common in plants like peas and wheat.
• Cross-pollination: The transfer of pollen from the anther of one flower to the stigma of another flower on a different individual of the same species. This promotes genetic diversity and is seen in species like maize and sunflowers.
• Agents of Pollination:
• Biotic Agents: These include living organisms such as bees, butterflies, birds, and bats. Insects are the most common pollinators, often attracted by brightly coloured petals or scent.
• Abiotic Agents: These involve non-living forces such as wind (Anemophily) and water (Hydrophily). Wind-pollinated plants, like grasses and conifers, usually produce massive amounts of lightweight pollen.
• Double Fertilisation: This is a characteristic feature of Angiosperms (flowering plants). One male gamete fuses with the egg to form the zygote, while the second male gamete fuses with two polar nuclei to form the triploid endosperm, which provides nourishment to the developing embryo.

Important Points

• Post-Fertilisation Changes: After successful pollination and fertilisation, the flower undergoes significant changes. The sepals, petals, and stamens usually wither and fall off.
• Seed and Fruit Development: The ovule develops into the seed, which contains the embryo (the future plant) and is encased in a protective seed coat (Option 1 refers to this post-fertilisation structure). Simultaneously, the ovary ripens and develops into the fruit.
• Significance: Pollination is not only essential for the survival of plant species but is also critical for global food security. Approximately 75% of the world’s food crops depend, at least in part, on pollination.
• Economic Value: Many agricultural sectors, including the production of almonds, apples, and coffee, rely heavily on honeybee pollination services to ensure high yields and quality seeds.

The correct answer is Persistence of sound due to multiple reflections of sound waves.

Key Points

• Reverberation is defined as the persistence of sound in a particular space after the original sound source has stopped. It occurs because sound waves undergo multiple reflections from surfaces such as walls, ceilings, and floors in an enclosed area.
• In a large room or auditorium, a sound created will continue to be heard by repeated reflection until its intensity is reduced to a level where it is no longer audible.
• The human ear has a persistence of hearing of approximately 0.1 seconds. When reflected sound waves arrive at the ear in intervals shorter than this, the brain perceives them as a single, continuous, and prolonged sound rather than distinct echoes. Hence, statement 2 is correct.
• While some reverberation can enhance the richness of music, excessive reverberation is undesirable in cinemas and lecture halls because it causes sounds to overlap and become garbled, making speech unintelligible.
• The other options provided, such as the spreading of sound (diffusion/diffraction), change in pitch (Doppler Effect), and loss of energy (attenuation), describe different acoustic phenomena. Hence, statements 1, 3, and 4 are incorrect.

Additional Information

• Reverberation Time:
• This is the duration required for the sound pressure level to decay by 60 decibels (dB) after the source is silenced.
• Wallace Clement Sabine, a pioneer in architectural acoustics in the late 19th century, formulated Sabine’s Law, which calculates this time based on the volume of the room and the absorption coefficients of its surfaces.
• Echo vs. Reverberation:
• An echo is a distinct, separate repetition of sound caused by reflection from a distant object.
• To hear a clear echo, the minimum distance between the source and the reflecting surface must be roughly 17.2 meters (assuming the speed of sound is 344 m/s in air). In contrast, reverberation involves reflections from much closer surfaces arriving rapidly.
• Acoustic Treatment:
• To control reverberation, architects use sound-absorbent materials. Common examples include compressed fiberboard, glass wool, heavy curtains, and carpets.
• Even the upholstery of seats in theaters is specifically chosen for its sound-absorbing properties to ensure clarity whether the hall is empty or full.

Important Points

• Multiple Reflection Applications:
• Megaphones and Horns: These are designed to use multiple reflections to direct sound waves in a specific forward direction, preventing them from spreading out.
• Stethoscopes: This medical device transmits the sound of a patient's heartbeat to the doctor’s ears through multiple reflections within the tube.
• Curved Ceilings: In concert halls and conference rooms, ceilings are often curved so that sound, after reflection, is distributed uniformly across the entire width of the hall.
• Speed of Sound: The speed varies depending on the medium (solid, liquid, or gas) and temperature. In air at 20°C, it is approximately 344 m/s, which is a critical constant used in calculating reflection timings.

The correct answer is Both A and R are true, and R is the correct explanation for A.

Key Points

• Assertion (A): When iron is exposed to moist air (i.e., air containing both oxygen and water vapor), a reddish-brown layer forms on its surface. This layer is commonly referred to as Rust. Therefore, Assertion (A) is correct.
• Reason (R): The primary reason for the formation of this reddish-brown substance is that iron undergoes a chemical reaction with the oxygen and moisture present in the air. This reaction results in the formation of Hydrated Iron(III) Oxide (Fe2O3·nH2O). Therefore, Reason (R) is correct, and it provides the correct explanation for Assertion (A).
• Rusting is an Oxidation Reaction.
• In this process, iron loses electrons and combines with oxygen to form an oxide.
• This process not only affects the surface of the iron but, over time, weakens the metal from within.
• Both air (oxygen) and moisture (water) are essential for rusting to occur. If either of these is absent, the rusting process will not take place.

Additional Information

• Corrosion: The gradual deterioration of metals due to environmental factors such as air, moisture, or acids is known as corrosion. In the specific case of iron, this process is referred to as rusting.
• Methods for Preventing Rust:
• Painting: Applying paint to iron creates a barrier that prevents air and moisture from coming into direct contact with the metal.
• Applying Grease or Oil: Applying grease to machine parts provides protection against moisture.
• Galvanization: The process of coating iron or steel with a thin layer of the metal known as Zinc is called galvanization. Since zinc reacts more readily than iron, it protects the iron.
• Electroplating: The process of coating iron with corrosion-resistant metals—such as chromium or nickel—using electricity.
• Alloys: Iron is combined with carbon, chromium, and nickel to produce Stainless Steel, which does not rust.

Important Points

• Chemical Equation: The rusting process is generally represented as 4Fe + 3O2 + 6H2O → 4Fe(OH)3, or ultimately as Fe2O3·xH2O (hydrated ferric oxide).
• Environmental Impact: In coastal or industrial regions, the rate of rusting is significantly higher than in ordinary areas due to the elevated levels of salts and moisture present in the air.
• Mass Change: When iron rusts, its weight gradually increases as oxygen and water combine with the metal.
• Historical Example: The Iron Pillar located within the precincts of the Qutub Minar in Delhi stands as a testament to ancient Indian metallurgical expertise; despite having been constructed nearly 1600 years ago, it remains free of rust.

The correct answer is Potassium.

Key Points

• Potassium (chemical symbol K, from Neo-Latin kalium) is a chemical element with Atomic Number 19. It is a silvery-white metal that is soft enough to be easily cut with a knife with little force.
• It is characterized by an extremely low density of approximately 0.862 g/cm³ at 20°C, which is less than that of water (1.0 g/cm³). This physical property causes the metal to float on the surface of water while it reacts. Hence, the correct answer is Potassium.
• As a member of the Alkali Metals (Group 1 of the periodic table), it possesses only one valence electron in its outer shell. This single electron is involved in metallic bonding, which is relatively weak, leading to the characteristic softness and low melting point (approx 63.5°C) of the metal.
• Due to its extreme chemical reactivity, especially with moisture and air, it is never found as a free element in nature and is typically stored under mineral oil or kerosene to prevent oxidation.

Additional Information

• Copper (Cu):
• It is a transition metal located in Group 11 of the periodic table.
• Unlike potassium, copper has a high density of approximately 8.96 g/cm³ and is significantly harder. It is known for its excellent electrical conductivity and thermal conductivity. Hence, option 1 is incorrect.
• Aluminium (Al):
• It is a post-transition metal in Group 13.
• While aluminium is known for being lightweight compared to steel (density 2.70 g/cm³), it is much denser than potassium and is a structural metal, not "soft" in the chemical sense of being cut like wax. Hence, option 2 is incorrect.
• Magnesium (Mg):
• It is an alkaline earth metal found in Group 2 of the periodic table with Atomic Number 12.
• Its density is approximately 1.74 g/cm³. While it is the lightest structural metal, it is harder than alkali metals like potassium and sodium because it has two valence electrons, strengthening its metallic lattice. Hence, option 4 is incorrect.

Important Points

• Periodic Trends: In the periodic table, alkali metals (Lithium, Sodium, Potassium, Rubidium, Cesium) generally show a decrease in hardness and melting points as you move down the group due to the increasing atomic radius and weakening of metallic bonds.
• Density Comparisons: Only three metals in the periodic table are light enough to float on water: Lithium (lightest solid element), Sodium, and Potassium.
• Applications: Potassium is vital for biological functions in the human body, specifically for nerve impulses and muscle contractions. Industrially, it is widely used in the production of fertilizers (as potash) and in specialty glasses.
• Identification: Potassium can be identified through a flame test, where it emits a characteristic lilac or light purple color.

Shortcut Trick

Solve the first few terms: (1 − 1/2) = 1/2; (1 − 1/3) = 2/3; (1 − 1/4) = 3/4.

Notice the cancellation pattern: The denominator of one term cancels the numerator of the next.

In this telescoping product, only the first numerator and the last denominator remain.

Result = First Numerator / Last Denominator = 1 / m.

∴ The correct answer is 1/m.

Alternate Method

Given: A product of terms in the form (1 − 1/n) from n = 2 to m.

Formula Used: (1 − 1/n) = (n − 1)/n

⇒ Term 1: (1 − 1/2) = 1/2

⇒ Term 2: (1 − 1/3) = 2/3

⇒ Term 3: (1 − 1/4) = 3/4

⇒ Term m: (1 − 1/m) = (m − 1)/m

⇒ Product = (1/2) × (2/3) × (3/4) × (4/5) × ... × (m−1)/m

⇒ After cross-cancellation of intermediate terms: 1/m

∴ The correct answer is 1/m.

Additional Information

Telescoping Series

A sequence where most terms cancel out, leaving only the first and last parts. For addition: ∑(a_n − a_n+1) = a₁ − a_last.

Successive Fractions Product

In products like (1 + 1/n), the pattern is (n+1)/n. Here, denominators cancel numerators differently: result is (m+1)/initial_denominator.

Pattern Recognition

Always simplify the first three terms of an infinite or large finite mathematical series to identify the cancellation behavior.

Shortcut Trick

Use the direct relation between Marked Price (MP) and Cost Price (CP): MP ÷ CP = (100 + Gain%) ÷ (100 − Discount%).

MP ÷ 391 = (100 + 68) ÷ (100 − 31) = 168 ÷ 69.

MP = (168 ÷ 69) × 391.

Dividing 391 by 69: 69 × 5 = 345, 391 − 345 = 46 (not direct). Simplification: 391 ÷ 23 = 17 and 69 ÷ 23 = 3.

MP = (168 ÷ 3) × 17 = 56 × 17 = 952.

∴ The correct answer is ₹952.

Alternate Method

Given → CP = ₹391, Gain% = 68%, Discount% = 31%. Formulas: SP = CP × (100 + P%) ÷ 100 | MP = SP × 100 ÷ (100 − D%).

⇒ Step 1: Find Selling Price (SP). ⇒ SP = 391 × (168 ÷ 100) = 391 × 1.68 = 656.88. ⇒ Step 2: Find Marked Price (MP). ⇒ MP × (100 − 31) ÷ 100 = 656.88. ⇒ MP × 0.69 = 656.88. ⇒ MP = 656.88 ÷ 0.69 = 952.

∴ The correct answer is ₹952.

Additional Information

Relation between CP and MP

When both profit percentage and discount percentage are known, the ratio of MP to CP is always (100 + Profit%) / (100 − Discount%).

Marked Price & Discount

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

Shortcut Trick

Combined efficiency of A and B = 1/3

Efficiency of B alone = 1/12

Efficiency of A = 1/3 − 1/12 = (4 − 1)/12 = 3/12 = 1/4

Time for A to do single work = 4 days

Time for A to do double the work = 4 × 2 = 8 days

∴ The correct answer is 8 days.

Alternate Method

Given: Time taken by (A + B) = 3 days; Time taken by B = 12 days.

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

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

⇒ Efficiency of A = 4 − 1 = 3 units/day

⇒ Double the work = 12 × 2 = 24 units

⇒ Time taken by A alone = 24 ÷ 3 = 8 days

∴ The correct answer is 8 days.

Additional Information

Efficiency-Time Relationship

Efficiency is inversely proportional to the time taken to complete the work. Efficiency = Work ÷ Time.

Work-Time Formula

If A can do a work in x days and B in y days, together they take (xy)/(x+y) days.

Work & Wages

Wages are distributed in the ratio of the efficiency of the workers or the amount of work done.

Shortcut Trick

The age difference between Juli and her daughter is always constant at 24 years.

In 6 years, the ratio of their ages will be 2 : 1.

Difference in ratio parts = 2 − 1 = 1 unit.

1 unit → 24 years. Therefore, Juli's age in 6 years = 2 × 24 = 48 years.

Present age of Juli = 48 − 6 = 42 years.

∴ The correct answer is 42 years.

Alternate Method

Let the present age of the daughter be x years.

Given: Juli is 24 years older than her daughter.

⇒ Juli's present age = x + 24

⇒ In 6 years, Juli's age = (x + 24) + 6 = x + 30

⇒ In 6 years, Daughter's age = x + 6

⇒ According to the problem: x + 30 = 2 × (x + 6)

⇒ x + 30 = 2x + 12

⇒ 2x − x = 30 − 12

⇒ x = 18

⇒ Juli's present age = 18 + 24 = 42

∴ The correct answer is 42 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two people remains constant throughout their lives, regardless of how many years pass.

Ratio Method for Age

If the ratio of ages at two different times is known, equate the parts of the ratio to the constant age difference to find the multiplier.

Linear Equations in Ages

Always define present age as the base variable (x) to simplify calculations for future (+n years) or past (−n years) conditions.

Shortcut Trick

Apply the successive percentage formula: a + b + (a × b)/100

Here, a = −5% (decrease) and b = +8% (increase).

Effective % = −5 + 8 + (−5 × 8)/100 = 3 − 0.4 = 2.6%

Positive result indicates an increase.

∴ The correct answer is 2.6%.

Alternate Method

Given: First week decrease = 5%, Second week increase = 8%.

Assume initial price = 100 units

⇒ Price after 5% decrease = 100 − 5 = 95 units

⇒ Price after 8% increase = 95 + (8/100 × 95)

⇒ Price after 8% increase = 95 + 7.6 = 102.6 units

⇒ Net change = 102.6 − 100 = 2.6 units

⇒ Effective percentage increase = (2.6/100) × 100 = 2.6%

∴ The correct answer is 2.6%.

Additional Information

Successive Percentage Change

Used when a value is changed multiple times consecutively. Formula: Net % = a + b + (ab/100). Use positive for increase and negative for decrease.

Multiplying Factor Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100). For 5% decrease and 8% increase, factor = 0.95 × 1.08 = 1.026.

Base Change Property

If price increases by x%, then to restore it, consumption must decrease by [x/(100+x)] × 100%.

Shortcut Trick

Assume all 6 fruits are bananas: Cost = 6 × ₹10 = ₹60.

Total difference from actual cost = ₹80 − ₹60 = ₹20.

Extra cost of an apple compared to a banana = ₹20 − ₹10 = ₹10.

Number of apples = 20 ÷ 10 = 2.

Number of bananas = 6 − 2 = 4.

Difference (Bananas − Apples) = 4 − 2 = 2.

∴ The correct answer is 2.

Alternate Method

Given:

Cost of 1 apple = ₹20

Cost of 1 banana = ₹10

Total fruits = 6

Total amount paid = ₹80

Let the number of apples be 'a' and the number of bananas be 'b'.

⇒ a + b = 6 --- (Equation 1)

⇒ 20a + 10b = 80 --- (Equation 2)

Divide Equation 2 by 10:

⇒ 2a + b = 8 --- (Equation 3)

Subtract Equation 1 from Equation 3:

⇒ (2a + b) − (a + b) = 8 − 6

⇒ a = 2

Substitute a = 2 in Equation 1:

⇒ 2 + b = 6

⇒ b = 4

The number of bananas is 4 and the number of apples is 2.

⇒ Difference = b − a = 4 − 2 = 2

∴ The correct answer is 2.

Additional Information

Linear Equations in Two Variables

A system of two equations like ax + by = c can be solved using substitution, elimination, or cross-multiplication methods.

Alligation Method

The ratio of the quantities can be found using (Price_higher − Average) : (Average − Price_lower). Here, Average = 80 ÷ 6 = ₹13.33.

Unitary Method Application

Calculating the cost by assuming an extreme case (e.g., all items are of one type) helps identify the quantity of the other type through the residual cost.

Shortcut Trick

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

⇒ Teacher's Age = 20 + (30 × (20 − 19))

⇒ Teacher's Age = 20 + 30 = 50

∴ The correct answer is 50 years.

Alternate Method

Given:

Initial number of students = 30

Initial average age = 19 years

New number of people (students + teacher) = 31

New average age = 20 years

Formula Used:

Sum of ages = Average × Number of persons

⇒ Sum of ages of 30 students = 30 × 19 = 570 years

⇒ Sum of ages of 31 persons (including teacher) = 31 × 20 = 620 years

⇒ Age of the teacher = Total sum − Sum of students

⇒ Age of the teacher = 620 − 570 = 50 years

∴ The correct answer is 50 years.

Additional Information

Average Concept

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

Inclusion Rule

When a new person joins and the average increases, the new person's value = Old Average + (New Total Strength × Increase in Average).

Exclusion Rule

When a person leaves and the average decreases, the excluded person's value = Old Average + (Initial Strength × Decrease in Average).

Shortcut Trick

First, calculate the semi-perimeter (s): s = (34 + 18 + 20) ÷ 2 = 36 cm.

Find the Area using Heron's formula: Area = √(36 × (36−34) × (36−18) × (36−20)) = √(36 × 2 × 18 × 16).

Area = √(36 × 36 × 16) = 6 × 6 × 4 = 144 cm².

The altitude (h) corresponding to the 18 cm base: h = (2 × Area) ÷ Base = (2 × 144) ÷ 18 = 16 cm.

∴ The correct answer is 16 cm.

Alternate Method

Given:

Sides of the triangle: a = 34 cm, b = 18 cm, c = 20 cm.

Formula Used:

1. Heron's Formula for Area: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2.
2. General Area formula: Area = (1/2) × Base × Altitude.

Calculations:

⇒ Semi-perimeter (s) = (34 + 18 + 20) ÷ 2

⇒ s = 72 ÷ 2 = 36 cm

⇒ Area = √[36 × (36 − 34) × (36 − 18) × (36 − 20)]

⇒ Area = √[36 × 2 × 18 × 16]

⇒ Area = √[36 × 36 × 16]

⇒ Area = 6 × 6 × 4 = 144 cm²

⇒ Area = (1/2) × 18 × Altitude

⇒ 144 = 9 × Altitude

⇒ Altitude = 144 ÷ 9

⇒ Altitude = 16 cm

∴ The correct answer is 16 cm.

Additional Information

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side (a + c).

Smallest Side and Altitude

In any triangle, the longest altitude always corresponds to the shortest side, and the shortest altitude corresponds to the longest side.

Area via Sine of Angle

Area can also be calculated if an angle is known: Area = (1/2) × a × b × sin(C), where C is the angle between sides a and b.

Shortcut Trick

Apply the algebraic identity: x² + y² = (x + y)² − 2xy.

Substitute the given values: Sum = 4 and Product = 2.

Calculation: 4² − 2 × 2 = 16 − 4.

Result = 12.

∴ The correct answer is 12.

Alternate Method

Given:

Sum of variables (x + y) = 4

Product of variables (xy) = 2

Formula Used:

(x + y)² = x² + y² + 2xy

Calculations:

⇒ (x + y)² = x² + y² + 2xy

⇒ (4)² = x² + y² + 2 × 2

⇒ 16 = x² + y² + 4

⇒ x² + y² = 16 − 4

⇒ x² + y² = 12

∴ The correct answer is 12.

Additional Information

Square of a Difference

The identity for the square of a subtraction is (x − y)² = x² + y² − 2xy. This can be used to find x² + y² if the difference and product are known.

Difference of Two Squares

The formula x² − y² = (x − y)(x + y) is used to factorize the difference of two squared terms.

Relationship between Identities

The two basic square identities are related by the formula: (x + y)² − (x − y)² = 4xy.

Shortcut Trick

Let the number be x.

20% × 10% × 2/3 × x = 697

1/5 × 1/10 × 2/3 × x = 697

1/75 × x = 697 ⇒ x = 697 × 75 = 52275

∴ The correct answer is 52275.

Alternate Method

Given: Result of the calculation = 697

⇒ 20% of 10% of 2/3 × number = 697

⇒ (20/100) × (10/100) × (2/3) × number = 697

⇒ (1/5) × (1/10) × (2/3) × number = 697

⇒ (2/150) × number = 697

⇒ (1/75) × number = 697

⇒ number = 697 × 75

⇒ number = 52275

∴ The correct answer is 52275.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 20% = 1/5 and 10% = 1/10.

Successive Percentage

When applying successive percentages, multiply the fraction equivalents directly: a% of b% of X = (a/100) × (b/100) × X.

Reverse Calculation

If the final value is known, find the initial number by multiplying the final value by the reciprocal of each factor.

Shortcut Trick

Let the total registered voters be 100%.

Voters who did not vote = 8%, so voters who cast votes = 92%.

Winner received 48% of total voters. Runner-up received (92% − 48%) − 120 blank votes = 44% − 120.

Victory Margin = 48% − (44% − 120) = 4% + 120.

Given 4% + 120 = 780 ⇒ 4% = 660 ⇒ 100% = 16,500.

∴ The correct answer is 16,500.

Alternate Method

Given:

⇒ Let Total Voters = x

⇒ Votes Cast = 92% of x = 0.92x

⇒ Winner's Votes = 48% of x = 0.48x

⇒ Runner-up's Votes = (Total Cast − Blank − Winner) = (0.92x − 120 − 0.48x) = 0.44x − 120

Calculations:

⇒ Winner − Runner-up = 780

⇒ 0.48x − (0.44x − 120) = 780

⇒ 0.04x + 120 = 780

⇒ 0.04x = 660

⇒ x = 660 ÷ 0.04

⇒ x = 16,500

∴ The correct answer is 16,500.

Additional Information

Election Base Percentage

Always identify if the percentage is of total registered voters or valid votes cast to set the correct base for calculations.

Valid vs Invalid Votes

Valid Votes = (Total Cast − Invalid/Blank). The sum of votes of all candidates must equal the Total Valid Votes.

Victory Margin Formula

Margin = Votes of Winner − Votes of Runner-up. In percentage terms: (Winner% − Runner-up%) × Total = Margin.

Shortcut Trick

Relative Speed in same direction = 87 − 51 = 36 kmph.

Convert speed to m/s: 36 × (5/18) = 10 m/s.

Total distance (2L) = Speed × Time = 10 × 175 = 1750 m.

Length of one train (L) = 1750 ÷ 2 = 875 m.

∴ The correct answer is 875 m.

Alternate Method

Given: Speed of Rajdhani Express = 87 km/hr, Speed of Garib Rath Express = 51 km/hr, Time taken = 175 seconds, Lengths are equal (L₁ = L₂ = L).

Formula Used: Relative Speed (Same Direction) = S₁ − S₂; Total Distance = L₁ + L₂; Distance = Speed × Time.

⇒ Relative Speed = 87 − 51 = 36 km/hr

⇒ Relative Speed in m/s = 36 × (5/18) = 10 m/s

⇒ Total distance to be covered = L + L = 2L

⇒ 2L = 10 × 175

⇒ 2L = 1750 m

⇒ L = 1750 ÷ 2 = 875 m

∴ The correct answer is 875 m.

Additional Information

Relative Speed Concepts

When moving in the same direction, subtract speeds (S₁ − S₂). When moving in opposite directions, add speeds (S₁ + S₂).

Unit Conversion

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

Train Crossing Rules

The distance covered when one train crosses another is always the sum of their lengths (L₁ + L₂), regardless of the direction.

Shortcut Trick

Any two consecutive integers are always co-prime to each other.

Since P and P + 1 are consecutive integers, their highest common factor (HCF) must be 1.

∴ The correct answer is 1.

Alternate Method

Given values: P is a prime number.

The consecutive number is P + 1.

⇒ Let us assume a prime number, P = 3.

⇒ Then, P + 1 = 3 + 1 = 4.

⇒ Factors of 3 are 1 and 3.

⇒ Factors of 4 are 1, 2, and 4.

⇒ Common factor is 1.

⇒ HCF(3, 4) = 1.

∴ The correct answer is 1.

Additional Information

Co-prime Numbers

Two numbers are said to be co-prime if their highest common factor (HCF) is equal to 1.

HCF of Prime Numbers

The HCF of any two distinct prime numbers is always 1, because prime numbers only have 1 and themselves as factors.

LCM of Co-prime Numbers

The Least Common Multiple (LCM) of any two co-prime numbers, such as P and P + 1, is their product, which is P × (P + 1) = P² + P.

Shortcut Trick

The effective interest rate for 2 years at 14% = 14 + 14 + (14 × 14 ÷ 100) = 29.96%.

Total Interest = 2500 × 29.96 ÷ 100 = ₹749.

Total Amount = Principal + Interest = 2500 + 749 = ₹3249.

∴ The correct answer is ₹3249.

Alternate Method

Given:

Principal (P) = ₹2,500

Rate (R) = 14% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R ÷ 100)ⁿ

Calculations:

⇒ A = 2500 × (1 + 14 ÷ 100)²

⇒ A = 2500 × (114 ÷ 100) × (114 ÷ 100)

⇒ A = 2500 × (57 ÷ 50) × (57 ÷ 50)

⇒ A = 2500 × (3249 ÷ 2500)

⇒ A = ₹3249

∴ The correct answer is ₹3249.

Additional Information

Compound Interest Formula

The interest is calculated on the principal plus the accumulated interest. CI = A − P.

Effective Rate of Interest

For a two-year period at rate R%, the effective percentage is 2R + (R² ÷ 100)%.

CI vs SI Difference

For 2 years, the difference between CI and SI is given by P × (R ÷ 100)².

Shortcut Trick

Start with the innermost square root: √(11 − 4√7) = √(√7 − 2)² = √7 − 2.

Substitute this value back into the intended expression: √(51 − √7 + √7 − 2).

The irrational terms −√7 and +√7 cancel each other out perfectly.

This simplifies the remaining expression to: √(51 − 2) = √49.

Finally, the square root of 49 is 7.

∴ The correct answer is 7.

Alternate Method

Given:

Expression: √(51 − √7 + √(11 − 4√7))

Formula Used:

(a − b)² = a² + b² − 2ab

Calculation:

⇒ First, evaluate the innermost radical: √(11 − 4√7)

⇒ The term inside can be rewritten as: 11 − 4√7 = 7 + 4 − 2 × 2 × √7

⇒ Applying the algebraic identity, we get: (√7)² + (2)² − 2 × √7 × 2

⇒ This perfectly forms a square: (√7 − 2)²

⇒ Therefore, the square root simplifies to: √(√7 − 2)² = √7 − 2

⇒ Now, substitute this result into the complete expression:

⇒ √(51 − √7 + √7 − 2)

⇒ The −√7 and +√7 terms offset and cancel each other out.

⇒ The expression now becomes: √(51 − 2)

⇒ √49

⇒ 7

∴ The correct answer is 7.

Additional Information

Simplifying Surds of the Form √(a ± √b)

To simplify such surds, we convert the inner expression into a perfect square (x ± y)² = x² + y² ± 2xy, where the rational part equals x² + y² and the irrational part matches 2xy.

Addition and Subtraction of Surds

Only like surds (surds having the same index and same radicand) can be added or subtracted. For example, m√x ± n√x = (m ± n)√x.

Nested Radicals and BODMAS

When solving expressions with nested radicals, always begin simplification from the innermost radical and work your way outward step by step.

Shortcut Trick

Calculate semi-perimeter (s): s = (87 + 108 + 75) ÷ 2 = 135 cm.

Calculate Area using Heron's: Area = √(135 × (135−87) × (135−108) × (135−75)) = √(135 × 48 × 27 × 60).

Area = √(3240²) = 3240 cm².

Altitude (h) on base 108: h = (2 × Area) ÷ base = (2 × 3240) ÷ 108 = 60 cm.

∴ The correct answer is 60 cm.

Alternate Method

Given: Sides of triangle a = 87 cm, b = 108 cm, c = 75 cm.

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

⇒ Semi-perimeter s = (a + b + c) ÷ 2

⇒ s = (87 + 108 + 75) ÷ 2 = 135 cm

⇒ Area = √[135 × (135 − 87) × (135 − 108) × (135 − 75)]

⇒ Area = √[135 × 48 × 27 × 60]

⇒ Area = √[5 × 27 × 3 × 16 × 27 × 5 × 12]

⇒ Area = √[5² × 27² × 4² × 6²]

⇒ Area = 5 × 27 × 4 × 6 = 3240 cm²

⇒ Area = (1/2) × base × altitude

⇒ 3240 = (1/2) × 108 × altitude

⇒ 3240 = 54 × altitude

⇒ altitude = 3240 ÷ 54 = 60 cm

∴ The correct answer is 60 cm.

Additional Information

Heron's Formula

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

Altitude of a Triangle

The perpendicular distance from a vertex to the opposite side. The length of the altitude can be found using h = (2 × Area) ÷ Base.

Inradius of a Triangle

The radius of the inscribed circle is given by r = Area ÷ s, where s is the semi-perimeter.

Shortcut Trick

Total amount = ₹10,279

Sum of deductions = 67 + 33 + 13 = ₹113

Remaining amount = 10,279 − 113 = ₹10,166

Ratio after deduction (A : B : C) = 4 : 2 : 20 (Total units = 26)

Value of 1 unit = 10,166 ÷ 26 = ₹391

Difference (C − B) = (20 − 2) units + (13 − 33) = 18 × 391 − 20 = 7,038 − 20 = ₹7,018

∴ The correct answer is ₹7018.

Alternate Method

Given: Total amount = ₹10,279; Deductions: A = ₹67, B = ₹33, C = ₹13; Ratio = 4 : 2 : 20.

Formula: New Share = Original Share − Deduction; Ratio sum × x = Total remaining amount.

⇒ Total remaining = 10,279 − (67 + 33 + 13) = 10,166

⇒ 4x + 2x + 20x = 10,166

⇒ 26x = 10,166

⇒ x = 10,166 ÷ 26 = 391

⇒ B's original share = 2 × 391 + 33 = 782 + 33 = ₹815

⇒ C's original share = 20 × 391 + 13 = 7,820 + 13 = ₹7,833

⇒ Difference = 7,833 − 815 = 7,018

∴ The correct answer is ₹7018.

Additional Information

Ratio and Proportion

Ratio represents the comparison between two or more quantities. For a total S divided in ratio a:b:c, the value of one part is S / (a+b+c).

Deduction Method

When fixed amounts are subtracted from shares, subtract the total deduction from the grand total before applying the ratio distribution.

Shortcut Trick

Mean = (2 ÷ 3) × Mode = (2 ÷ 3) × 3 = 2.

Using the empirical formula: Mode = 3 × Median − 2 × Mean.

3 = 3 × Median − 2(2) ⇒ 3 = 3 × Median − 4 ⇒ Median = 7/3.

Product = 2 × (7/3) = 14/3.

Square = (14/3)² = 196/9 = 21 7/9.

∴ The correct answer is 21 7/9.

Alternate Method

Given:

Mode = 3

Mode = (3/2) × Mean

Formula Used:

Empirical relationship: Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 3 = (3/2) × Mean

⇒ Mean = (3 × 2) ÷ 3 = 2

⇒ Now, substitute values in the empirical formula:

⇒ 3 = 3 × Median − 2(2)

⇒ 3 = 3 × Median − 4

⇒ 7 = 3 × Median

⇒ Median = 7/3

⇒ Required value = (Median × Mean)²

⇒ Result = (7/3 × 2)²

⇒ Result = (14/3)² = 196/9

⇒ Result = 21 7/9

∴ The correct answer is 21 7/9.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship is approximately Mode = 3 × Median − 2 × Mean.

Symmetric Distribution

In a perfectly symmetrical distribution, the values of Mean, Median, and Mode are all equal (Mean = Median = Mode).

Positively Skewed Distribution

When the tail of the distribution is on the right side, it is called positively skewed, where Mean > Median > Mode.

Negatively Skewed Distribution

When the tail of the distribution is on the left side, it is called negatively skewed, where Mode > Median > Mean.

Shortcut Trick

Convert times to fractions: Pipe P = 75/2 minutes, Pipe Q = 45 minutes.

Let Total Capacity (LCM of 75 and 45) = 225 units.

Efficiency P = 225 ÷ (75/2) = 6 units/min; Efficiency Q = 225 ÷ 45 = 5 units/min.

Pipe P works for the full 30 minutes: 6 × 30 = 180 units.

Remaining work for Pipe Q = 225 − 180 = 45 units.

Time x = 45 ÷ 5 = 9 minutes.

∴ The correct answer is 9 minutes.

Alternate Method

Given: Time P = 37.5 min = 75/2 min, Time Q = 45 min, Total time = 30 min.

Formula Used: Total Work = Time × Efficiency

⇒ Let Pipe Q be closed after x minutes.

⇒ Work done by P in 30 mins + Work done by Q in x mins = 225

⇒ (6 × 30) + (5 × x) = 225

⇒ 180 + 5x = 225

⇒ 5x = 45

⇒ x = 9 minutes

∴ The correct answer is 9 minutes.

Additional Information

Pipes and Cisterns Efficiency

Efficiency is inversely proportional to time. If a pipe fills a tank in T hours, its efficiency is 1/T per hour.

Inlet and Outlet Pipes

Inlet pipes have positive efficiency (filling), while outlet/leakage pipes have negative efficiency (emptying).

Combined Work Formula

If pipe A fills in x and pipe B fills in y, time to fill together is (x × y) ÷ (x + y).

Shortcut Trick

In a 30°−60°−90° triangle, the sides are in the ratio 1 : √3 : 2.

The side adjacent to 60° (base) is 1 unit, and the hypotenuse (ladder) is 2 units.

Given: 1 unit = 5.6 m

⇒ 2 units = 5.6 × 2 = 11.2 m

∴ The correct answer is 11.2 m.

Alternate Method

Given:

Angle of elevation (θ) = 60°

Distance from wall (Base) = 5.6 m

Formula Used:

cos(θ) = Base ÷ Hypotenuse

Calculations:

⇒ cos(60°) = 5.6 ÷ Length of ladder

⇒ 1/2 = 5.6 ÷ Length of ladder

⇒ Length of ladder = 5.6 × 2

⇒ Length of ladder = 11.2 m

∴ The correct answer is 11.2 m.

Additional Information

Trigonometric Ratios

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

Standard Values

Key values to remember: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.

Pythagoras Theorem

In any right-angled triangle, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Total Perimeter = 2 × Straight Length + 2 × π × Radius

400 = 2(100) + 2 × (22/7) × r

200 = (44/7) × r

r = 1400 ÷ 44 = 31.818...

∴ The correct answer is 31.82 metres.

Alternate Method

Given:

Length of straight portions (L) = 100 m

Total Perimeter (P) = 400 m

Value of π = 22/7

Formula Used:

Perimeter of Track = (2 × Length of straight part) + (Circumference of a full circle formed by two semicircles)

Perimeter = 2L + 2πr

Calculations:

⇒ 400 = (2 × 100) + (2 × 22/7 × r)

⇒ 400 = 200 + (44/7) × r

⇒ 400 − 200 = (44/7) × r

⇒ 200 = (44/7) × r

⇒ r = (200 × 7) ÷ 44

⇒ r = 1400 ÷ 44

⇒ r = 31.818181...

⇒ r ≈ 31.82 m

∴ The correct answer is 31.82 metres.

Additional Information

Area of the Athletic Track

The total area is the sum of the rectangular area and the area of the full circle: Area = (L × 2r) + πr².

Circumference of a Semicircle

The distance around the curved part of a semicircle is πr. The total perimeter of a closed semicircle is πr + 2r.

Standard Running Track

A standard 400m outdoor track usually has a radius of approximately 36.5 metres for the innermost lane.

Shortcut Trick

Total distance for one complete round = 4 × Side = 4 × 25 = 100 m.

Speed in m/s = 4 × (5/18) = 10/9 m/s.

Time = Distance ÷ Speed = 100 ÷ (10/9) = 10 × 9 = 90 seconds.

∴ The correct answer is 90 seconds.

Alternate Method

Given:

Side of the square field = 25 m

Speed of the boy = 4 km/hr

Formula Used:

Perimeter of square = 4 × side

Time = Distance ÷ Speed

Conversion: 1 km/hr = 5/18 m/s

Calculations:

⇒ Total Distance = Perimeter = 4 × 25 = 100 m

⇒ Speed in m/s = 4 × (5/18) = 20/18 = 10/9 m/s

⇒ Time taken = 100 ÷ (10/9)

⇒ Time taken = 100 × (9/10)

⇒ Time taken = 10 × 9 = 90 seconds

∴ The correct answer is 90 seconds.

Additional Information

Unit Conversion

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

Square Properties

The distance covered in one complete round is the Perimeter (4a). The area is side² (a²).

Basic Speed Formula

Speed = Distance ÷ Time | Distance = Speed × Time | Time = Distance ÷ Speed.

Shortcut Trick

Mean proportional (MP) formula: MP² = Product of numbers

⇒ 84² = X × 147

⇒ X = (84 × 84) ÷ 147

⇒ X = 48

∴ The correct answer is 48.

Alternate Method

Given:

First term = X

Third term = 147

Mean Proportional = 84

Formula Used:

If b is the mean proportional between a and c, then b² = ac or b = √(ac).

Calculations:

⇒ 84 = √(X × 147)

⇒ 84 × 84 = X × 147

⇒ 7056 = X × 147

⇒ X = 7056 ÷ 147

⇒ X = 48

∴ The correct answer is 48.

Additional Information

Third Proportional

If a and b are two numbers, then the third proportional c is given by the relation a:b :: b:c, which means c = b²/a.

Fourth Proportional

If a, b, and c are three numbers, the fourth proportional d is such that a:b :: c:d, which means d = (b × c)/a.

Continued Proportion

Three numbers a, b, c are in continued proportion if a/b = b/c, meaning b is the geometric mean of a and c.

Shortcut Trick

Profit per unit = Selling Price − Cost Price

Profit per kg = 100 − 85 = ₹15

Total Profit = Quantity × Profit per unit

Total Profit = 10 × 15 = ₹150

∴ The correct answer is ₹150.

Alternate Method

Given: Quantity = 10 kg, Cost Price (CP) = ₹85/kg, Selling Price (SP) = ₹100/kg.

Formulas: Total Profit = Total SP − Total CP | Total CP = Quantity × CP per unit | Total SP = Quantity × SP per unit

⇒ Total Cost Price = 10 × 85 = ₹850

⇒ Total Selling Price = 10 × 100 = ₹1000

⇒ Total Profit = 1000 − 850 = ₹150

∴ The correct answer is ₹150.

Additional Information

Profit Percentage

It is the ratio of profit to cost price expressed as a percentage: Profit% = (Profit ÷ CP) × 100.

Net Gain and Loss

Profit occurs when SP > CP, while loss occurs when CP > SP. Loss = CP − SP.

Shortcut Trick

Net discount in 'Buy x get y free' schemes is calculated as y ÷ (x + y) × 100.

Here, x = 30 and y = 11.

Discount % = 11 ÷ (30 + 11) × 100 = 11 ÷ 41 × 100.

Discount % = 1100 ÷ 41 = 26.829...

Rounding to two decimal places, we get 26.83%.

∴ The correct answer is 26.83%.

Alternate Method

Given: Buy 30 items, get 11 items free.

Formulas: Discount % = (Discount ÷ MP) × 100 | Total Items = Paid Items + Free Items.

⇒ Let the Marked Price (MP) of 1 item be 1.

⇒ Total items given to customer = 30 + 11 = 41.

⇒ Total MP of 41 items = 41.

⇒ Customer pays for only 30 items, so Selling Price (SP) = 30.

⇒ Discount = MP − SP = 41 − 30 = 11.

⇒ Discount % = (11 ÷ 41) × 100 = 26.829...%.

∴ The correct answer is 26.83%.

Additional Information

Marked Price & Discount

The price listed on the label is Marked Price. Discount is always calculated on the MP unless stated otherwise. SP = MP × (100 − D%) ÷ 100.

Free Goods Schemes

In "Buy A get B free", the effective discount rate is B ÷ (A + B) × 100. The shopkeeper bears the cost of (A+B) items but receives revenue for only A items.

Given: O, P, Q, R, W, X and Y are sitting in a straight line facing north.

Only two people sit to the right of X.

Only two people sit between X and Q.

Only two people sit between O and P.

P sits to the immediate left of X.

R sits to the immediate right of Y.

Now, the third position from the left end is W.

Hence, the correct answer is "Option 3".

The logic followed here is:

thus, "109" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

Given: A, B, C, D, E and F sit around a circular table facing the centre.

E sits third to the right of F.

B sits second to the left of E.

C sits second to the right of A.

D sits to the immediate right of C.

So, only one person sits between E and D.

Hence, the correct answer is "Option 1".

The logic followed here is:

For, NOK → JKG

For, EFB → ABX

Now, check the options using the same pattern:

Option 1) TUQ → PQM

Option 2) TTO → PQM

Option 3) TTO → OPL

Option 4) TUQ → OQL

Hence, the correct answer is Option 1.

The logic followed here is:

Logic: Second Number = (First Number − Third Number) × 2.

For, (44, 52, 18)

⇒ 44 − 18 = 26

⇒ 26 × 2 = 52

For, (50, 56, 22)

⇒ 50 − 22 = 28

⇒ 28 × 2 = 56

Now, checking the options using the same pattern:

Option 1) (41, 82, 1)

⇒ 41 − 1 = 40

⇒ 40 × 2 = 80 ≠ 82

Option 2) (42, 78, 4)

⇒ 42 − 4 = 38

⇒ 38 × 2 = 76 ≠ 78

Option 3) (19, 14, 13)

⇒ 19 − 13 = 6

⇒ 6 × 2 = 12 ≠ 14

Option 4) (30, 22, 19)

⇒ 30 − 19 = 11

⇒ 11 × 2 = 22

Hence, the correct answer is Option 4.

The logic followed here is based on the alphabetical position pattern:

Option 1: TVW

Option 2: OQR

Option 3: BEE

Option 4: EGH

Thus, BEE does not follow the pattern.

Hence, the odd one out is Option 3.

The logic followed here is:

For, NK 16 → QN 12

For, PS 20 → SV 16

Now, apply the same to BX 30:

So, BX 30 → EA 26

Hence, the correct answer is Option 2.

The logic followed here is:

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

Hence, the correct answer is "Option 1"

The logic followed here is:

thus, "276" will replace the question mark in the given series.

Hence, the correct option is "Option 1".

Given: Seven boxes B, D, E, P, S, T and W are kept one over the other but not necessarily in the same order.

W is kept second from the bottom.

Only three boxes are kept between W and B.

P is kept at one of the positions below W.

Only D is kept between B and S.

E is kept at one of the positions above B.

Between P and E → B, D, S, T, W

Hence, the correct answer is Option 2.

Given: (Left) 3 @ © 6 5 € € 3 3 8 € % 2 7 6 # 7 * £ € 9 $ (Right)

Now,

Symbols, which are immediately preceded by a number and also immediately followed by a number.

So, Needed pattern is: Number - Symbol - Number

(Left) 3 @ © 6 5 € € 3 3 8 € % 2 7 6 # 7 * £ € 9 $ (Right)

So, total valid symbols = 1

Hence, the correct answer is Option 3.

BODMAS Table: 

Decoding the given information:

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

⇒ 16 ÷ 4 − 8 × 3 + 28 = ?

⇒ 4 − 24 + 28 = ?

⇒ 32 − 24 = ?

⇒ 8

Hence, the correct option is "Option 3".

The logic followed here is:

Given number: 1245367

Logic:

Add 1 to each odd digit

Subtract 2 from each even digit

Original number: 1 2 4 5 3 6 7

New number: 2 0 2 6 4 4 8

Digits appearing more than once = 2, 4

Total such digits = 2

Hence, the correct answer is Option 3.

The logic followed here is:

Given:

Mr. Z starts at Point A (0, 0)

13 km South

15 km East (left from South)

27 km North

31 km West

14 km South

So To reach Point A again from P: Move 16 km East

Hence, the correct answer is Option 1.