RRB Group D Official Paper (Held On: 09 Jan, 2026 Shift 1) (Previous Year Paper)

The correct answer is A-2; B-1; C-3; D-4.

Key Points

• Acids and Hydrogen Ions (A-2): According to the Arrhenius theory, an acid is a substance that dissociates in water to yield hydrogen ions (H+). Since H+ ions cannot exist alone, they combine with water molecules to form hydronium ions (H3O+). Hence, A matches with 2 is correct.
• Bases and Hydroxide Ions (B-1): A base is defined as a substance that, when dissolved in water, increases the concentration of hydroxide ions (OH-). Bases that are soluble in water are specifically called alkalis. Hence, B matches with 1 is correct.
• Neutralization Reaction (C-3): When an acid reacts with a base, the H+ ions of the acid react with the OH- ions of the base to form water (H2O) and a salt. This chemical reaction is known as Neutralization and is typically exothermic in nature. Hence, C matches with 3 is correct.
• Electrical Conductivity (D-4): Both acids and bases in their aqueous solutions are capable of conducting electricity. This conductivity is attributed to the presence of free ions (cations and anions) in the solution, which act as charge carriers. Pure distilled water does not conduct electricity, but the addition of an acid or base provides the necessary electrolytes. Hence, D matches with 4 is correct.
• Hence, the correct answer is A-2; B-1; C-3; D-4.

Additional Information

• Characteristics of Acids:
• Acids have a sour taste and turn blue litmus paper red.
• They have a pH value of less than 7 on the pH scale.
• Common examples include Hydrochloric acid (HCl), Sulphuric acid (H2SO4), and Citric acid found in lemons.
• Strong acids like HCl ionize completely in water, while weak acids like acetic acid (vinegar) ionize only partially.
• Characteristics of Bases:
• Bases have a bitter taste and a soapy or slippery feel to the touch.
• They turn red litmus paper blue and have a pH value of greater than 7.
• Common examples include Sodium hydroxide (NaOH), Calcium hydroxide (Ca(OH)2), and Potassium hydroxide (KOH).
• Bases are used extensively in the manufacture of soaps, detergents, and antacids.
• The pH Scale:
• The pH scale was introduced by S.P.L. Sorensen in 1909 to measure the acidity or alkalinity of a solution.
• The scale ranges from 0 to 14, where 7 is neutral (the pH of pure water).
• A lower pH signifies a higher concentration of hydrogen ions, making the solution more acidic.

Important Points

• Indicators: Substances like Phenolphthalein and Methyl Orange are used to test for acids and bases. Phenolphthalein remains colorless in acid but turns pink in basic solutions.
• Universal Indicator: This is a mixture of several indicators that shows different colors at different concentrations of hydrogen ions in a solution, helping determine the approximate pH value.
• Strength of Solutions: The strength of an acid or base depends on the number of H+ ions or OH- ions produced, respectively. For instance, 1 Molar HCl is a stronger acid than 1 Molar CH3COOH because it produces more H+ ions.

The correct answer is Momentum has both magnitude and direction.

Key Points

• Momentum is a fundamental concept in physics, defined as the product of an object's mass and its velocity. It is typically represented by the symbol p.
• Statement 1: Momentum is a vector quantity. By definition, a vector quantity is one that has both a magnitude (how much) and a direction (where it is going). The direction of the momentum vector is the same as the direction of the velocity vector. Hence, statement 1 is correct.
• Statement 2: The Standard International (SI) unit of momentum is the kilogram-meter per second, written as kg·m/s. The statement claims the unit is kilogram per second, which is incorrect as it lacks the meter (distance) component required for velocity. Hence, statement 2 is incorrect.
• Statement 3: Mathematically, momentum is calculated as p = m × v, where m is mass and v is velocity. It is the product of these two variables, not the sum. Hence, statement 3 is incorrect.
• Statement 4: Since momentum is derived from velocity (which is a vector), it must have a direction. Only scalar quantities, such as speed or mass, have magnitude only. Hence, statement 4 is incorrect.

Additional Information

• Conservation of Momentum: This is a major law of physics stating that the total momentum of an isolated system remains constant if no external forces act on it. This principle is vital in analyzing collisions and explosions.
• Newton's Second Law of Motion: Sir Isaac Newton originally defined his second law in terms of momentum. He stated that the force (F) acting on an object is equal to the rate of change of momentum (dp/dt) over time.
• Impulse: The change in momentum is known as impulse. It is the product of the average force and the time interval during which it acts. The units for impulse are Newton-seconds (N·s), which are equivalent to kg·m/s.
• Relativistic Momentum: In Einstein’s theory of special relativity, as an object's speed approaches the speed of light, its momentum increases more rapidly than its velocity, requiring a modified formula that includes the Lorentz factor.

Important Points

• Dimensional Formula: The dimensional representation of momentum is [MLT⁻¹], where M is Mass, L is Length, and T is Time.
• Mass vs. Velocity Influence: A very heavy object (like a ship) moving slowly can have the same momentum as a light object (like a bullet) moving very fast.
• Practical Application: Safety features in vehicles, such as airbags and crumple zones, are designed to increase the time it takes for momentum to change during a crash, thereby reducing the impact force on the passengers.
• CGS Unit: In the Centimetre-Gram-Second system, the unit of momentum is the gram-centimetre per second (g·cm/s).

The correct answer is Fossilisation of plants and animals under high temperature and pressure over millions of years.

Key Points

• Fossil fuels like coal and petroleum are formed through the biological and chemical transformation of ancient organic matter.
• Coal primarily originates from the remains of terrestrial plants and trees that grew in vast swampy forests approximately 300 to 360 million years ago during the Carboniferous Period. When these plants died, they were buried under layers of water and dirt, preventing full decay. Hence, statement 4 is the most accurate explanation.
• Petroleum (crude oil) and natural gas are derived from the remains of microscopic marine organisms, such as plankton and algae. These organisms settled on the ocean floor and were covered by sediments like silt and mud.
• The transformation process requires anaerobic conditions (absence of oxygen), which ensures that the organic matter does not rot completely but undergoes slow chemical changes.
• Over millions of years, the weight of the accumulating sediment layers creates high pressure and generates high temperature from the Earth's interior, which eventually converts the buried organic debris into hydrocarbons.

Additional Information

• Classification of Coal: Based on the degree of compression and carbon content, coal is divided into four main types: Peat (lowest carbon), Lignite (brown coal), Bituminous (most common), and Anthracite (hardest coal with over 85% carbon).
• Petroleum Refining: Crude oil extracted from the ground is a mixture of various hydrocarbons. It is processed in refineries using fractional distillation to produce petrol, diesel, kerosene, asphalt, and liquefied petroleum gas (LPG).
• Non-Renewable Resources: Because the formation of these fuels takes millions of years, they are classified as non-renewable energy sources. Their current rate of consumption far exceeds the rate of natural replenishment.
• Environmental Impact: The combustion of fossil fuels releases carbon dioxide (CO2), sulfur dioxide, and nitrogen oxides, which contribute to the greenhouse effect and global warming.

Important Points

• Carbonisation: The slow chemical process of conversion of dead vegetation into coal is specifically known as carbonisation. This happens under conditions of extreme pressure and lack of air.
• Oil Windows: Petroleum formation typically occurs within a specific temperature range called the "oil window" (usually 50°C to 150°C). If the temperature is higher, the organic matter is further broken down into natural gas (methane).
• Geological Traps: Petroleum is often found trapped in porous reservoir rocks (like sandstone or limestone) situated beneath an impermeable cap rock (like shale or salt) that prevents the oil from escaping to the surface.

The correct answer is Electrolytic reduction of their molten chlorides or oxides.

Key Points

• Metals located at the top of the reactivity series, such as Sodium (Na), Magnesium (Mg), Calcium (Ca), and Aluminium (Al), are highly reactive and never found in nature as free elements.
• These metals cannot be reduced by carbon or carbon monoxide because they have a much higher affinity for oxygen than carbon does. Consequently, carbon cannot pull oxygen away from these metal oxides. Hence, statement 4 is incorrect.
• To extract these highly electropositive metals, the process of electrolytic reduction is employed. This involves passing an electric current through the molten (liquid) state of their chlorides or oxides. Hence, statement 3 is correct.
• During the electrolysis of molten sodium chloride (NaCl), the sodium metal is deposited at the cathode (the negatively charged electrode), while chlorine gas is liberated at the anode (the positively charged electrode).
• In the case of Aluminium, it is extracted from its molten oxide (Alumina - Al2O3) using the Hall-Héroult process, where cryolite is often added to lower the melting point and increase conductivity.

Additional Information

• Roasting:
• This is the process of heating sulphide ores strongly in the presence of excess air to convert them into metal oxides.
• It is typically used for metals of medium reactivity like Zinc (Zn) and Mercury (Hg). For example, Cinnabar (HgS) is roasted to obtain mercury. Hence, statement 1 is incorrect in this context.
• Calcination:
• This involves heating carbonate ores strongly in the absence or limited supply of air to turn them into oxides.
• An example is the calcination of Zinc Carbonate (ZnCO3) to produce Zinc Oxide (ZnO). This method is used for ores found as carbonates in the middle of the reactivity series. Hence, statement 2 is incorrect in this context.
• Reduction with Carbon:
• Metals of moderate reactivity, such as Iron (Fe), Zinc (Zn), Lead (Pb), and Tin (Sn), are extracted by reducing their oxides with coke (carbon) in a furnace.
• In a Blast Furnace, Haematite (Fe2O3) is reduced by carbon/carbon monoxide to produce molten iron.
• Reactivity Series Overview:
• High Reactivity (K, Na, Ca, Mg, Al): Extracted by Electrolysis.
• Medium Reactivity (Zn, Fe, Pb, Cu): Extracted by Reduction using Carbon after roasting/calcination.
• Low Reactivity (Ag, Au): Found in native/free state; Mercury (Hg) is extracted by simple heating of its ore.

Important Points

• The Reactivity Series is an arrangement of metals in descending order of their chemical reactivity. Potassium (K) is at the top, while Gold (Au) is at the bottom.
• Electrolytic refining is a different process used for purifying impure metals like Copper (Cu) and Gold (Au) after they have been extracted by other chemical methods.
• For Aluminium extraction, the molten mixture of Alumina and Cryolite (Na3AlF6) is electrolyzed at approximately 950°C to 1000°C.
• The choice of extraction method is purely determined by the position of the metal in the activity series and the chemical nature of its ore.

The correct answer is It has a focal length of 1 metre..

Key Points

• The power of a lens is defined as the ability of a lens to converge or diverge light rays. Mathematically, it is the reciprocal of its focal length.
• The SI unit of power of a lens is the dioptre, which is denoted by the symbol D.
• The relationship between power (P) and focal length (f) is given by the formula P = 1/f, where the focal length must be measured in metres.
• By definition, 1 dioptre is the power of a lens whose focal length is 1 metre (1 D = 1 m⁻¹). Hence, the statement in the question indicates a focal length of 1 metre. Hence, the correct answer is option 2.
• The power of a lens is positive for a converging (convex) lens and negative for a diverging (concave) lens.

Additional Information

• Focal Length (f):
• It is the distance between the optical centre of the lens and the principal focus.
• If the focal length is given in centimetres, the power formula is modified to P = 100/f (in cm). For example, a lens with a focal length of 25 cm has a power of +4 D.
• Sign Convention and Lens Types:
• A convex lens has a real focus and thus a positive focal length, leading to a positive power.
• A concave lens has a virtual focus and a negative focal length, resulting in a negative power.
• Power of a Combination of Lenses:
• When two or more thin lenses are placed in contact, the total power (P) of the system is the algebraic sum of the individual powers: P = P1 + P2 + P3 + ....
• This principle is used by optometrists to design corrective spectacles and by engineers in telescopes and microscope objectives.
• Magnification (m):
• Unlike power, magnification is the ratio of the height of the image to the height of the object (m = h'/h) or the ratio of image distance to object distance (m = v/u).
• It is a unitless quantity, whereas power has the unit of m⁻¹ or dioptre.

Important Points

• Corrective Lenses: A person suffering from Myopia (nearsightedness) is prescribed a lens with negative power (concave), while someone with Hypermetropia (farsightedness) requires a lens with positive power (convex).
• Optical Instruments: The use of dioptre helps in the standardized manufacturing of eye-glasses and contact lenses globally. The least distance of distinct vision for a normal human eye is approximately 25 cm.
• Refractive Index: The power of a lens also depends on the refractive index of the material it is made of and the radii of curvature of its surfaces, as described by the Lens Maker's Formula.

The correct answer is P = I × R².

Key Points

• Electric Power (P) is defined as the rate at which electrical energy is consumed or dissipated in an electric circuit. It is the rate of doing work.
• The fundamental expression for Electric Power is the product of Potential Difference (V) and Current (I), represented as P = V × I. Hence, statement 1 is correct.
• According to Ohm's Law, the relationship between voltage, current, and resistance is given by V = I × R. By substituting V in the power formula (P = VI), we get P = (I × R) × I, which results in P = I² × R. Hence, statement 2 is correct.
• Using Ohm's Law again, we can express current as I = V / R. By substituting I in the fundamental power formula (P = VI), we get P = V × (V / R), which results in P = V² / R. Hence, statement 4 is correct.
• The expression P = I × R² does not follow from any physical derivation of electrical laws and is dimensionally inconsistent with the units of power. Hence, statement 3 is incorrect.
• Therefore, P = I × R² is NOT a correct expression for electric power.

Additional Information

• SI Unit of Power: The SI unit of electric power is the Watt (W). It is named after the Scottish inventor James Watt. 1 Watt is defined as the power consumed by a device when 1 Ampere of current flows through it at a potential difference of 1 Volt.
• Ohm's Law: Proposed by Georg Simon Ohm in 1827, it states that the current flowing through a conductor is directly proportional to the voltage across it, provided physical conditions like temperature remain constant (V = IR).
• Components of the Formula:
• V stands for Voltage or Potential Difference, measured in Volts (V).
• I stands for Electric Current, measured in Amperes (A).
• R stands for Resistance, measured in Ohms (Ω).
• Commercial Unit of Energy: While the SI unit of energy is the Joule (J), the commercial unit is the Kilowatt-hour (kWh). 1 kWh is the energy consumed by an appliance of 1000 Watts in one hour. Numerically, 1 kWh = 3.6 × 10⁶ Joules.

Important Points

• Joule's Law of Heating: This law states that the heat produced in a resistor is directly proportional to the square of the current, the resistance, and the time for which the current flows (H = I²Rt). This is directly related to the power formula P = I²R.
• Power in Series vs. Parallel:
• In a series circuit, the current (I) remains constant. Thus, P = I²R is more convenient for calculations, showing power is directly proportional to resistance.
• In a parallel circuit, the voltage (V) remains constant. Thus, P = V² / R is more convenient, showing power is inversely proportional to resistance.
• Practical Applications: These formulas are used to determine the power rating of household appliances like bulbs, heaters, and irons. For example, a bulb rated 100W, 220V uses these relations to define its operational resistance.

The correct answer is negative.

Key Points

• The work done by a force on an object is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement.
• Mathematically, work is expressed as W = F × d × cosθ, where F is the force magnitude, d is the displacement, and θ is the angle between the force vector and the displacement vector.
• Friction is a force that resists the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.
• When an object moves on a surface, kinetic friction acts in a direction opposite to the direction of the object's displacement.
• Because the force of friction and the displacement are in opposite directions, the angle θ is 180 degrees.
• Since the cosine of 180 degrees is -1, the resulting value of work is negative. Hence, the correct answer is negative.

Additional Information

• Types of Work:
• Positive Work: Occurs when the force and displacement are in the same direction (angle between 0 and 90 degrees). For example, work done by a person pulling a cart.
• Negative Work: Occurs when the force and displacement are in opposite directions (angle between 90 and 180 degrees). Friction is the most common example.
• Zero Work: Occurs when the force is perpendicular to the displacement (angle is 90 degrees) or if the displacement is zero. For instance, the centripetal force acting on a satellite does zero work because it is perpendicular to the motion.
• Nature of Friction:
• Friction is a non-conservative force, meaning the work done against it depends on the path taken rather than just the initial and final positions.
• The coefficient of friction (represented by the Greek letter μ) is a dimensionless scalar value that describes the ratio of the force of friction between two bodies and the normal force pressing them together.
• In almost all practical mechanical systems, friction converts kinetic energy into thermal energy (heat), leading to energy dissipation.

Important Points

• While work done by friction is usually negative, it can be positive in certain frames of reference or specific scenarios. For example, if a block A is placed on top of block B and block B is accelerated, the friction exerted by B on A causes A to move. In this case, friction does positive work on block A because the force and displacement are in the same direction.
• The S.I. unit of work is the Joule (J), which is equivalent to 1 Newton-meter (N·m).
• The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy. Since friction usually does negative work, it typically decreases the kinetic energy of a moving object, causing it to slow down.

The correct answer is Synapse.

Key Points

• A Synapse is a functional junction between two neurons (nerve cells) or between a neuron and an effector organ, such as a muscle or gland.
• The actual physical gap between the membranes of the two communicating neurons is specifically known as the synaptic cleft, which typically measures about 20 to 40 nanometers in width.
• Information is transmitted across this gap through the release of signaling molecules called neurotransmitters, which bridge the distance between the presynaptic and postsynaptic cells.
• The concept of the synapse was first introduced and named by the Nobel Prize-winning physiologist Sir Charles Sherrington in 1897.
• This structure is vital for the unidirectional flow of nerve impulses, ensuring that signals travel from the axon terminal of one neuron toward the dendrites or cell body of the next.

Additional Information

• Chemical Synapses:
• These are the most common type of synapses in the human nervous system.
• When an electrical impulse (action potential) reaches the end of an axon, it triggers the influx of calcium ions.
• This causes synaptic vesicles to fuse with the membrane and release neurotransmitters like acetylcholine, dopamine, or serotonin into the cleft.
• Electrical Synapses:
• In these synapses, the membranes of the two neurons are very close together, connected by special proteins called gap junctions.
• Electrical current flows directly from one cell to another, making transmission much faster than chemical synapses. They are common in the heart and smooth muscle where synchronized contraction is required.
• Synaptic Plasticity:
• This refers to the ability of synapses to strengthen or weaken over time in response to increases or decreases in their activity.
• It is the fundamental biological basis for learning and memory in the brain.

Important Points

• Stroma: In biology, this term refers to the supportive, connective tissue framework of an organ (like the ovary or kidney) or the fluid-filled space surrounding the thylakoids inside a chloroplast. Hence, option 1 is incorrect.
• Intermembrane space: This is the region between the inner and outer membranes of a double-membrane organelle, such as mitochondria or chloroplasts. It plays a key role in oxidative phosphorylation. Hence, option 3 is incorrect.
• Lumen: This refers to the inside space of a tubular structure or a hollow organ, such as the interior of an artery, the small intestine, or the endoplasmic reticulum. Hence, option 4 is incorrect.
• A single neuron can have anywhere from 1,000 to 10,000 synapses, allowing for a massive network of trillions of connections within the human brain.

The correct answer is capillaries; gases.

Key Points

• The alveoli are the functional units of the respiratory system where the actual process of gas exchange occurs between the lungs and the blood.
• Each alveolus is wrapped in a dense, mesh-like network of capillaries, which are the smallest and thinnest blood vessels in the human body.
• The primary function of this structural arrangement is to facilitate the exchange of gases, specifically oxygen (O2) and carbon dioxide (CO2), through the process of diffusion.
• The walls of both the alveoli and the capillaries are composed of a single layer of epithelial cells, making them extremely thin (about 0.5 micrometers) to allow gases to pass through easily.
• During inhalation, oxygen moves from the alveoli into the capillaries to be carried to the rest of the body, while carbon dioxide moves from the capillaries into the alveoli to be exhaled.

Additional Information

• Surface Area for Gas Exchange: The human lungs contain approximately 300 to 500 million alveoli. If spread out, they would cover a surface area of about 70 to 100 square meters, which is roughly the size of a tennis court.
• Diffusion Gradient: The movement of gases is driven by partial pressure differences. Oxygen pressure is higher in the alveoli than in the deoxygenated blood within the capillaries, causing it to flow inward.
• Role of Hemoglobin: Once oxygen diffuses into the capillaries, it binds to hemoglobin molecules inside red blood cells (RBCs) for transport to various tissues and organs.
• Blood Flow Direction: Deoxygenated blood reaches the capillaries via the pulmonary artery from the right ventricle of the heart, and oxygenated blood returns to the left atrium via the pulmonary vein.
• Respiratory Membrane: The combination of the alveolar wall and the capillary wall is often referred to as the respiratory membrane or the blood-air barrier.

Important Points

• Pulmonary Surfactant: The alveoli are lined with a lipoprotein substance called surfactant, which reduces surface tension and prevents the tiny air sacs from collapsing during exhalation.
• Mechanism of Action: The exchange of gases is a purely passive process, meaning it does not require cellular energy (ATP) to occur.
• Effect of Diseases: Conditions such as emphysema (often caused by smoking) lead to the destruction of alveolar walls, significantly reducing the surface area available for gas exchange and causing shortness of breath.
• Capillary Transit Time: Under normal resting conditions, a red blood cell spends about 0.75 seconds in the pulmonary capillaries, which is more than enough time for oxygen saturation to reach 100%.

The correct answer is TT × tt.

Key Points

• A cross between TT (homozygous dominant) and tt (homozygous recessive) results in all offspring having the genotype Tt. This means 100% of the F1 generation will be heterozygous. Hence, statement 3 is correct.
• In the cross tt × tt, both parents are homozygous recessive. They can only pass the t allele to their offspring, resulting in 100% homozygous recessive (tt) individuals. Hence, statement 1 is incorrect.
• In the cross TT × TT, both parents are homozygous dominant. They can only pass the T allele to their offspring, resulting in 100% homozygous dominant (TT) individuals. Hence, statement 2 is incorrect.
• In the cross Tt × Tt, both parents are heterozygous. According to Mendelian genetics, this cross produces a genotypic ratio of 1:2:1 (25% TT, 50% Tt, and 25% tt). Since only 50% are heterozygous, they are not "all" heterozygous. Hence, statement 4 is incorrect.
• This principle was first established by Gregor Mendel in his 1865 experiments with pea plants (Pisum sativum), forming the basis of the Law of Segregation.

Additional Information

• Heterozygous: This refers to an individual having two different alleles for a particular gene, such as Tt. In such cases, the dominant allele (T) typically masks the expression of the recessive allele (t).
• Homozygous: This term describes an organism that has two identical alleles for a trait, which can be either homozygous dominant (TT) or homozygous recessive (tt).
• Genotype and Phenotype: The genotype is the actual genetic makeup (e.g., Tt), while the phenotype is the observable physical characteristic (e.g., Tall height). In an F1 generation of Tt, the phenotype is usually 100% dominant.
• Punnett Square: This is a graphical representation used to calculate the probability of an offspring having a particular genotype. It was devised by Reginald Punnett in 1905.
• Monohybrid Cross: This is a genetic cross between two individuals involving one pair of contrasting characters. The cross of TT × tt is a classic example of a monohybrid cross looking at plant height.

Important Points

• Law of Dominance: Gregor Mendel observed that in the F1 generation, only one of the parental traits (the dominant one) appears. For the cross TT × tt, although the offspring are heterozygous, they all appear Tall.
• Pure-breeding Strains: The organisms TT and tt are considered pure-breeding or true-breeding because when self-pollinated, they always produce offspring with the same trait. Crossing two different true-breeding parents is the only way to ensure 100% heterozygosity in the first generation.
• Numerical Ratios: In the F2 generation (produced by crossing Tt × Tt), the phenotypic ratio is 3:1, while the genotypic ratio is 1:2:1.

The correct answer is Sound travels slower than light.

Key Points

• The fundamental reason we see lightning before hearing thunder is the vast difference in the speed at which light and sound travel through the atmosphere.
• Light travels at an incredible speed of approximately 300,000,000 meters per second (or 3 x 10^8 m/s) in a vacuum and slightly slower in air. This speed is so high that for terrestrial distances, light appears to travel instantaneously.
• Sound, on the other hand, travels much slower, at about 343 meters per second in dry air at a temperature of 20 degrees Celsius (68 degrees Fahrenheit).
• When a lightning strike occurs, both the optical flash and the acoustic shockwave (thunder) are created at the exact same moment. However, because light is nearly 874,000 times faster than sound, the visual information reaches the observer almost immediately, while the sound waves take several seconds to cover the same distance. Hence, the correct answer is Sound travels slower than light.

Additional Information

• Nature of Light:
• Light is an electromagnetic wave that does not require a medium to propagate. It can travel through the vacuum of outer space.
• In the context of a storm, the photons emitted by the electrical discharge of lightning move through the air with minimal resistance to their speed.
• Nature of Sound:
• Sound is a mechanical wave (specifically a longitudinal wave) that requires a medium like air, water, or solids to travel.
• The speed of sound is highly dependent on the density and temperature of the medium. For instance, sound travels faster in water and even faster in solids like steel compared to its speed in air.
• Distance Estimation:
• Observers can estimate the distance of a storm using the "flash-to-bang" method. Since sound travels about 1 kilometer every 3 seconds (or 1 mile every 5 seconds), counting the delay between the flash and the thunder allows one to calculate how far away the lightning struck.
• If the delay is 9 seconds, the lightning strike is approximately 3 kilometers away.

Important Points

• Temperature Effect: The speed of sound increases as the temperature of the air increases. At 0 degrees Celsius, the speed of sound is about 331 m/s, whereas at 30 degrees Celsius, it increases to approximately 349 m/s.
• Sonic Boom: When an object, such as a supersonic jet, travels faster than the speed of sound (Mach 1), it creates a shockwave similar to thunder, known as a sonic boom.
• Lightning Composition: A lightning bolt can heat the surrounding air to 30,000 Kelvin (five times hotter than the surface of the sun). This sudden thermal expansion of air is what generates the sound wave we call thunder.
• Refraction: Sometimes, lightning can be seen but thunder is not heard; this is often called "heat lightning." This occurs because the sound waves are refracted (bent) upward by different temperature layers in the atmosphere before they can reach the observer's ears.

The correct answer is Filtration.

Key Points

• Filtration is a physical method used to separate insoluble solid particles from a liquid or gas by passing the mixture through a porous medium, such as filter paper or a membrane. Since the particles in a suspension are large enough to be trapped, this method is highly effective. Hence, option 2 is correct.
• A suspension is a heterogeneous mixture containing solid particles large enough for sedimentation. These particles are usually larger than 1000 nanometers and are visible to the naked eye.
• In the filtration process, the clear liquid that passes through the filter is known as the filtrate, while the solid particles that remain on the filter medium are referred to as the residue.
• Crystallization is a technique used to separate a pure solid from a solution in the form of its crystals. It is commonly used for purification rather than simple mechanical separation of a suspension. Hence, option 1 is incorrect.
• Distillation is used to separate components of a mixture containing miscible liquids that boil without decomposition and have a significant difference in their boiling points (usually more than 25 K). Hence, option 3 is incorrect.
• Evaporation is a process used to separate a non-volatile solute (like salt) from a volatile solvent (like water) by heating the mixture until the liquid turns into vapor. Hence, option 4 is incorrect.

Additional Information

• Types of Mixtures:
• Solution: A homogeneous mixture of two or more substances where the particle size is less than 1 nm. They do not scatter light and cannot be separated by filtration.
• Colloid: A mixture where the particle size (between 1 nm and 1000 nm) is intermediate between a solution and a suspension. While they appear homogeneous, they are heterogeneous and exhibit the Tyndall effect.
• Suspension: A heterogeneous mixture where particles eventually settle down due to gravity if left undisturbed, making the mixture unstable.
• Other Separation Techniques:
• Centrifugation: Used when the solid particles are too small to be caught by a filter. The mixture is spun rapidly, forcing denser particles to the bottom. Examples include separating cream from milk or diagnostic blood tests.
• Chromatography: A technique used to separate solutes that dissolve in the same solvent, often used to separate pigments from natural colors or drugs from blood.
• Sublimation: Used to separate mixtures containing a sublimable volatile component (like Ammonium chloride or Camphor) from a non-sublimable impurity.

Important Points

• The Tyndall Effect is the scattering of a beam of light by medium particles. It is clearly visible in suspensions and colloids but not in true solutions.
• The effectiveness of Filtration depends on the pore size of the filter medium and the particle size of the suspended matter.
• Unlike Chemical separation, methods like Filtration, Evaporation, and Distillation are Physical separation methods because they do not alter the chemical identity of the substances involved.
• Common examples of Suspensions include Chalk powder in water, Muddy water, and Flour in water.

The correct answer is The density of the fluid.

Key Points

• The buoyant force, also known as upthrust, is the upward force exerted by a fluid that opposes the weight of a partially or fully immersed object.
• According to Archimedes' Principle, the magnitude of the buoyant force is exactly equal to the weight of the fluid displaced by the object.
• The formula for buoyant force is Fb = ρ × V × g, where ρ (rho) represents the density of the fluid, V represents the volume of the displaced fluid, and g represents the acceleration due to gravity (approximately 9.8 m/s²).
• The density of the fluid is a critical factor; for a fixed volume, a denser fluid (like saltwater compared to freshwater) will exert a greater upward force. Hence, option 1 is correct.
• The weight of the object does not determine the magnitude of the buoyant force, though it determines whether the object will sink or float when compared to the buoyant force. Hence, option 2 is incorrect.
• The color of the object is a physical appearance property and has no physical interaction with fluid mechanics or pressure differentials. Hence, option 3 is incorrect.
• The shape of the object does not affect the buoyant force directly; only the total volume of the displaced fluid matters. Two objects of different shapes but the same volume will experience the same buoyant force if fully submerged. Hence, option 4 is incorrect.

Additional Information

• Archimedes' Principle: Discovered in the 3^rd Century BC by the Greek mathematician Archimedes of Syracuse, this principle is the fundamental law of buoyancy used in ship design and submarines.
• Sinking and Floating: An object sinks if its density is greater than the density of the fluid. It floats if its density is less than the fluid's density. If the densities are equal, it exhibits neutral buoyancy.
• Fluid Pressure: Buoyancy arises because fluid pressure increases with depth. The pressure on the bottom of an submerged object is higher than the pressure on the top, resulting in a net upward force.
• Real-world Example: In the Dead Sea, the extremely high salt concentration increases the density of the water to about 1240 kg/m³. This high density creates a massive buoyant force, allowing humans to float effortlessly on the surface.
• Applications: This concept is used in Hydrometers (to measure fluid density), Lactometers (to check the purity of milk), and the operation of Hot Air Balloons (where hot air is less dense than the surrounding cool air).

Important Points

• Apparent Weight: When an object is immersed in a fluid, it appears to lose weight. The apparent weight is calculated as Actual Weight - Buoyant Force.
• Volume Displacement: For a fully submerged object, the volume of displaced fluid is equal to the total volume of the object. For a floating object, it is equal to the volume of the submerged portion only.
• Relative Density: Also known as Specific Gravity, it is the ratio of the density of a substance to the density of a reference substance (usually water at 4 degrees Celsius, which is 1000 kg/m³).
• Center of Buoyancy: This is the centroid of the displaced volume of fluid. For a floating body to be stable, the relative positions of the Center of Gravity and the Center of Buoyancy are crucial.

The correct answer is Pure chemical compounds.

Key Points

• The Law of Constant Proportions, also frequently referred to as the Law of Definite Proportions, is a fundamental law of chemistry which states that a pure chemical compound always contains the same elements combined together in the same fixed proportion by mass.
• This law was first proposed by the French chemist Joseph Proust in 1799. His observations were based on the analysis of various compounds, including cupric carbonate, where he found the composition of elements to be the same regardless of whether the sample was natural or synthetic.
• It applies exclusively to pure chemical compounds because they are formed through chemical bonds between atoms in specific stoichiometric ratios. Hence, option 1 is correct.
• Elements are substances consisting of only one type of atom. The law describes how different elements interact to form compounds; it does not apply to a single element in isolation. Hence, option 2 is incorrect.
• Mixtures do not follow this law because their components can be mixed in any arbitrary ratio. For example, a saltwater solution can be prepared with 1 gram of salt or 10 grams of salt in the same volume of water without changing its fundamental nature as a mixture. Hence, options 3 and 4 are incorrect.
• A classic example is Water (H2O). In any sample of pure water, the ratio of the mass of Hydrogen to the mass of Oxygen is always 1:8 (approximately 11.1% Hydrogen and 88.9% Oxygen by mass).

Additional Information

• Law of Conservation of Mass:
• Established by Antoine Lavoisier in 1789.
• It states that mass is neither created nor destroyed during a chemical reaction.
• The total mass of the reactants must equal the total mass of the products in a closed system.
• Law of Multiple Proportions:
• Proposed by John Dalton in 1803.
• If two elements combine to form more than one compound, the masses of one element that combine with a fixed mass of the other element are in a ratio of small whole numbers.
• Example: In Carbon Monoxide (CO) and Carbon Dioxide (CO2), the masses of Oxygen that combine with 12g of Carbon are 16g and 32g, respectively, forming a 1:2 ratio.
• Dalton’s Atomic Theory:
• Published in 1808, this theory provided the first theoretical framework for the laws of chemical combination.
• Dalton postulated that atoms of different elements combine in simple whole-number ratios to form chemical compounds.
• Exceptions to the Law:
• Non-stoichiometric compounds (also known as berthollides), such as Wustite (FeO), where the ratio of elements can vary within a certain range.
• Isotopic variations: Since isotopes of the same element have different masses, compounds made with different isotopes (e.g., Heavy Water, D2O vs. Normal Water, H2O) will have different mass ratios.

Important Points

• Compounds are distinct from mixtures because compounds have fixed properties, specific boiling/melting points, and their constituents cannot be separated by physical methods like filtration or evaporation.
• The percentage composition of a compound is calculated by dividing the molar mass of the element by the total molecular mass of the compound and multiplying by 100.
• In Ammonia (NH3), Nitrogen and Hydrogen are always present in a mass ratio of 14:3.
• The discovery of this law was a major step toward the atomic theory of matter, as it suggested that matter is composed of discrete units that combine in specific ways.

The correct answer is Plasmolysis observed when living plant cells lose water.

Key Points

• Plasmolysis is the process in which cells lose water in a hypertonic solution. When a living plant cell loses water through osmosis, there is shrinkage or contraction of the contents of the cell away from the cell wall.
• This phenomenon specifically demonstrates the role and presence of the cell wall, as the rigid outer boundary remains intact while the protoplasm (the living part of the cell) shrinks. Hence, statement 4 is correct.
• The cell wall is a rigid, non-living outer layer found in plant cells, fungi, and bacteria. It provides structural support and protection, and its interaction with the plasma membrane during water loss is what makes plasmolysis visible.
• Exchange of gases through stomata of leaves primarily involves the process of diffusion and transpiration. While turgor pressure (regulated by osmosis) opens and closes stomata, it is not the primary phenomenon used to describe the structural role of the cell wall in osmosis compared to plasmolysis. Hence, statement 1 is incorrect.
• Diffusion of gases through the cell membrane is a passive process that occurs based on concentration gradients and does not specifically highlight the mechanical or structural role of the cell wall. Hence, statement 2 is incorrect.
• The absorption of light by chlorophyll is a photochemical process central to photosynthesis and is unrelated to the osmotic movement of water or the structural function of the cell wall. Hence, statement 3 is incorrect.

Additional Information

• Osmosis in Plant Cells:
• Osmosis is the movement of water molecules from a region of higher water concentration to a region of lower water concentration through a semi-permeable membrane.
• In plant cells, the cell wall prevents the cell from bursting when it takes up water (endosmosis) in a hypotonic solution. The pressure exerted by the swelling protoplast against the wall is called turgor pressure.
• Types of Osmotic Solutions:
• Hypertonic Solution: A solution with a higher solute concentration than the cell. Water moves out, leading to plasmolysis.
• Hypotonic Solution: A solution with a lower solute concentration than the cell. Water moves in, making the cell turgid.
• Isotonic Solution: A solution where the solute concentration is equal to the cell. There is no net movement of water, and the cell becomes flaccid.
• The Cell Wall:
• In plants, the cell wall is composed mainly of cellulose, hemicellulose, and pectin.
• It is fully permeable to water and substances in solution, unlike the plasma membrane, which is selectively permeable.
• Its primary function is to provide mechanical strength and maintain the shape of the plant, allowing it to grow to great heights.

Important Points

• Significance of Plasmolysis:
• It proves that the cell membrane is a semi-permeable boundary.
• It is a reversible process; if a plasmolysed cell is placed in a hypotonic solution (like pure water) before permanent damage occurs, water enters the cell, and the protoplasm again touches the cell wall. This reverse process is called deplasmolysis.
• Real-world applications include the salting of meat or addition of sugar to jams, which creates a hypertonic environment that causes plasmolysis in bacteria, thereby preventing spoilage.

The correct answer is The highest concentration of pesticides.

Key Points

• Biological Magnification, also known as Biomagnification, is the process where the concentration of non-biodegradable toxic substances increases at each successive trophic level of a food chain.
• In the provided food chain: Grass → Grasshopper → Frog → Snake → Hawk, the Hawk occupies the fifth trophic level as the top predator.
• Since pesticides are often non-biodegradable, they are not broken down by the metabolic processes of organisms nor are they excreted easily. Instead, they accumulate in the fatty tissues of the organisms.
• As a predator eats multiple prey items from the level below it, it ingests all the accumulated toxins from those individuals. Consequently, the organism at the highest trophic level always ends up with the maximum concentration of these chemicals.
• Therefore, in this specific chain, the Hawk will exhibit the highest concentration of pesticides compared to the grass, grasshopper, frog, or snake. Hence, the correct answer is option 3.

Additional Information

• Trophic Levels:
• First Trophic Level: Consists of Producers like Grass, which have the lowest concentration of toxins as they are the entry point into the food web.
• Intermediate Levels: Includes Primary Consumers (Grasshopper), Secondary Consumers (Frog), and Tertiary Consumers (Snake). The concentration increases step-by-step through these levels.
• Quaternary Consumers: Represented here by the Hawk, which acts as the apex predator in this sequence.
• Energy Flow vs. Biomagnification:
• According to the 10% Law proposed by Raymond Lindeman in 1942, only 10% of the energy is transferred to the next trophic level, while 90% is lost as heat.
• Contrarily, non-biodegradable chemicals do not follow this reduction; their total mass remains largely the same while the biomass of the higher level is smaller, leading to a much higher concentration.
• Common Pollutants:
• DDT (Dichlorodiphenyltrichloroethane): One of the most famous examples of a pesticide that causes Biomagnification. It was widely used in the 20^th century before being banned in many countries.
• Heavy Metals: Elements like Mercury, Lead, and Cadmium also undergo this process, often leading to conditions like Minamata disease in humans who consume high-level predatory fish.
• Ecological Impact:
• High concentrations of pesticides in birds of prey (like Hawks and Eagles) interfere with calcium metabolism, causing the thinning of eggshells. This leads to premature breaking of eggs and a significant decline in bird populations.

Important Points

• Biodegradable vs. Non-biodegradable: Only non-biodegradable substances (those that cannot be decomposed by biological processes) undergo Biomagnification. Biodegradable substances are broken down and do not accumulate.
• Bioaccumulation: This refers to the build-up of a substance in a single organism's body over time, whereas Biomagnification refers to the increase in concentration across multiple levels of the food chain.
• Human Impact: Humans, being omnivores and often occupying the top of the food chain, are highly susceptible to the effects of biological magnification through the consumption of meat, fish, and dairy.

The correct answer is X = V/I.

Key Points

• Ohm's Law, discovered by the German physicist Georg Simon Ohm in 1827, provides the fundamental relationship between voltage, current, and resistance in an electrical circuit.
• According to this law, the potential difference (V) across the ends of a given metallic wire in an electric circuit is directly proportional to the current (I) flowing through it, provided its temperature remains constant.
• Mathematically, this relationship is expressed as V = I × R, where R is a constant for the given metallic wire at a given temperature and is called its resistance.
• In the given question, the resistance is denoted by X. Therefore, the equation becomes V = I × X.
• To find the relation for X, we rearrange the formula as X = V / I. Hence, option 1 is correct.
• The relationship is linear, meaning if the potential difference is doubled, the current also doubles, keeping the ratio V/I (the resistance) constant.

Additional Information

• Resistance (X):
• It is the property of a conductor to oppose the flow of electric current.
• The SI unit of resistance is the Ohm, denoted by the Greek symbol Ω.
• 1 Ohm is defined as the resistance of a conductor when a potential difference of 1 Volt applied across its ends results in a current of 1 Ampere flowing through it.
• Factors Affecting Resistance:
• The resistance of a uniform metallic conductor is directly proportional to its length (L).
• It is inversely proportional to the area of cross-section (A).
• It depends on the nature of the material (resistivity) and the temperature of the conductor. For most metals, resistance increases with an increase in temperature.
• Potential Difference (V):
• It is the work done to move a unit charge from one point to another in an electric field. The SI unit is the Volt.
• Electric Current (I):
• It is the rate of flow of electric charges through a cross-section of a conductor. The SI unit is the Ampere.

Important Points

• V-I Graph: For a conductor that obeys Ohm's Law (an Ohmic conductor), the graph of potential difference (V) versus current (I) is a straight line passing through the origin. The slope of this line represents the resistance of the conductor.
• Ohmic vs. Non-Ohmic: Materials like metals are generally Ohmic at stable temperatures. However, components like diodes, transistors, and vacuum tubes do not follow a linear relationship and are called Non-Ohmic conductors.
• Application of the Formula: The formula V = IX is crucial for calculating the necessary voltage or current in electrical circuits, designing household wiring, and determining the power consumption of appliances.
• Formula Variations:
• To find Current: I = V / X
• To find Voltage: V = I × X
• To find Resistance: X = V / I

The correct answer is refraction and dispersion of light by raindrops.

Key Points

• A rainbow is a meteorological phenomenon that is caused by reflection, refraction, and dispersion of light in water droplets resulting in a spectrum of light appearing in the sky.
• When sunlight strikes a raindrop, it is first refracted as it enters the surface of the droplet. This causes the white light to slow down and bend.
• During this process, dispersion occurs, where the white sunlight is split into its component colors (VIBGYOR) because different wavelengths of light travel at different speeds within the water.
• The light then hits the back of the raindrop and undergoes internal reflection. After reflecting, the light is refracted once more as it exits the droplet, further spreading the colors.
• The combined effect of these optical processes—refraction, dispersion, and reflection—directs the light toward the observer's eye at specific angles (approximately 40 to 42 degrees), forming a circular arc of colors. Hence, the correct answer is refraction and dispersion of light by raindrops.

Additional Information

• The Visible Spectrum (VIBGYOR): The colors in a rainbow always appear in a specific order: Violet, Indigo, Blue, Green, Yellow, Orange, and Red. Red light has the longest wavelength and is refracted the least, appearing on the outer arc. Violet light has the shortest wavelength and is refracted the most, appearing on the inner arc.
• Conditions for Formation: For a rainbow to be visible, two conditions must be met simultaneously: the Sun must be shining in one part of the sky, and it must be raining (or water droplets must be present) in the opposite part of the sky. The observer must stand with their back to the Sun.
• Primary vs. Secondary Rainbows: A Primary Rainbow is formed by one internal reflection inside the raindrop. A Secondary Rainbow is formed when light undergoes two internal reflections. The secondary rainbow is fainter, appears outside the primary arc, and its color sequence is reversed (Red on the inside, Violet on the outside).
• Shape of a Rainbow: While we usually see a rainbow as an arc from the ground, it is actually a full circle. The ground usually obstructs the bottom half of the circle. However, observers in airplanes or at high altitudes can sometimes see a complete circular rainbow.
• Historical Context: Sir Isaac Newton was the first to demonstrate in the 17^th century (specifically around 1666) that white light is composed of a spectrum of colors by using a glass prism, which mimics the dispersive behavior of raindrops.

Important Points

• Refraction: This is the bending of light when it passes from one medium (like air) to another medium of different density (like water). The refractive index of water is approximately 1.33.
• Dispersion: This is the separation of white light into colors of different frequencies and wavelengths. This happens because the velocity of light in a medium depends on its wavelength.
• Total Internal Reflection: While a portion of light undergoes internal reflection at the back of the raindrop to form a rainbow, it is important to note that it is often technically partial reflection, though it is the primary reason the light returns toward the observer.

The correct answer is maximum, intermediate, minimum.

Key Points

• The strength of the intermolecular forces of attraction varies significantly across the three primary states of matter: solids, liquids, and gases.
• In solids, the particles are packed very closely together in a rigid, often crystalline structure. Because the inter-particle distance is very small, the forces of attraction between them are the maximum. This high attraction prevents the particles from moving out of their positions, resulting in a fixed shape and fixed volume.
• In liquids, the particles are slightly further apart compared to solids. The forces of attraction are intermediate; they are strong enough to keep the particles together (maintaining a fixed volume) but weak enough to allow them to slide past each other. This gives liquids the ability to flow and take the shape of their container.
• In gases, the particles are separated by very large distances relative to their size. The intermolecular forces are the minimum (almost negligible), which allows the particles to move independently and randomly in all directions. Consequently, gases have neither a fixed shape nor a fixed volume.
• The order of the force of attraction between particles is Solids > Liquids > Gases.

Additional Information

• Kinetic Energy and Temperature:
• The kinetic energy of particles is inversely related to the strength of attractive forces. In gases, the kinetic energy is maximum, which overcomes the attractive forces. In solids, kinetic energy is minimum, allowing attractive forces to dominate.
• As temperature increases, the kinetic energy of the particles increases, which can eventually overcome the attractive forces, leading to a change in state (e.g., melting or boiling).
• Compressibility:
• Gases are highly compressible because the intermolecular spaces are very large and the forces of attraction are minimum.
• Solids and liquids have very low compressibility because their particles are already very close due to maximum or intermediate forces.
• Diffusion:
• The rate of diffusion is fastest in gases because of the minimum force of attraction and high velocity of particles.
• In solids, diffusion is almost non-existent because the maximum force of attraction keeps particles locked in fixed positions.
• Density:
• Typically, solids have the highest density because the maximum forces of attraction pull the maximum number of particles into a unit volume. Gases have the lowest density due to the minimum forces.

Important Points

• The intermolecular forces mentioned are also known as Van der Waals forces or dipole-dipole interactions depending on the nature of the substance.
• The thermal energy of a substance works in opposition to the intermolecular forces. While attractive forces try to keep particles together, thermal energy tries to keep them apart by increasing their vibration or translation.
• The Inter-particle space follows an order exactly opposite to the forces of attraction: Gases (maximum) > Liquids (intermediate) > Solids (minimum).

The correct answer is Both Assertion (A) and Reason (R) are true, and (R) is the correct explanation of Assertion (A).

Key Points

• Assertion (A) states that sexual reproduction is essential in plants for maintaining the continuity of the species for many generations. Reproduction is a fundamental biological process that ensures life continues beyond the lifespan of an individual organism. Hence, Assertion (A) is correct.
• Reason (R) highlights that sexual reproduction leads to genetic variation, which enhances adaptability and evolutionary fitness. In this process, offspring inherit a unique combination of DNA from two different gametes. Hence, Reason (R) is correct.
• The genetic diversity created through sexual reproduction is the primary mechanism that allows a plant population to survive environmental changes, such as climate shifts, new pathogens, or predators.
• Because these variations allow the species to adapt and evolve over geological time, it directly enables the long-term continuity mentioned in the assertion. Without such variation, a species would be highly vulnerable to extinction. Therefore, Reason (R) is the correct explanation of Assertion (A).

Additional Information

• Mechanism of Variation: During sexual reproduction, variation arises primarily during meiosis through processes like crossing over (in Prophase I) and independent assortment of chromosomes. This reshuffles the genetic material, ensuring no two offspring (except identical twins) are exactly alike.
• Evolutionary Fitness: This term refers to an organism's ability to survive and reproduce in a specific environment. Natural selection acts on the genetic variation produced by sexual reproduction, preserving traits that improve a plant's chances of passing its genes to the next generation.
• Comparison with Asexual Reproduction: In asexual reproduction (such as vegetative propagation), offspring are genetically identical clones of the parent. While this is efficient for rapid colonization, the lack of diversity means the entire population could be wiped out by a single environmental stressor.
• Significance in Plants: For Angiosperms (flowering plants) and Gymnosperms, sexual reproduction results in the formation of seeds. Seeds often contain dormant embryos and protective coats, allowing the species to survive unfavorable seasons and disperse to new geographical locations.
• Pollination and Fertilization: These are the two critical steps in plant sexual reproduction. Pollination involves the transfer of pollen grains from the anther to the stigma, while fertilization is the fusion of male and female gametes to form a zygote.

Important Points

• DNA Recombination: This is the production of offspring with combinations of traits that differ from those found in either parent, serving as the raw material for evolution.
• Species Continuity: This refers to the uninterrupted lineage of a group of organisms. Sexual reproduction provides the flexibility required for a lineage to persist across millions of years despite the Earth's changing ecology.
• Biological Stability: While individual plants die, the process of meiosis and syngamy (fertilization) ensures that the gene pool remains dynamic and resilient.

The correct answer is It is change in distance divided by change in velocity.

Key Points

• Average speed is defined as the total distance traveled by an object divided by the total time interval taken to cover that distance. Hence, statement 1 is correct.
• The standard unit of measurement for speed in the International System of Units (SI) is meters per second (m/s), although it can also be expressed in km/h or miles per hour depending on the context. Hence, statement 2 is correct.
• Speed is a scalar quantity, meaning it possesses only magnitude and no directional component. Therefore, the average speed of an object does not change based on the direction in which the object is moving. Hence, statement 3 is correct.
• The mathematical expression for average speed is Total Distance / Total Time. Defining it as the change in distance divided by the change in velocity is scientifically incorrect and has no physical basis in kinematics. Hence, statement 4 is incorrect.
• Hence, the incorrect statement (and thus the correct answer) is It is change in distance divided by change in velocity.

Additional Information

• Scalar vs Vector Quantities:
• A scalar quantity like speed or distance is described solely by its numerical value.
• A vector quantity like velocity or displacement requires both magnitude and direction for a complete description.
• If a car travels in a circular path and returns to its starting point, its average velocity is 0 because the net displacement is 0, but its average speed will be a positive numerical value.
• Instantaneous Speed:
• This refers to the speed of an object at a specific point in time or at a particular instant.
• The speedometer in vehicles is the primary tool used to measure instantaneous speed.
• Unit Conversions:
• To convert speed from kilometers per hour (km/h) to meters per second (m/s), we multiply the value by 5/18.
• To convert m/s back to km/h, the value is multiplied by 18/5.

Important Points

• Dimensional Formula: The dimensional formula for speed is represented as [M0 L1 T-1], where M is mass, L is length, and T is time.
• Odometer: While the speedometer measures speed, the odometer is the device used to measure the total distance traveled by a vehicle.
• Uniform Motion: In uniform motion, an object covers equal distances in equal intervals of time, meaning its speed remains constant throughout the journey.
• Average Velocity Formula: Unlike average speed, average velocity is calculated as Total Displacement / Total Time. Because displacement is the shortest path between two points, the magnitude of average velocity is always less than or equal to average speed.

The correct answer is Ethanol.

Key Points

• Ethanol, also known as ethyl alcohol (chemical formula C2H5OH), is the most common alcohol used as a biofuel worldwide.
• It is a renewable energy source because it is primarily produced through the fermentation of sugars and starches found in plant materials such as sugarcane, corn, maize, and wheat.
• In many countries, including India, Brazil, and the USA, ethanol is blended with gasoline (petrol) to reduce the consumption of fossil fuels and lower carbon emissions.
• The Government of India has launched the Ethanol Blended Petrol (EBP) Programme, which initially targeted 5% blending but has been scaled up significantly over the years.
• As a fuel, ethanol has a higher octane rating than pure gasoline, which helps improve engine performance and prevents engine knocking.

Additional Information

• Methanol (CH3OH):
• Also known as wood alcohol, it can be used as a fuel, particularly in racing cars and high-performance engines.
• While it is cheaper to produce from coal or natural gas, it is highly toxic and more corrosive to engine components than ethanol.
• Butanol (C4H9OH):
• It is considered an advanced biofuel with an energy density very close to that of gasoline.
• Unlike ethanol, butanol is less hydroscopic (absorbs less water) and can be transported through existing oil pipelines, though its production cost remains high.
• Propanol (C3H7OH):
• Commonly used as a solvent and in antiseptics (rubbing alcohol), but it is not typically used as a bulk commercial fuel due to production costs and chemical properties.
• Flex-Fuel Vehicles (FFVs):
• These are specialized vehicles equipped with internal combustion engines designed to run on more than one fuel, usually gasoline blended with ethanol (up to 85% or E85).

Important Points

• India's E20 Target: The Government of India has advanced the target of achieving 20% ethanol blending in petrol from 2030 to 2025-26 to reduce import dependency on crude oil.
• National Policy on Biofuels (2018): This policy categorizes biofuels into First Generation (1G) bio-ethanol (from molasses, sugar beet, etc.) and Second Generation (2G) ethanol (from non-food crops and agricultural residues).
• Global Biofuels Alliance (GBA): Launched during the G20 Summit in 2023 under India’s presidency, this alliance aims to accelerate the global uptake of biofuels like ethanol to support the energy transition.
• Environmental Impact: Using ethanol-blended fuels significantly reduces the emission of Carbon Monoxide (CO), Hydrocarbons (HC), and Nitrogen Oxides (NOx), contributing to cleaner air in urban areas.

The correct answer is Sepal.

Key Points

• The Sepal is the outermost part of a flower that typically appears green and leaf-like. Its primary role is to enclose and protect the flower in its bud stage before it blooms.
• Collectively, all the sepals of a flower are referred to as the Calyx, which forms the first whorl of a typical flower structure.
• Beyond physical protection, sepals help prevent the delicate internal reproductive tissues from desiccation (drying out) and provide a barrier against predatory insects or harsh environmental conditions.
• In many plant species, once the flower opens, the sepals remain at the base of the petals, though in some cases, they may wither and fall off after the blooming process is complete.
• The presence of sepals is a characteristic feature of Angiosperms (flowering plants), helping ensure the survival of the reproductive organs until they are ready for pollination.

Additional Information

• Petal (Corolla):
• The Petals represent the second whorl of the flower. They are often brightly colored and scented to attract pollinators such as bees, birds, and butterflies.
• While they offer some secondary protection, their main evolutionary function is to facilitate the transfer of pollen.
• Stamen (Androecium):
• The Stamen is the male reproductive part of the flower. It consists of two main components: the Anther and the Filament.
• The Anther is responsible for the production of pollen grains, which contain the male gametes.
• Pistil (Gynoecium):
• The Pistil (also known as the Carpel) is the female reproductive part. It is usually located in the center of the flower.
• It comprises the Stigma (receptive tip), the Style (supporting tube), and the Ovary (which contains the ovules). After fertilization, the ovary develops into a fruit.
• Receptacle:
• This is the thickened part of the stem (pedicel) from which all the flower organs grow. It acts as the structural foundation for the Calyx, Corolla, Androecium, and Gynoecium.

Important Points

• Tepals: In some plants, such as Lilies or Tulips, the sepals and petals are indistinguishable in color and shape. In these instances, they are collectively referred to as Tepals (or the Perianth).
• Bisexual vs. Unisexual: A Complete Flower contains all four whorls (sepals, petals, stamens, and pistils). If any of these are missing, it is termed an Incomplete Flower.
• Pollination: This is the process where pollen is transferred from the Anther to the Stigma, eventually leading to fertilization and the production of seeds.
• Monocots vs. Dicots: Monocot flowers usually have parts in multiples of three, whereas Dicot flowers typically have parts in multiples of four or five.

The correct answer is Propanol.

Key Points

• Propanol is an alcohol with the chemical formula C3H8O. It consists of a three-carbon chain where one hydrogen atom is replaced by a hydroxyl group (-OH).
• In the structure of Propanol (specifically 1-propanol or 2-propanol), all the carbon-carbon and carbon-oxygen bonds are single covalent bonds (sigma bonds). There are no double bonds present in its molecular framework.
• Functional groups like ketones, aldehydes, and carboxylic acids contain a carbonyl group, which consists of a carbon atom double-bonded to an oxygen atom (C=O).
• Propanone, Propanal, and Propanoic acid all possess this C=O double bond, making Propanol the only option among the choices that is a saturated compound with respect to its functional group oxygen.
• Because it lacks double or triple bonds, Propanol is classified as a derivative of a saturated hydrocarbon (alkane).

Additional Information

• Propanone (Acetone):
• It is the simplest ketone with the formula CH3COCH3. It contains a carbonyl group ($C=O$) located on the second carbon of the chain. Hence, option 1 is incorrect.
• The carbon in the C=O bond is sp2 hybridized, which necessitates the presence of one pi bond (the double bond).
• Propanal (Propionaldehyde):
• It is an aldehyde with the formula CH3CH2CHO. The functional group is located at the end of the carbon chain. Hence, option 3 is incorrect.
• Similar to ketones, aldehydes contain a carbon-oxygen double bond as part of the formyl group.
• Propanoic Acid:
• It is a carboxylic acid with the formula CH3CH2COOH. Its functional group consists of both a carbonyl group (C=O) and a hydroxyl group (-OH) attached to the same carbon atom. Hence, option 4 is incorrect.
• The presence of the C=O bond means it contains a double bond in its structure.
• Alcohol Classification:
• Alcohols like Propanol are characterized by the -OH group. Depending on the position, it can be n-propanol (1-propanol) or isopropanol (2-propanol). In both isomers, only single bonds exist. Hence, option 2 is correct.

Important Points

• Hybridization: In Propanol, all three carbon atoms are sp3 hybridized, forming a tetrahedral geometry around each carbon. In contrast, the functional carbon in propanone, propanal, and propanoic acid is sp2 hybridized to accommodate the double bond.
• Saturation: A molecule is considered unsaturated if it contains double or triple bonds between any two atoms (often specifically carbon-carbon, but functional group double bonds like C=O also count towards the degree of unsaturation).
• Valency of Oxygen: While oxygen always maintains a valency of 2, it can satisfy this by forming two single bonds (as in alcohols and ethers) or one double bond (as in carbonyl compounds).
• Physical Properties: Due to the -OH group and the absence of double bonds, Propanol can engage in extensive hydrogen bonding, giving it a higher boiling point compared to Propanal or Propanone of similar molecular weight.

The correct answer is Muscle cells.

Key Points

• Plastids are double-membrane-bound organelles found primarily in the cells of plants and algae. They are responsible for manufacturing and storing important chemical compounds used by the cell.
• Muscle cells are animal cells, and one of the fundamental differences between plant and animal cells is the presence of plastids. Animal cells, including human muscle cells, lack plastids entirely because they do not perform photosynthesis or require specialized plant storage structures like leucoplasts. Hence, the correct answer is Muscle cells.
• Leaf cells (specifically the mesophyll cells) contain a high density of a specific type of plastid known as chloroplasts. These contain the pigment chlorophyll, which captures light energy for photosynthesis. Hence, statement 1 is incorrect as plastids are present.
• Stem cells in plants, particularly in herbaceous or young stems, contain plastids such as chloroplasts or proplastids (precursors to other plastids). These allow the stem to contribute to the plant's energy production. Hence, statement 3 is incorrect as plastids are present.
• Root cells also contain plastids, typically in the form of leucoplasts. Since roots are generally underground and not exposed to light, they do not require chloroplasts but use amyloplasts (a type of leucoplast) to store starch. Hence, statement 4 is incorrect as plastids are present.

Additional Information

• Classification of Plastids:
• Chloroplasts: These are the most well-known plastids, containing chlorophyll and carotenoid pigments. They are the site of photosynthesis, converting light energy into chemical energy in the form of glucose.
• Chromoplasts: These contain fat-soluble carotenoid pigments like carotene and xanthophylls. They provide the distinct yellow, orange, or red colors to flowers and ripening fruits, aiding in pollination and seed dispersal.
• Leucoplasts: These are colorless plastids of varied shapes and sizes that store nutrients. They are further divided into:
• Amyloplasts: Specialized for storing carbohydrates (starch), commonly found in potato tubers and grains.
• Elaioplasts: Responsible for storing oils and fats.
• Aleuroplasts (Proteinoplasts): Responsible for storing proteins.
• Semi-autonomous Nature:
• Like mitochondria, plastids are semi-autonomous organelles. They possess their own circular DNA and 70S ribosomes, allowing them to synthesize some of their own proteins and replicate independently within the cell.
• Endosymbiotic Theory:
• Scientific evidence suggests that plastids originated from cyanobacteria that entered into a symbiotic relationship with early eukaryotic cells roughly 1.5 billion years ago.

Important Points

• Internal Structure: The inside of a chloroplast is organized into a matrix called the stroma, which contains membrane-bound sacs called thylakoids. These thylakoids are stacked like coins to form grana (singular: granum).
• Pigment Distribution: While all plastids derive from proplastids, their final form depends on the tissue location and exposure to environmental factors like light. For instance, if a root (containing leucoplasts) is exposed to light, it may turn green as the plastids develop into chloroplasts.
• Functions: Beyond energy production and storage, plastids play vital roles in the synthesis of fatty acids, amino acids, and terpenes, which are essential for the plant's survival and defense mechanisms.

Shortcut Trick

Convert ratio 1/2 : 1/3 : 1/6 to integers by multiplying with LCM (6) → 3 : 2 : 1.

Total units = 3 + 2 + 1 = 6 units.

Value of 6 units = 602 bananas.

First monkey (3 units) = (602 ÷ 6) × 3 = 602 ÷ 2 = 301.

∴ The correct answer is 301 bananas.

Alternate Method

Given: Total bananas = 602. Ratio = 1/2 : 1/3 : 1/6.

⇒ Simplify Ratio: LCM of 2, 3, 6 = 6.

⇒ Ratio = (1/2 × 6) : (1/3 × 6) : (1/6 × 6) = 3 : 2 : 1.

⇒ Let parts be 3x, 2x, and 1x.

⇒ 3x + 2x + 1x = 602

⇒ 6x = 602

⇒ x = 602 ÷ 6 = 301/3

⇒ Share of first monkey = 3x = 3 × (301/3) = 301.

∴ The correct answer is 301 bananas.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d to integers, multiply both terms by the LCM of denominators b and d.

Inverse Ratio

The inverse ratio of a : b is 1/a : 1/b, which simplifies to b : a for two terms.

Proportional Division

Share of a part = (Specific Ratio Part ÷ Sum of Ratio Parts) × Total Value.

Shortcut Trick

Number of 4-legged animals (goats) = (Total feet − 2 × Total heads) ÷ 2

⇒ Number of goats = (82 − 2 × 33) ÷ 2

⇒ Number of goats = (82 − 66) ÷ 2 = 16 ÷ 2 = 8

∴ The correct answer is 8 goats.

Alternate Method

Given:

Total animal heads = 33

Total animal feet = 82

Number of feet for one hen = 2

Number of feet for one goat = 4

Formula Used:

Total feet = (2 × Number of hens) + (4 × Number of goats)

Calculations:

Let the number of goats be G and the number of hens be H.

⇒ G + H = 33 → (Equation 1)

⇒ 4G + 2H = 82 → (Equation 2)

Divide Equation 2 by 2:

⇒ 2G + H = 41 → (Equation 3)

Subtract Equation 1 from Equation 3:

⇒ (2G + H) − (G + H) = 41 − 33

⇒ G = 8

∴ The correct answer is 8 goats.

Additional Information

Alligation Method

You can find the ratio of hens to goats by placing 2 and 4 at the ends and the average feet per head (82/33) in the center.

Assumption Method

If we assume all animals are hens, total feet = heads × 2. The extra feet (Total − Assumed) are due to goats, where each goat contributes 2 extra feet.

Linear Variables

For any problem with two types of items having different counts (like 2-legged/4-legged or ₹2/₹5 coins), use x + y = Total Quantity and ax + by = Total Value.

Shortcut Trick

Apply the direct Algebraic Identity: a³ + b³ + c³ = (a + b + c)[(a + b + c)² − 3(ab + bc + ca)] + 3abc.

Substitute the given values: 14 × [14² − 3(47)] + 3(15).

Simplify: 14 × [196 − 141] + 45 = 14 × 55 + 45.

Calculate: 770 + 45 = 815.

∴ The correct answer is 815.

Alternate Method

Given: a + b + c = 14, ab + bc + ca = 47, and abc = 15.

Formula Used:

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

2. a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − (ab + bc + ca))

Calculations:

⇒ 14² = a² + b² + c² + 2(47)

⇒ 196 = a² + b² + c² + 94

⇒ a² + b² + c² = 196 − 94 = 102

⇒ a³ + b³ + c³ − 3(15) = 14 × (102 − 47)

⇒ a³ + b³ + c³ − 45 = 14 × 55

⇒ a³ + b³ + c³ − 45 = 770

⇒ a³ + b³ + c³ = 770 + 45

⇒ a³ + b³ + c³ = 815

∴ The correct answer is 815.

Additional Information

Special Case of the Cubic Identity

If a + b + c = 0, then the identity simplifies significantly to a³ + b³ + c³ = 3abc.

Alternative Form of Identity

The expression a² + b² + c² − ab − bc − ca can also be written as ½[(a − b)² + (b − c)² + (c − a)²], which is useful when differences between variables are known.

Square of a Trinomial

The basic expansion used to find the sum of squares is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).

Shortcut Trick

For a number N with n distinct prime factors, the number of pairs of coprime factors (a, b) such that a × b = N is 2ⁿ⁻¹.

Prime factorization of 180 = 2² × 3² × 5¹.

Number of distinct prime factors (n) = 3 (which are 2, 3, and 5).

Number of coprime pairs = 2³⁻¹ = 2² = 4.

∴ The correct answer is 4.

Alternate Method

Given: Product of two coprime integers is 180.

Formula Used: Two numbers are coprime if their Highest Common Factor (HCF) is 1.

⇒ Prime factorization of 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5¹.

⇒ Let the two coprime numbers be x and y. Since they are coprime, any prime power (like 2²) must stay together in one of the numbers.

⇒ The distinct prime power blocks are (2²), (3²), and (5¹), i.e., 4, 9, and 5.

⇒ Possible pairs (x, y):

⇒ 1) (1, 180)

⇒ 2) (4, 9 × 5) = (4, 45)

⇒ 3) (9, 4 × 5) = (9, 20)

⇒ 4) (5, 4 × 9) = (5, 36)

∴ The correct answer is 4.

Additional Information

Coprime Numbers

Two numbers a and b are coprime or relatively prime if HCF(a, b) = 1. They do not need to be prime themselves.

Number of Coprime Pairs Formula

For a number N = p₁^a × p₂^b × ... × p_n^k, the number of ways to write N as a product of two coprime integers is 2ⁿ⁻¹.

Total Number of Factors

The total number of factors of N is (a+1)(b+1)...(k+1). For 180, total factors = (2+1)(2+1)(1+1) = 18.

Shortcut Trick

For two successive discounts of a% and b%, the single equivalent discount is given by the formula: (a + b − ab ÷ 100)%.

Here, a = 16 and b = 7.

Equivalent discount = (16 + 7 − (16 × 7) ÷ 100)%

= (23 − 112 ÷ 100)%

= 23 − 1.12 = 21.88%.

∴ The correct answer is 21.88%.

Alternate Method

Given: First discount = 16%, Second discount = 7%.

Formula: Single Equivalent Discount = 100 − [100 × (100 − d₁)% × (100 − d₂)%]

⇒ Let the Marked Price (MP) be 100.

⇒ Price after first discount = 100 − (16% of 100) = 84.

⇒ Price after second discount = 84 − (7% of 84) = 84 − 5.88 = 78.12.

⇒ Total discount = 100 − 78.12 = 21.88.

⇒ Equivalent discount % = (21.88 ÷ 100) × 100 = 21.88%.

∴ The correct answer is 21.88%.

Additional Information

Successive Discount

When a series of discounts are applied one after another on the reducing balance, they are called successive discounts. The order of applying discounts does not change the final result.

Three Successive Discounts

For three discounts a, b, and c, first find the equivalent of a and b (say x), then find the equivalent of x and c.

Shortcut Trick

For any quadrilateral WXYZ, the angle formed by the bisectors of two adjacent angles (∠Y and ∠Z) is equal to half the sum of the other two angles (∠W and ∠X).

∠ZOY = (∠W + ∠X) ÷ 2

∠ZOY = (74° + 58°) ÷ 2

∠ZOY = 132° ÷ 2 = 66°

∴ The correct answer is 66°.

Alternate Method

Given:

In quadrilateral WXYZ:

∠W = 74°

∠X = 58°

The bisectors of ∠Y and ∠Z meet at O.

Formula Used:

Sum of interior angles of a quadrilateral = 360°

Sum of angles of a triangle = 180°

Calculations:

In quadrilateral WXYZ, the sum of all interior angles is 360°:

⇒ ∠W + ∠X + ∠Y + ∠Z = 360°

Substitute the given values of ∠W and ∠X:

⇒ 74° + 58° + ∠Y + ∠Z = 360°

⇒ 132° + ∠Y + ∠Z = 360°

⇒ ∠Y + ∠Z = 360° − 132°

⇒ ∠Y + ∠Z = 228°

Since ZO and YO are the angle bisectors of ∠Z and ∠Y respectively:

⇒ ∠OZY = ∠Z ÷ 2

⇒ ∠OYZ = ∠Y ÷ 2

In triangle ZOY, the sum of angles is 180°:

⇒ ∠ZOY + ∠OZY + ∠OYZ = 180°

⇒ ∠ZOY + (∠Z ÷ 2) + (∠Y ÷ 2) = 180°

⇒ ∠ZOY + (∠Z + ∠Y) ÷ 2 = 180°

Substitute the value of (∠Z + ∠Y) = 228°:

⇒ ∠ZOY + 228° ÷ 2 = 180°

⇒ ∠ZOY + 114° = 180°

⇒ ∠ZOY = 180° − 114°

⇒ ∠ZOY = 66°

Option Analysis:

• Option (1) 68°: Incorrect. If ∠ZOY = 68°, then the sum of ∠Y and ∠Z would be 2 × (180° − 68°) = 224°, which implies ∠W + ∠X = 360° − 224° = 136°. This contradicts the given sum of 132°.

• Option (2) 60°: Incorrect. If ∠ZOY = 60°, then the sum of ∠Y and ∠Z would be 2 × (180° − 60°) = 240°, which implies ∠W + ∠X = 360° − 240° = 120°. This contradicts the given sum of 132°.

• Option (3) 66°: Correct. As mathematically proven, ∠ZOY = 180° − (∠Y + ∠Z) ÷ 2 = 66°.

• Option (4) 59°: Incorrect. If ∠ZOY = 59°, then the sum of ∠Y and ∠Z would be 2 × (180° − 59°) = 242°, which implies ∠W + ∠X = 360° − 242° = 118°. This contradicts the given sum of 132°.

∴ The correct answer is 66°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining two angles. Formula: ∠ZOY = (∠W + ∠X) ÷ 2.

Sum of Interior Angles of a Polygon

For any n-sided polygon, the sum of interior angles is given by (n − 2) × 180°. For a quadrilateral (n = 4), the sum is always 360°.

Angle Bisectors of Opposite Angles

The angle between the bisectors of two opposite angles of a quadrilateral is equal to half the difference of the other two opposite angles.

Shortcut Trick

Convert percentages and fractions into a single multiplying factor.

80% × 50% × 3/8 → 4/5 × 1/2 × 3/8 = 3/20

If 3 units = 537, then 1 unit = 179.

Total value (20 units) = 179 × 20 = 3580.

∴ The correct answer is 3580.

Alternate Method

Given: 80% of 50% of 3/8 of a number is 537.

Formula: Result = (P₁/100) × (P₂/100) × Fraction × Number

Let the number be x.

⇒ (80/100) × (50/100) × (3/8) × x = 537

⇒ (4/5) × (1/2) × (3/8) × x = 537

⇒ (12/80) × x = 537

⇒ (3/20) × x = 537

⇒ x = (537 × 20) / 3

⇒ x = 179 × 20

⇒ x = 3580

∴ The correct answer is 3580.

Additional Information

Percentage to Fraction Conversion

To simplify calculations, remember common fractions: 80% = 4/5, 50% = 1/2, 25% = 1/4, and 12.5% = 1/8.

Successive Percentage

When applying multiple percentages to a single base, the net percentage is the product of their decimal equivalents.

Base Value Concept

Always identify the 'of' term as the denominator or base value in ratio-based solving.

Shortcut Trick

Since Mode = 22.75, the Modal Class is 21−28.

Using Mode formula: 22.75 = 21 + [(11 − 9) / (2 × 11 − 9 − X)] × 7

⇒ 1.75 = [2 / (13 − X)] × 7 ⇒ 13 − X = 14 / 1.75 = 8

⇒ X = 5

Total Frequency = 1 + 7 + 9 + 11 + 5 + 3 = 36.

∴ The correct answer is 36.

Alternate Method

Given:

Mode = 22.75

Modal Class = 21−28 (since 22.75 lies in this range)

Lower limit (L) = 21

Frequency of modal class (f₁) = 11

Frequency of class preceding modal class (f₀) = 9

Frequency of class succeeding modal class (f₂) = X

Class width (h) = 7

Formula Used:

Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

Calculations:

⇒ 22.75 = 21 + [(11 − 9) / (2 × 11 − 9 − X)] × 7

⇒ 22.75 − 21 = [2 / (22 − 9 − X)] × 7

⇒ 1.75 = [2 / (13 − X)] × 7

⇒ 1.75 = 14 / (13 − X)

⇒ 13 − X = 14 / 1.75

⇒ 13 − X = 8

⇒ X = 13 − 8 = 5

Total Frequency (Sum of all frequencies):

⇒ N = 1 + 7 + 9 + 11 + 5 + 3

⇒ N = 36

∴ The correct answer is 36.

Additional Information

Empirical Relationship

For a moderately skewed distribution: Mode = 3 × Median − 2 × Mean.

Modal Class Identification

The class interval with the highest frequency is called the Modal Class.

Median Formula

Median = L + [((N/2) − cf) / f] × h, where cf is the cumulative frequency of the preceding class.

Shortcut Trick Scheme i: Total Discount = 37%. Scheme ii: Successive Discount = a+b−(ab÷100) = 33+43−(33×43÷100) = 76−14.19 = 61.81%. Scheme iii: Successive Discount = 46+22−(46×22÷100) = 68−10.12 = 57.88%. Compare the discounts: 37% (i) < 57.88% (iii) < 61.81% (ii). The least discount is the least beneficial to the customer. ∴ The correct answer is Scheme i.

Alternate Method Given: Three different discount schemes. To find the least beneficial (minimum discount percentage). Formulas: Successive Discount = a + b − (ab ÷ 100)%

⇒ Scheme i: Single discount = 37%. ⇒ Scheme ii: Single equivalent of 33% and 43% = [33 + 43 − (33 × 43)/100]% = [76 − 14.19]% = 61.81%. ⇒ Scheme iii: Single equivalent of 46% and 22% = [46 + 22 − (46 × 22)/100]% = [68 − 10.12]% = 57.88%. ⇒ Comparing all three: 37% is the minimum value among the available options. ∴ The correct answer is Scheme i.

Additional Information Customer Perspective vs. Seller Perspective For a customer, the most beneficial scheme is the one offering the highest discount. Conversely, the least beneficial is the one offering the lowest discount. Successive Discount Calculation When two successive discounts x% and y% are given, the net discount is always less than the sum (x + y)% due to the reduction in the base amount for the second discount.

Shortcut Trick

Difference in annual interest rate = 7% − 3.5% = 3.5%

Total interest percentage difference for 4 years = 3.5% × 4 = 14%

Positive difference in ₹ = 14% of ₹4,80,000

Calculation: 14/100 × 4,80,000 = 14 × 4,800 = 67,200

∴ The correct answer is ₹67,200.

Alternate Method

Given:

Principal (P) = ₹4,80,000

Time (T) = 4 years

Rate of Bank A (R_A) = 3.5% p.a.

Rate of Bank B (R_B) = 7% p.a.

Formula Used:

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

Calculations:

⇒ SI paid to Bank A = (4,80,000 × 3.5 × 4) ÷ 100

⇒ SI paid to Bank A = 4,800 × 14 = ₹67,200

⇒ SI paid to Bank B = (4,80,000 × 7 × 4) ÷ 100

⇒ SI paid to Bank B = 4,800 × 28 = ₹1,34,400

⇒ Positive Difference = ₹1,34,400 − ₹67,200

⇒ Positive Difference = ₹67,200

∴ The correct answer is ₹67,200.

Additional Information

Total Amount Formula

Amount (A) = Principal (P) + Simple Interest (SI). It represents the total money repaid to the lender.

Direct Difference Rule

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

Simple Interest Nature

Simple Interest remains constant every year if the principal and rate do not change, unlike Compound Interest which increases annually.

Shortcut Trick

Total time = 1 + 1.5 = 2.5 hours.

Total distance = Average speed × Total time = 55 × 2.5 = 137.5 km.

Distance covered in 1^st part = 70 × 1 = 70 km.

Distance covered in 2^nd part = 137.5 − 70 = 67.5 km.

Speed p = 67.5 ÷ 1.5 = 45 km/hr.

∴ The correct answer is 45 km/hr.

Alternate Method

Given:

Speed 1 = 70 km/hr, Time 1 = 1 hour

Speed 2 = p km/hr, Time 2 = 1.5 hours

Average Speed = 55 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Distance = Speed × Time

Calculations:

⇒ Total Distance = (70 × 1) + (p × 1.5)

⇒ Total Time = 1 + 1.5 = 2.5 hours

⇒ Average Speed = (70 + 1.5p) ÷ 2.5

⇒ 55 = (70 + 1.5p) ÷ 2.5

⇒ 55 × 2.5 = 70 + 1.5p

⇒ 137.5 = 70 + 1.5p

⇒ 1.5p = 137.5 − 70

⇒ 1.5p = 67.5

⇒ p = 67.5 ÷ 1.5 = 45

∴ The correct answer is 45 km/hr.

Additional Information

Average Speed Formula

For different speeds v₁, v₂ over different time intervals t₁, t₂: Avg Speed = (v₁t₁ + v₂t₂) ÷ (t₁ + t₂).

Harmonic Mean Case

If distances are equal, Average Speed = 2v₁v₂ ÷ (v₁ + v₂).

Unit Conversion

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

Shortcut Trick

Given cot θ = 14 / 3 = Base / Perpendicular.

Let Base (B) = 14 and Perpendicular (P) = 3.

Hypotenuse (H) = √(14² + 3²) = √(196 + 9) = √205.

Since cos θ = Base / Hypotenuse = 14 / √205.

∴ The correct answer is 14/√205.

Alternate Method

Given: cot θ = 14/3

Formula Used:

⇒ cot θ = Base / Perpendicular

⇒ cos θ = Base / Hypotenuse

⇒ Hypotenuse² = Perpendicular² + Base²

Calculations:

⇒ Base (B) = 14, Perpendicular (P) = 3

⇒ Hypotenuse (H) = √(14² + 3²)

⇒ H = √(196 + 9) = √205

⇒ cos θ = B / H = 14 / √205

∴ The correct answer is 14/√205.

Additional Information

Trigonometric Identities

Relation between functions: cosec²θ − cot²θ = 1 and sec²θ − tan²θ = 1.

Reciprocal Relations

cot θ = 1/tan θ and cos θ = 1/sec θ.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: H² = P² + B².

Shortcut Trick

Effective Multiplier = 0.55 × 0.40 × 0.45 × 1.50

Effective Multiplier = 0.22 × 0.675 = 0.1485

Percentage Change = (0.1485 − 1) × 100 = −85.15%

∴ The correct answer is Decreases by 85.15%.

Alternate Method

Given:

First Month Change = 45% Decrease

Second Month Change = 60% Decrease

Third Month Change = 55% Decrease

Fourth Month Change = 50% Increase

Let the Original Price be 100.

⇒ Price after 1^st month = 100 × (100 − 45)/100 = 55

⇒ Price after 2^nd month = 55 × (100 − 60)/100 = 55 × 0.4 = 22

⇒ Price after 3^rd month = 22 × (100 − 55)/100 = 22 × 0.45 = 9.9

⇒ Price after 4^th month = 9.9 × (100 + 50)/100 = 9.9 × 1.5 = 14.85

⇒ Total Percentage Change = [(Final Price − Original Price) ÷ Original Price] × 100

⇒ Percentage Change = [(14.85 − 100) ÷ 100] × 100 = −85.15%

∴ The correct answer is Decreases by 85.15%.

Additional Information

Successive Percentage Formula

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

Multiplier Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ... × (1 ± r_n/100).

Percentage Decrease

If the final value is less than the initial value, the percentage change is negative, indicating a decrease.

Shortcut Trick

Effective Cost Price (CP) = 200 + 10% of 200 = 200 + 20 = ��220.

To earn 25% profit, use the multiplier: Selling Price (SP) = CP × 1.25.

SP = 220 × (5 ÷ 4) = 55 × 5 = ₹275.

∴ The correct answer is ₹275.

Alternate Method

Given: CP = ₹200, Overhead = 10% of CP, Profit% = 25%.

Formulas: Total CP = CP + Overhead | SP = Total CP × (100 + P%) ÷ 100

⇒ Overhead Expenses = 10% of 200 = (10 ÷ 100) × 200 = ₹20.

⇒ Total Cost Price = 200 + 20 = ₹220.

⇒ Selling Price = 220 × (100 + 25) ÷ 100.

⇒ SP = 220 × (125 ÷ 100) = 220 × (5 ÷ 4).

⇒ SP = 55 × 5 = ₹275.

∴ The correct answer is ₹275.

Additional Information

Overhead Expenses

Any extra expenditure incurred on goods like freight, transportation, repair, or labor is added to the initial Cost Price to calculate the Total Cost Price.

Profit Calculation

Profit or Loss is always calculated on the Total Cost Price (CP + Overheads), not just the buying price. Profit = SP − Total CP.

Shortcut Trick

Required percentage = (Value being compared ÷ Base value) × 100

Compare B's marks (50) with A's marks (75) as the base.

Required % = (50 ÷ 75) × 100 = 2/3 × 100 = 66.666...

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by A = 75

Marks obtained by B = 50

Maximum marks = 95 (Not required for this calculation as the base is A's marks)

Formula Used:

Percentage = (B ÷ A) × 100

Calculations:

⇒ Required Percentage = (50 ÷ 75) × 100

⇒ Required Percentage = (2 ÷ 3) × 100

⇒ Required Percentage = 200 ÷ 3

⇒ Required Percentage = 66.6666...

⇒ Rounding off to two decimal places = 66.67%

∴ The correct answer is 66.67%.

Additional Information

Base Value Identification

In percentage problems, the value following the word 'of' or 'than' usually acts as the denominator (base).

Percentage Change

Percentage increase/decrease = (Difference ÷ Original Value) × 100.

Fraction to Percentage Conversion

1/3 = 33.33%, 2/3 = 66.67%, 1/4 = 25%, 1/5 = 20%.

Shortcut Trick

Fourth Proportional of 1, 3, 6 → (3 × 6) ÷ 1 = 18.

Mean Proportional of 12, 3 → √(12 × 3) = √36 = 6.

Required Ratio → 18 : 6 = 3 : 1.

∴ The correct answer is 3 : 1.

Alternate Method

Given: Numbers for Fourth Proportional (1, 3, 6) and Mean Proportional (12, 3).

Formula Used: Fourth Proportional of a, b, c is (b × c) / a. Mean Proportional of a, b is √(a × b).

⇒ Fourth Proportional = (3 × 6) ÷ 1 = 18.

⇒ Mean Proportional = √(12 × 3) = √36 = 6.

⇒ Required Ratio = 18 : 6.

⇒ Dividing both sides by 6, we get 3 : 1.

∴ The correct answer is 3 : 1.

Additional Information

Third Proportional

The third proportional of two numbers a and b is defined as c such that a : b :: b : c, which gives c = b² / a.

Continued Proportion

Three numbers a, b, c are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third (a/b = b/c).

Invertendo and Alternendo

If a : b = c : d, then Invertendo states b : a = d : c and Alternendo states a : c = b : d.

Shortcut Trick

Separate the integers and decimals for faster mental calculation.

⇒ (36 − 30) + (300.30 − 0.26)

⇒ 6 + 300.04

⇒ 306.04

∴ The correct answer is 306.04.

Alternate Method

Given:

36 − 30 + 300.3 − 0.26

Formula Used:

Group all positive values and negative values together to simplify addition and subtraction.

Calculations:

⇒ Sum of positive values = 36 + 300.3 = 336.3

⇒ Sum of negative values = −30 − 0.26 = −30.26

⇒ Total = 336.3 − 30.26

⇒ Add a trailing zero to match decimal places: 336.30 − 30.26

⇒ Total = 306.04

∴ The correct answer is 306.04.

Additional Information

BODMAS Rule

It specifies the correct order of mathematical operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Addition and Subtraction hold the same priority and can be grouped or calculated strictly from left to right.

Decimal Addition and Subtraction

Always align numbers exactly by their decimal point. Insert trailing zeros to the right of the decimal so all values have the exact same number of decimal places before computing.

Checking with Decimal Digits

In decimal simplifications with distinct options, evaluating only the final decimal digits (e.g., .30 − .26 = .04) can frequently help you pinpoint the correct option immediately without having to complete the full arithmetic sequence.

Shortcut Trick

Let the Total Capacity of the cistern be the LCM of 12 and 14 = 84 units.

Efficiency of (P + Q) = 84 ÷ 12 = 7 units/hour.

Efficiency of P = 84 ÷ 14 = 6 units/hour.

Efficiency of Q = 7 − 6 = 1 unit/hour.

Time for Q to fill 1/4^th cistern = (84 ÷ 4) ÷ 1 = 21 hours.

∴ The correct answer is 21 hours.

Alternate Method

Given:

Time taken by P and Q = 12 hours

Time taken by P alone = 14 hours

Formula Used:

Total Work = Time × Efficiency

Calculations:

⇒ Efficiency of Q = Efficiency of (P + Q) − Efficiency of P

⇒ Efficiency of Q = 7 − 6 = 1 unit/hour

⇒ One-fourth of the cistern capacity = 84 ÷ 4 = 21 units

⇒ Time taken by Q alone = 21 ÷ 1 = 21 hours

∴ The correct answer is 21 hours.

Additional Information

Pipes & Cisterns Efficiency

If a pipe fills a tank in 'x' hours and another empties it in 'y' hours, the net work done in one hour is (1/x − 1/y).

Work & Wages

Wages are always distributed in the ratio of the efficiency or the work done by the individuals, not the time taken.

Shortcut Trick

Assume an estimated mean of 60 for the given observations.

Calculate deviations: +35, −15, +25, −15, −9, +25, +24, +6, −13.

Sum of deviations = 35 − 15 + 25 − 15 − 9 + 25 + 24 + 6 − 13 = 63.

Average deviation = 63 ÷ 9 = 7.

Actual Mean = Assumed Mean + Average Deviation = 60 + 7 = 67.

∴ The correct answer is 67.

Alternate Method

Given: Observations are 95, 45, 85, 45, 51, 85, 84, 66, and 47.

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

Calculations:

⇒ Sum = 95 + 45 + 85 + 45 + 51 + 85 + 84 + 66 + 47

⇒ Sum = 603

⇒ Total count (n) = 9

⇒ Arithmetic Mean = 603 ÷ 9

⇒ Arithmetic Mean = 67

∴ The correct answer is 67.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of items from their arithmetic mean is always zero: Σ(x − x̄) = 0.

Empirical Relationship

For moderately skewed distributions: Mode = 3 × Median − 2 × Mean.

Weighted Arithmetic Mean

When observations have different importance, use Mean = Σ(w × x) ÷ Σw, where w is the weight of each observation.

Shortcut Trick

Ratio 10 years ago (F:S) = 3 : 1 (Difference = 2)

Ratio 10 years hence (F:S) = 2 : 1 (Difference = 1)

To equalize difference, multiply second ratio by 2 → 4 : 2

Increase in units = 4 − 3 = 1 unit. Total time gap = 10 + 10 = 20 years.

∴ 1 unit = 20 years. Son's age 10 years ago = 1 unit = 20 years.

Present age of son = 20 + 10 = 30 years.

∴ The correct answer is 30 years.

Alternate Method

Given: Ratio 10 years ago = 3:1, Ratio 10 years later = 2:1.

Formula: Age after n years = Age + n.

⇒ Let Son's age 10 years ago be x and Father's age be 3x.

⇒ Present ages: Son = x + 10, Father = 3x + 10.

⇒ Ages after 10 years: Son = x + 20, Father = 3x + 20.

⇒ According to question: (3x + 20) = 2 × (x + 20)

⇒ 3x + 20 = 2x + 40

⇒ x = 20

⇒ Son's present age = x + 10 = 20 + 10 = 30 years.

∴ The correct answer is 30 years.

Additional Information

Ratio Difference Method

When the time interval added to two people is the same, the numerical difference between their ratio parts must be equalized to solve directly.

Age Constant Property

The difference between the ages of two persons remains constant throughout their lives, regardless of the number of years passed.

Shortcut Trick

The perimeter of a triangle formed by joining the mid-points of the sides of a given triangle is always half of the perimeter of the original triangle.

Perimeter of triangle ABC = 5 + 6 + 7 = 18 cm.

Required Perimeter = 18 ÷ 2 = 9 cm.

∴ The correct answer is 9 cm.

Alternate Method

Given: Sides of ΔABC are AB = 5 cm, BC = 6 cm, and CA = 7 cm.

According to the Mid-point Theorem, the segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length.

Let D, E, and F be the mid-points of sides AB, BC, and CA respectively.

⇒ Side DE = CA ÷ 2 = 7 ÷ 2 = 3.5 cm

⇒ Side EF = AB ÷ 2 = 5 ÷ 2 = 2.5 cm

⇒ Side FD = BC ÷ 2 = 6 ÷ 2 = 3.0 cm

⇒ Perimeter of ΔDEF = DE + EF + FD

⇒ Perimeter = 3.5 + 2.5 + 3.0 = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Mid-point Theorem

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Area of Mid-point Triangle

The area of the triangle formed by joining the mid-points is exactly 1/4^th of the area of the original triangle ABC.

Similarity Property

The triangle formed by joining the mid-points is similar to the original triangle with a side ratio of 1:2.

Shortcut Trick

Let Total Work (LCM of 18, 12, 36) = 36 units.

Efficiency of Pipe A = 36 ÷ 18 = 2 units/hr.

Efficiency of Pipe B = 36 ÷ 12 = 3 units/hr.

Efficiency of Leakage = 36 ÷ 36 = −1 unit/hr.

Combined Efficiency = 2 + 3 − 1 = 4 units/hr.

Time to fill 1/3^rd capacity = (36 × 1/3) ÷ 4 = 12 ÷ 4 = 3 hours.

∴ The correct answer is 3 hr.

Alternate Method

Given: Filling time for Pipe A = 18 hr, Pipe B = 12 hr, Emptying time for Leakage = 36 hr.

Calculations:

⇒ Total Efficiency when all are active = 2 + 3 − 1 = 4 units/hr

⇒ Capacity to be filled = 1/3 × 36 = 12 units

⇒ Required Time = 12 ÷ 4 = 3 hr

∴ The correct answer is 3 hr.

Additional Information

Pipes and Cisterns Efficiency

Inlet pipes have positive efficiency while outlet pipes or leakages have negative efficiency in calculations.

Combined Work Formula

Total Time = (Total Work to be done) ÷ (Sum of individual efficiencies).

LCM Method

Total work is assumed as the Least Common Multiple of individual time periods to avoid calculations with fractions.

Shortcut Trick

Convert all numbers into same base

8 = 2³ ⇒ 2^3m × 2³ = 2⁹ ⇒ 3m + 3 = 9 ⇒ m = 2

9 = 3² ⇒ 3²ⁿ × 3² = 3⁶ ⇒ 2n + 2 = 6 ⇒ n = 2

m/n = 2/2 = 1

∴ Answer = 1

Alternate Method

Given:

2^3m × 8 = 2⁹ and 3²ⁿ × 9 = 3⁶

Formula used:

a^x × a^y = a^{(x + y)}

Calculation:

⇒ 2^3m × 2³ = 2⁹

⇒ 2^{(3m + 3)} = 2⁹

⇒ 3m + 3 = 9

⇒ 3m = 6 ⇒ m = 2

⇒ 3²ⁿ × 3² = 3⁶

⇒ 3^{(2n + 2)} = 3⁶

⇒ 2n + 2 = 6

⇒ 2n = 4 ⇒ n = 2

⇒ m/n = 2/2 = 1

∴ The correct answer is 1.

Additional Information

Laws of Exponents

a^x × a^y = a^{(x + y)}, (a^x)^y = a^(xy)

Equal Bases Rule

If bases are equal, then powers must be equal.

Power Conversion

Convert numbers like 8 = 2³ and 9 = 3² to simplify equations.

Shortcut Trick

The Volume of a Cuboid is calculated by the product of its three dimensions.

Volume = 34 × 10 × 26

Perform multiplication: (34 × 26) × 10 = 884 × 10 = 8840 m³.

∴ The correct answer is 8840 m³.

Alternate Method

Given:

Length (L) of the tank = 34 m

Breadth (B) of the tank = 10 m

Height (H) of the tank = 26 m

Formula Used:

Volume of a Cuboid = L × B × H

Calculations:

⇒ Volume = 34 × 10 × 26

⇒ Volume = 340 × 26

⇒ Volume = 8840 m³

∴ The correct answer is 8840 m³.

Additional Information

Total Surface Area of Cuboid

The total surface area is the sum of the areas of all 6 faces: TSA = 2(LB + BH + HL).

Diagonal of a Cuboid

The length of the longest rod that can be placed inside a cuboid is given by √(L² + B² + H²).

Volume of a Cube

For a cube where all sides are equal (L = B = H = a), the volume is simply a³.

Shortcut Trick

Time = Distance ÷ Speed

Time = 60 ÷ 12 = 5

∴ The correct answer is 5 hr.

Alternate Method

Given:

Total Distance = 60 km

Speed of the cyclist = 12 km/hr

Formula Used:

Time = Distance / Speed

Calculations:

⇒ Time taken = 60 / 12

⇒ Time taken = 5 hr

∴ The correct answer is 5 hr.

Additional Information

Basic Speed Formula

Speed is defined as the distance covered per unit time. Speed = Distance ÷ Time.

Unit Conversion

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

Average Speed

Average Speed = (Total Distance) ÷ (Total Time taken).

Given: 

Number of people between = (Higher Position - Lower Position) - 1.

Total number of people in the row = 62

Shlok’s position from the left end = (Total - Position from Right) + 1 = (62 - 20) + 1 = 43

Biplov’s position from the left end = 45

Shlok → 43

Biplov → 45

Number of people between = (45 - 43) - 1 = 2 - 1 = 1

So, the number of people between Shlok and Biplov is 1.

Hence, the correct answer is Option 4.

The logic followed here is:

Given:

Town G is to the north of Town F

Town H is to the east of Town F

Town I is to the north of Town H

Town J is to the east of Town I

Town K is to the east of Town J

 

So, Town K is to the North-East of Town F.

Hence, the correct answer is Option 3.

The logic followed here is:

Given:

Nishant starts from Point A and drives 45 km North

He takes a right turn → 44 km East

He takes a right turn → 60 km South

He takes a right turn → 56 km West

He takes a final right turn → 15 km North

 

So, To reach Point A again, he must travel 12 km East.

Hence, the correct answer is Option 3.

Given: (Left) 7 7 & & # 3 3 # 9 1 4 € 4 # # 7 £ $ 4 © 9 6 (Right)

Now,

We need to find numbers which are immediately preceded by a symbol and also immediately followed by a symbol.

So, needed pattern is: Symbol - Number - Symbol

(Left) 7 7 & & # 3 3 # 9 1 4 € 4 # # 7 £ $ 4 © 9 6 (Right)

So, total valid numbers = 3

Hence, the correct answer is Option 4.

The logic followed here is:

thus, "42" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

Given: Six people P, Q, R, S, T and U are sitting around a circular table facing towards the centre of the table.

Exactly two people sit between S and T.

U sits to the immediate left of T.

R is immediate neighbour of S and Q both.

So, P is an immediate neighbour of both S and U.

Hence, the correct answer is "Option 2".

The logic followed here is:

FO-HQ-KM

HQ-JS-MO

Now, checking the options:

Option 1) JS-LU-OR

Option 2) IS-LV-OR

Option 3) IS-LU-OR

Option 4) JS-LU-OQ

Hence, the correct answer is Option 4.

Given: Seven friends L, M, N, O, Q, R and S are sitting in a straight line facing north.

Only three people sit between L and O.

No one sits to the right of N.

Only one person sits between N and O.

M is an immediate neighbour of both Q and L.

R sits to the immediate left of O.

So, only four persons sits between S and L.

Hence, the correct answer is "Option 4".