RRB Group D Official Paper (Held On: 29 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is A pollen tube grows through the style to reach the ovary.

Key Points

• The process of plant reproduction begins with pollination, which is the transfer of pollen grains from the anther to the stigma of a flower.
• Once a compatible pollen grain lands on the stigma, it absorbs moisture and sugar-rich secretions from the stigmatic surface, leading to its germination.
• During germination, a pollen tube emerges from one of the germ pores of the pollen grain and grows downward through the internal tissues of the style. Hence, statement 4 is correct.
• The pollen tube acts as a conduit to transport the male gametes (sperm cells) toward the ovule located inside the ovary.
• The growth of the tube is chemotropic, meaning it is guided by chemical signals secreted by the synergid cells within the embryo sac.
• Leaves do not begin photosynthesis specifically because of pollination; photosynthesis is a continuous metabolic process occurring in chlorophyll-containing tissues. Hence, statement 1 is incorrect.
• Petals usually wither or fall off after successful pollination rather than opening wider, as their primary role of attracting pollinators is complete. Hence, statement 2 is incorrect.
• Pollen does not develop into a seed directly; a seed only forms after the complex process of fertilization occurs within the ovule. Hence, statement 3 is incorrect.

Additional Information

• Siphonogamy: This refers to the process in seed plants where a pollen tube is used to transfer male gametes to the egg cell, removing the requirement for external water for fertilization.
• Double Fertilization: A unique feature of Angiosperms (flowering plants) where one male gamete fuses with the egg to form a diploid zygote, and the second male gamete fuses with two polar nuclei to form the triploid primary endosperm nucleus (PEN).
• Structure of the Pistil: The female reproductive part, or carpel, consists of the stigma (receptive tip), the style (stalk), and the ovary (swollen base containing ovules).
• Post-Fertilization Changes: After successful fertilization, the ovule matures into a seed, and the ovary wall develops into the pericarp or the fruit.

Important Points

• Pollen-Pistil Interaction: This is a dynamic recognition process where the pistil evaluates the pollen to ensure it is "compatible" (belonging to the same species and not self-sterile) before allowing the pollen tube to grow.
• Role of Nutrients: Essential elements like Boron and Calcium are critical for the successful germination and steady elongation of the pollen tube through the style.
• Generative and Vegetative Cells: The pollen grain typically consists of a vegetative cell (which forms the tube) and a generative cell (which divides to form two sperm cells).

The correct answer is When strong acid reacts with strong base.

Key Points

• A neutral salt is formed when a strong acid reacts with a strong base. In this chemical reaction, the hydrogen ions (H+) from the acid and hydroxyl ions (OH-) from the base combine to form water (H2O), leaving behind a salt that has no acidic or basic properties. Hence, statement 3 is correct.
• When a strong acid reacts with a weak base, the resulting salt is acidic in nature. This happens because the conjugate acid of the weak base is strong enough to donate protons to water, resulting in a pH value of less than 7. Hence, statement 1 is incorrect.
• When a weak acid reacts with a strong base, a basic salt is formed. The conjugate base of the weak acid reacts with water to produce hydroxyl ions, making the solution alkaline with a pH value greater than 7. Hence, statement 2 is incorrect.
• When an acid reacts with a metal, it generally produces a salt and hydrogen gas. While a salt is formed, this specific interaction is categorized as a single displacement reaction rather than a standard neutralization reaction intended to define the neutrality of a salt. Hence, statement 4 is incorrect.
• The pH of a neutral salt solution, such as sodium chloride (NaCl) or potassium nitrate (KNO3), is exactly 7 at 25 degrees Celsius.

Additional Information

• Neutralization Reaction:
• It is a chemical reaction in which an acid and a base react quantitatively with each other.
• The general formula is: Acid + Base → Salt + Water.
• This reaction is usually exothermic, meaning it releases heat energy to the surroundings.
• Strength of Acids and Bases:
• Strong Acids: Acids that ionize completely in aqueous solutions, such as Hydrochloric acid (HCl), Sulfuric acid (H2SO4), and Nitric acid (HNO3).
• Strong Bases: Bases that dissociate completely in water, such as Sodium hydroxide (NaOH) and Potassium hydroxide (KOH).
• Weak Acids: Acids that only partially ionize, like Acetic acid (CH3COOH) and Carbonic acid (H2CO3).
• Weak Bases: Bases that do not ionize fully, such as Ammonium hydroxide (NH4OH).
• pH Scale:
• The pH scale was introduced by S.P.L. Sorensen in 1909 to measure the acidity or alkalinity of a solution.
• It ranges from 0 to 14. A pH of 7 is neutral, below 7 is acidic, and above 7 is basic.

Important Points

• Common Examples of Salts:
• Sodium Chloride (NaCl): Formed from HCl (Strong Acid) and NaOH (Strong Base). It is a neutral salt used in food preservation and as common table salt.
• Ammonium Chloride (NH4Cl): Formed from HCl (Strong Acid) and NH4OH (Weak Base). It is an acidic salt used in dry cell batteries.
• Sodium Carbonate (Na2CO3): Formed from H2CO3 (Weak Acid) and NaOH (Strong Base). It is a basic salt used in glass manufacturing and water softening.
• Salt Hydrolysis:
• This is the process where salt ions react with water to produce acidity or alkalinity. Neutral salts derived from strong acids and strong bases do not undergo significant hydrolysis, which is why their pH remains 7.

The correct answer is Stamens – production of pollen grains.

Key Points

• Stamens represent the male reproductive part of a flower. They are responsible for the production of pollen grains, which contain the male gametes necessary for fertilization. Hence, statement 3 is correct.
• The Pistil (also known as the carpel) is the female reproductive part. Its primary biological function is to receive pollen at the stigma and house the ovules within the ovary for fertilization. While it is involved in reproduction, the attraction of pollinators is primarily the function of the petals. Hence, statement 1 is incorrect.
• Sepals are the outermost green leaf-like structures of a flower, collectively known as the Calyx. Their primary function is to protect the flower during its bud stage. They do not develop ovules; ovules develop inside the ovary of the pistil. Hence, statement 2 is incorrect.
• Petals are the brightly colored parts of the flower, collectively called the Corolla. Their primary function is the attraction of pollinators like insects and birds through their color and scent. They do not produce female gametes; these are produced within the embryo sac inside the ovule. Hence, statement 4 is incorrect.
• A typical flower is organized into four whorls: the calyx, corolla, androecium (stamens), and gynoecium (pistil).

Additional Information

• Structure of Stamen:
• It consists of two main parts: the Anther and the Filament.
• The Anther is a bilobed structure where microsporangia develop to produce pollen grains through the process of microsporogenesis.
• The Filament is a long, slender stalk that supports the anther and attaches it to the thalamus or petal.
• Structure of Pistil:
• The Stigma serves as a landing platform for pollen grains.
• The Style is the elongated tube that connects the stigma to the ovary.
• The Ovary is the basal swollen part containing one or more ovules attached to a flattened, cushion-like placenta.
• Pollination and Fertilization:
• Pollination is the physical transfer of pollen grains from the anther to the stigma. This can be facilitated by wind, water, or biological agents.
• After pollination, the pollen grain germinates, and a pollen tube grows through the style to reach the ovule for fertilization.

Important Points

• Essential vs. Non-essential Whorls: The Androecium and Gynoecium are called essential whorls because they are directly involved in sexual reproduction. The Calyx and Corolla are accessory or non-essential whorls as they assist in protection and attraction.
• Flower Types:
• Bisexual (Hermaphrodite): Flowers that possess both stamens and pistils, such as Lily, Rose, and Hibiscus.
• Unisexual: Flowers that possess either stamens (staminate) or pistils (pistillate), such as Papaya, Corn, and Cucumber.
• Post-Fertilization Changes: After successful fertilization, the ovule develops into a seed, and the ovary matures into a fruit. The petals and sepals often wither and fall off.

The correct answer is formation of bluish-green colour solution of copper chloride.

Key Points

• When copper oxide reacts with dilute hydrochloric acid, a chemical reaction occurs that transforms the black solid powder into a soluble salt.
• Copper (II) oxide (CuO) is a basic oxide, and its reaction with hydrochloric acid (HCl) is a type of neutralization reaction (Metal Oxide + Acid → Salt + Water).
• The chemical equation for this process is: CuO + 2HCl → CuCl2 + H2O.
• The black colour of the copper oxide disappears as it dissolves in the acid, and the solution turns bluish-green.
• This characteristic bluish-green colour is due to the formation of Copper (II) chloride (CuCl2) in the aqueous solution.
• Statement 1 suggests the formation of a reddish-brown solution of copper sulphate. This is incorrect because the acid used is hydrochloric acid, not sulphuric acid, and copper sulphate typically forms a blue solution. Hence, statement 1 is incorrect.
• Statement 2 suggests a reddish-brown colour for copper chloride. While copper chloride is formed, its aqueous state is bluish-green, not reddish-brown. Hence, statement 2 is incorrect.
• Statement 3 states the formation of a bluish-green colour solution of copper chloride. This accurately describes the visual and chemical result of the reaction. Hence, statement 3 is correct.
• Statement 4 mentions a reddish-brown solution of cupric chloride. "Cupric chloride" is another name for copper (II) chloride, but the colour description is wrong. Hence, statement 4 is incorrect.

Additional Information

• Copper (II) Oxide (CuO): It is an inorganic compound, often appearing as a black solid. It is a mineral known as tenorite and acts as a precursor to many copper-containing products.
• Copper (II) Chloride (CuCl2): In its anhydrous form, it is yellowish-brown, but it commonly exists as a dihydrate (CuCl2·2H2O) which is bright green. In dilute aqueous solutions, the copper ions coordinate with water to produce the bluish-green tint.
• Nature of Metal Oxides: Most metal oxides are basic in nature. When they react with mineral acids like HCl, H2SO4, or HNO3, they form the corresponding metal salts and water.
• Neutralization Reaction: This specific reaction is a double displacement or neutralization where the hydrogen ions from the acid combine with the oxygen from the oxide to form water, while the metal cation and acid anion form the salt.
• Reactivity: While copper metal itself does not react with dilute hydrochloric acid because it is below hydrogen in the reactivity series, its oxide (CuO) reacts readily because the reaction is driven by the formation of water.

Important Points

• Oxidation State: In this reaction, copper remains in the +2 oxidation state. CuO is copper (II) oxide, and the resulting CuCl2 is copper (II) chloride (also known as cupric chloride).
• Visual Indicators: The transition from a black powder to a clear bluish-green liquid is a definitive chemical change.
• Safety: When performing this in a laboratory, stirring is essential to ensure the black CuO particles react completely with the dilute HCl.
• Product Identification: If the solution were evaporated, green crystals of copper chloride dihydrate would be obtained.

The correct answer is Electrolytic refining.

Key Points

• Electrolytic refining is a technique used for the purification of crude metals to obtain them in a highly pure state. It is the most widely used method for metals like Copper (Cu), Zinc (Zn), Tin (Sn), Nickel (Ni), Silver (Ag), and Gold (Au).
• In this process, a thick block of the impure metal acts as the anode (the positive electrode), and a thin strip of the pure metal acts as the cathode (the negative electrode).
• A water-soluble salt of the same metal is used as the electrolyte. For example, in the refining of Copper, acidified Copper Sulfate (CuSO4) solution is used as the electrolyte.
• When an electric current is passed through the electrolyte, the metal ions from the anode dissolve into the solution, while an equivalent amount of pure metal from the solution is deposited onto the cathode. Hence, the statement regarding electrolytic refining is correct.
• During this process, the soluble impurities go into the electrolyte solution, whereas the insoluble impurities settle down at the bottom of the anode and are known as anode mud.

Additional Information

• Alloying:
• This is the process of mixing a metal with one or more other elements (metals or non-metals) to improve its physical properties like hardness, tensile strength, and corrosion resistance.
• Examples include Brass (Copper and Zinc), Bronze (Copper and Tin), and Steel (Iron and Carbon). It is a method of modification, not purification.
• Roasting:
• Roasting is a metallurgical process where sulfide ores are heated strongly in the presence of excess air or oxygen.
• The goal is to convert the sulfide ore into a metal oxide, which is easier to reduce. For example, Zinc Blende (ZnS) is roasted to form Zinc Oxide (ZnO).
• Smelting:
• Smelting is a form of extractive metallurgy used to produce a metal from its ore. It involves heating the ore beyond its melting point, usually in the presence of a reducing agent like Carbon (Coke).
• This process removes the oxygen from the metal oxide, leaving behind the crude metal.
• Calcination:
• Similar to roasting, calcination involves heating the ore strongly, but in the absence or limited supply of air. This is typically used for carbonate or hydrated ores to remove moisture and volatile impurities.

Important Points

• Efficiency: Electrolytic refining can produce metals with a purity level as high as 99.99%, which is essential for electrical applications where high conductivity is required.
• Economic Value: The anode mud collected during the refining of Copper often contains precious metals like Antimony, Selenium, Tellurium, Silver, Gold, and Platinum. The recovery of these metals often covers the cost of the entire refining process.
• Oxidation and Reduction: In the electrolytic cell, oxidation occurs at the anode (loss of electrons by metal atoms), and reduction occurs at the cathode (gain of electrons by metal ions).

The correct answer is A train starting from rest.

Key Points

• Non-uniform linear velocity occurs when an object travels in a straight line but its speed changes over time, or when its direction changes while moving. In the case of linear motion, the direction remains constant, so non-uniformity arises specifically from a change in magnitude (speed).
• A train starting from rest implies that its initial velocity is zero. As the train begins to move, it must accelerate to gain speed. Because the magnitude of its velocity is increasing from 0 m/s to a higher value, the velocity is not constant. Hence, statement 4 is correct.
• A ball rolling at a uniform rate suggests that the ball is covering equal distances in equal intervals of time. If it is rolling in a single direction, both its speed and direction are constant, representing uniform velocity. Hence, statement 1 is incorrect.
• A body moving in a straight line at fixed speed is the classic definition of uniform linear velocity. Since the direction is fixed (straight line) and the magnitude is fixed (fixed speed), there is no change in the velocity vector. Hence, statement 2 is incorrect.
• A car moving at constant speed, if assumed to be on a straight path, maintains the same velocity throughout its motion. Without a change in rate of motion or path, it does not exhibit non-uniform behavior. Hence, statement 3 is incorrect.

Additional Information

• Velocity vs. Speed: Speed is a scalar quantity that refers to "how fast an object is moving." Velocity is a vector quantity that refers to "the rate at which an object changes its position." Velocity must specify both magnitude and direction.
• Uniform Motion: An object is said to be in uniform motion if it moves along a straight line with a constant velocity. This means it covers equal displacements in equal intervals of time, no matter how small these time intervals may be.
• Non-Uniform Motion: This occurs when an object covers unequal distances in equal intervals of time. It is also known as accelerated motion. A common example is a freely falling body or a vehicle moving through heavy traffic where brakes and accelerators are used frequently.
• Acceleration: This is defined as the rate of change of velocity with respect to time. If the velocity of an object changes from an initial value u to a final value v in time t, the acceleration a is given by (v - u) / t. In the case of a train starting from rest, u is always zero.
• Linear Motion: Also known as rectilinear motion, it is motion along a straight line. It can be uniform (constant velocity) or non-uniform (variable velocity/acceleration).

Important Points

• Change in Velocity: Velocity can be changed in three ways: by changing the speed, by changing the direction, or by changing both. For linear velocity, we primarily focus on the change in speed.
• Circular Motion Context: Even if a body moves at a constant speed in a circle, its velocity is non-uniform because the direction of motion is changing at every point. This is known as uniform circular motion, but the velocity vector itself is non-uniform.
• SI Units: The SI unit for both speed and velocity is meters per second (m/s), while the SI unit for acceleration is meters per second squared (m/s²).

The correct answer is Bos indicus.

Key Points

• Bos indicus, popularly known as Zebu cattle, is the scientific name for the indigenous humped cattle breeds found in India and other parts of South Asia and Africa.
• These cattle are distinctively characterized by a prominent fatty hump over their shoulders, a large dewlap (the fold of skin hanging below the neck), and sometimes drooping ears.
• The species is highly adapted to the tropical environment, showing remarkable resistance to high temperatures, humidity, and ectoparasites such as ticks.
• Historically, Bos indicus is believed to have originated in the Indian subcontinent and was domesticated as early as 6000 BCE.
• They are primarily used for dairy, draught power (ploughing and transport), and manure in traditional Indian agriculture. Hence, the correct answer is Bos indicus.

Additional Information

• Bos taurus:
• This refers to the humpless cattle species native to Europe and Northern Asia.
• Common examples include high milk-yielding breeds like Holstein Friesian, Jersey, and Brown Swiss.
• Unlike Bos indicus, these breeds are best suited for temperate climates and are more susceptible to heat stress and tropical diseases.
• Bubalus bubalis:
• This is the scientific name for the Domestic Water Buffalo, which is a different genus altogether from the cow.
• India possesses the largest population of water buffaloes in the world, with famous breeds like the Murrah, Surti, and Jaffrabadi.
• They are a major source of milk production in India, contributing significantly to the White Revolution.
• Bos bubalis:
• This term is often used as a synonym for Bubalus bubalis in older biological texts, though modern taxonomy classifies the water buffalo under the genus Bubalus.

Important Points

• A2 Milk Significance: Indian indigenous breeds (Bos indicus) are known for producing A2 beta-casein milk, which is scientifically considered easier to digest and healthier than the A1 milk typically produced by Bos taurus breeds.
• Major Indian Breeds: Some of the most notable Bos indicus breeds in India include the Gir (Gujarat), Sahiwal (Punjab/Haryana), Red Sindhi, Tharparkar (Rajasthan), and Ongole (Andhra Pradesh).
• Draught Breeds: Breeds like the Hallikar and Amrit Mahal from Karnataka are specifically bred for their immense physical strength and endurance in agricultural labor.
• Taxonomic Classification: Both Bos indicus and Bos taurus belong to the family Bovidae and the subfamily Bovinae, under the order Artiodactyla.
• Government Initiatives: To protect these indigenous species, the Government of India launched the Rashtriya Gokul Mission in 2014 for the conservation and scientific development of indigenous cattle breeds.

The correct answer is magnitude and direction.

Key Points

• A vector quantity is defined in physics as a physical quantity that possesses both a numerical value (magnitude) and a specified direction.
• Momentum, represented by the symbol p, is the product of an object's mass (m) and its velocity (v). Since velocity is a vector quantity, the resulting momentum must also be a vector quantity.
• The magnitude of momentum refers to "how much" motion an object has, which depends on how heavy the object is and how fast it is moving.
• The direction of momentum is always the same as the direction of the object's velocity. For example, if a car is moving North, its momentum vector is also directed toward the North.
• In contrast, scalar quantities such as mass, time, and speed only have magnitude and no direction. Hence, the correct answer is magnitude and direction.

Additional Information

• SI Unit and Dimensions:
• The Standard International (SI) unit of momentum is the kilogram-meter per second (kg·m/s).
• In terms of dimensional analysis, momentum is expressed as [M L T⁻¹], where M is mass, L is length, and T is time.
• Newton's Second Law of Motion:
• Sir Isaac Newton established that the force (F) acting on an object is equal to the rate of change of its momentum over time (t).
• This relationship is mathematically represented as F = dp/dt. If the mass remains constant, this simplifies to F = ma (force equals mass times acceleration).
• Law of Conservation of Momentum:
• This fundamental principle states that in an isolated system where no external forces are acting, the total linear momentum of the system remains constant (conserved).
• This law is crucial in analyzing collisions and explosions, such as the recoil of a gun when a bullet is fired.
• Impulse:
• Impulse is the change in momentum of an object when a force acts upon it for an interval of time.
• It is also a vector quantity and is measured in Newton-seconds (N·s), which is equivalent to kg·m/s.
• Relativistic Momentum:
• At speeds approaching the speed of light (c), the classical formula for momentum (p=mv) is modified according to Einstein's Theory of Relativity to account for Lorentz factor adjustments.

The correct answer is remains the same.

Key Points

• The relationship between wave speed (v), frequency (f), and wavelength (λ) is given by the wave equation: v = f × λ. This formula indicates that the speed of a wave is the product of its frequency and its wavelength.
• To find the frequency, the equation can be rearranged as f = v / λ. This shows that frequency is directly proportional to the wave speed and inversely proportional to the wavelength.
• In the given problem, the speed of the wave becomes three times its original value (3v), and the wavelength also becomes three times its original value (3λ).
• Substituting these new values into the frequency formula: New Frequency (f') = (3v) / (3λ). The numerical factor of 3 in the numerator and denominator cancels out, resulting in f' = v / λ.
• Since the new frequency f' is equal to the original frequency f, the frequency remains the same. Hence, the correct answer is remains the same.

Additional Information

• Wave Speed (v):
• It is the distance traveled by a wave per unit of time. The SI unit for wave speed is meters per second (m/s).
• The speed of a wave is primarily determined by the properties of the medium through which it travels (e.g., density, elasticity, or temperature).
• Wavelength (λ):
• It is the distance between two consecutive identical points of a wave, such as from crest to crest or trough to trough in a transverse wave.
• The SI unit of wavelength is the meter (m). In a given medium, if the frequency increases, the wavelength must decrease to maintain a constant speed.
• Frequency (f):
• It is the number of waves (or cycles) that pass a fixed point in a given amount of time, usually one second.
• The SI unit of frequency is the Hertz (Hz), named after the physicist Heinrich Hertz. 1 Hz is equal to 1 cycle per second.
• Time Period (T):
• It is the time taken for one complete cycle of the wave to pass a point. It is the reciprocal of frequency, expressed as T = 1 / f.
• The SI unit for the time period is the second (s).

Important Points

• Source Dependency: The frequency of a wave is determined solely by the source of the vibration. Once a wave is generated, its frequency does not change even if it moves into a different medium.
• Medium Dependency: When a wave enters a different medium (e.g., light traveling from air into glass), its speed and wavelength change, but its frequency remains constant.
• Electromagnetic Waves: For electromagnetic waves in a vacuum, the speed is a constant denoted by c (approximately 3 × 10^8 m/s). In this case, c = fλ, meaning frequency and wavelength are always perfectly inversely proportional.
• Sound Waves: In sound, frequency is perceived as pitch. A high-frequency sound has a high pitch, while a low-frequency sound has a low pitch.

The correct answer is Measuring electrical resistance of a wire.

Key Points

• Archimedes' Principle states that when a body is immersed fully or partially in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the body.
• Calculating Buoyant Force: This is the most direct application of the principle. By determining the volume of the displaced fluid and multiplying it by the density of the fluid and the acceleration due to gravity, the magnitude of the buoyant force acting on any object (floating or submerged) can be found. Hence, statement 1 is an application of the principle.
• Determining Density of Irregular Solids: Archimedes' principle is used to find the volume of an irregularly shaped object by measuring the volume of fluid it displaces when submerged. Once the volume is known, the density of the solid can be calculated by dividing its mass by that volume. Hence, statement 2 is an application of the principle.
• Submarines and Floating: Submarines utilize this principle to dive and surface. They have ballast tanks that can be filled with either water (increasing the overall weight/density) or air (decreasing the overall weight/density) to control the buoyancy relative to the surrounding water. Hence, statement 4 is an application of the principle.
• Measuring Electrical Resistance: This process involves Ohm's Law and the physical properties of a conductor, such as its resistivity, length, and cross-sectional area. It is measured using devices like a Wheatstone bridge or a multimeter and has no relation to fluid mechanics or buoyancy. Hence, statement 3 is NOT an application of the principle.

Additional Information

• Lactometers: These are specialized instruments used to determine the purity of milk. They operate on Archimedes' principle by measuring the density of the milk sample to detect the presence of added water.
• Hydrometers: These devices are used to measure the relative density or specific gravity of liquids. A hydrometer sinks deeper in a liquid of lower density and less in a liquid of higher density, following the laws of buoyancy.
• Ship Designing: Ships are designed with large hollow structures to ensure they displace a weight of water equal to or greater than their own mass. This allows heavy metal ships to remain afloat on the surface of the ocean.
• Hot Air Balloons: The principle also applies to gases. A hot air balloon rises because the hot air inside the envelope is less dense than the surrounding cool air, creating a buoyant force greater than the balloon's weight.

Important Points

• Historical Context: The principle was discovered by Archimedes of Syracuse in the 3rd century BC. According to legend, he discovered it while stepping into a bathtub and realized the water level rose as he entered.
• Buoyancy Formula: The buoyant force (Fb) can be expressed as Fb = V x rho x g, where V is the volume of displaced fluid, rho is the density of the fluid, and g is the acceleration due to gravity (approximately 9.8 m/s²).
• Apparent Weight: When an object is submerged, it appears to weigh less. This apparent weight is equal to the actual weight in air minus the buoyant force provided by the fluid.
• Criterion for Sinking: An object will sink if its average density is greater than the density of the fluid. It will float if its average density is less than that of the fluid.

The correct answer is Connect the ammeter in series with correct polarity.

Key Points

• An ammeter is a measuring instrument used to measure the electric current in a circuit. To measure the current passing through a component like a bulb, the ammeter must be connected in series so that the entire current of the circuit flows through it.
• The polarity of the ammeter is a critical precaution; the positive terminal (usually marked red or with a plus sign) must be connected to the wire coming from the positive terminal of the battery, and the negative terminal (usually black) should be directed towards the negative terminal. Hence, statement 2 is correct.
• Connecting an ammeter without a battery or power source is futile because there is no Electromotive Force (EMF) to drive the flow of electrons, resulting in a zero reading. Hence, statement 1 is incorrect.
• If an ammeter is connected with reversed polarity, the pointer in an analog meter will try to move in the opposite direction (below zero), which can cause mechanical damage to the needle or provide a negative reading on a digital multimeter. Hence, statement 3 is incorrect.
• Connecting an ammeter in parallel with a load (like a bulb) is dangerous because ammeters have very low internal resistance. A parallel connection creates a short circuit, allowing a massive amount of current to flow through the ammeter, which can burn the coil or blow the internal fuse. Hence, statement 4 is incorrect.

Additional Information

• Ideal Ammeter Characteristics:
• An ideal ammeter is defined as having zero resistance. This ensures that its inclusion in the circuit does not alter the total resistance and, consequently, does not change the current it is intended to measure.
• In practice, real ammeters have a very small but non-zero resistance, which causes a negligible voltage drop across them.
• Galvanometer to Ammeter Conversion:
• An ammeter is typically constructed by connecting a low resistance wire, known as a Shunt, in parallel with a sensitive Galvanometer.
• The shunt resistance allows the majority of the current to bypass the delicate galvanometer coil, enabling the measurement of large currents without damaging the instrument.
• Units and History:
• The unit of electric current is the Ampere (A), which is one of the seven SI base units. It is defined as the flow of one Coulomb of charge per second.
• The instrument is named after the French physicist André-Marie Ampère, who made significant contributions to the study of electromagnetism in the 1820s.
• Comparison with Voltmeter:
• Unlike an ammeter, a voltmeter is used to measure potential difference and is always connected in parallel.
• A voltmeter possesses high resistance to ensure it draws minimal current from the main circuit, whereas an ammeter requires low resistance to allow maximum current through it.

Important Points

• Safety Protocols: Before turning on the power, always verify the range of the ammeter. If the current exceeds the maximum scale deflection, the device may be permanently damaged.
• Types of Ammeters: Modern science uses various types, including Moving Coil Ammeters (for DC), Moving Iron Ammeters (for both AC and DC), and Digital Ammeters which use analog-to-digital converters (ADC) for high precision.
• Circuit symbols: In a standard circuit diagram, an ammeter is represented by a circle with the letter 'A' inside it, placed in the path of the current flow.

The correct answer is at the centre of curvature itself, real and of the same size.

Key Points

• When an object is placed at the centre of curvature (C) of a concave mirror, the image is formed at the centre of curvature (C) itself.
• The nature of the image formed in this specific position is real and inverted.
• The size of the image is exactly the same as the size of the object, meaning the linear magnification (m) is -1. Hence, statement 4 is correct.
• To understand this formation, consider two rays: one ray parallel to the principal axis passes through the Focus (F) after reflection, and another ray passing through the Focus (F) becomes parallel to the principal axis after reflection. Both rays meet at point C.
• If the object is placed between the focus (F) and the pole (P), the image is formed behind the mirror and is virtual, erect, and magnified. Hence, statement 1 is incorrect.
• If the object is placed at infinity, the image is formed at the focus (F) and is real, inverted, and highly diminished (point-sized). Hence, statement 3 is incorrect.
• If the object is placed at the focus (F), the image is formed at infinity and is real, inverted, and highly magnified. Hence, statement 2 is incorrect.

Additional Information

• Concave Mirror: It is a spherical mirror whose reflecting surface is curved inwards (towards the center of the sphere). It is also known as a converging mirror because it converges light rays at a single point.
• Mirror Formula: The relationship between the object distance (u), image distance (v), and focal length (f) is given by 1/f = 1/v + 1/u.
• Magnification (m): It is the ratio of the height of the image (h') to the height of the object (h), expressed as m = h'/h = -v/u. A negative sign indicates a real image, while a positive sign indicates a virtual image.
• Uses of Concave Mirrors: They are widely used in shaving mirrors and makeup mirrors to see a larger image, by dentists to see large images of teeth, and in solar furnaces to concentrate sunlight.
• Sign Convention: According to the New Cartesian Sign Convention, the focal length of a concave mirror is always negative, and the object distance (u) is always negative.

Important Points

• Radius of Curvature (R): The distance from the pole (P) to the centre of curvature (C). For spherical mirrors with small apertures, the Radius of Curvature is twice the focal length, mathematically represented as R = 2f.
• Principal Focus (F): For a concave mirror, it is the point on the principal axis where all rays parallel to the axis actually converge after reflection.
• Aperture: The diameter of the reflecting surface of the spherical mirror is called its aperture.
• Comparison with Convex Mirror: Unlike concave mirrors, a convex mirror always forms a virtual, erect, and diminished image regardless of the object's position.

The correct answer is Mass remains unchanged in reactions.

Key Points

• The Law of Conservation of Mass states that in an isolated system, mass is neither created nor destroyed by chemical reactions or physical transformations.
• According to the law, the mass of the products in a chemical reaction must equal the mass of the reactants. This means that the total number of atoms of each element is preserved throughout the process.
• The law was established by Antoine Lavoisier in 1789, marking a significant transition from qualitative to quantitative chemistry.
• Statement (1) suggests that the mass of reactants is always lost. This contradicts the principle that matter is conserved and not destroyed. Hence, statement 1 is incorrect.
• Statement (2) asserts that mass remains unchanged during reactions. This is the fundamental definition of the law, where the total mass remains constant before and after the reaction. Hence, statement 2 is correct.
• Statement (3) mentions that energy can be converted into mass. While true in relativistic physics and nuclear reactions, it does not define the classical Law of Conservation of Mass used in standard chemistry. Hence, statement 3 is incorrect.
• Statement (4) claims that the mass of products is always greater. This is impossible under the law as no new matter can be created from nothing. Hence, statement 4 is incorrect.

Additional Information

• Antoine Lavoisier: Known as the Father of Modern Chemistry, he used meticulous experimental methods and precision balances to prove that mass does not change during combustion or other reactions.
• Stoichiometry: The Law of Conservation of Mass is the foundation for balancing chemical equations. Since atoms are only rearranged and not lost, the coefficients in an equation must ensure the same number of atoms on both sides.
• Isolated vs. Open Systems: In an open system, mass might appear to change (e.g., a burning log losing mass as carbon dioxide and water vapor escape), but in a closed container, the total mass remains perfectly constant.
• Chemical vs. Physical Change: The law applies to both. For example, when 10 grams of ice melts into liquid water, the resulting mass is exactly 10 grams.
• Historical Context: Before 1789, many scientists believed in the Phlogiston theory, which argued that a substance called phlogiston was released during burning, explaining the apparent loss of weight in some substances.

Important Points

• Mass-Energy Equivalence: In modern physics, Albert Einstein expanded this concept with the formula E=mc², showing that mass and energy are interchangeable. However, in ordinary chemical reactions, the amount of mass converted to energy is so minuscule that it cannot be detected by standard instruments.
• Application: This law is crucial for industrial chemistry, environmental science, and pharmacology to calculate the exact yield of products from specific amounts of raw materials.
• Year of Discovery: The formal publication of this law occurred in 1789 in Lavoisier's textbook, "Traité Élémentaire de Chimie".

The correct answer is Cu.

Key Points

• The provided chemical equation is CuO + H2 -> Cu + H2O, which represents a classic Redox (Reduction-Oxidation) reaction occurring under Heat.
• Reduction is defined as the process of loss of oxygen from a substance or the gain of electrons by a species.
• In this reaction, Copper(II) oxide (CuO) reacts with Hydrogen (H2). The CuO loses its oxygen atom to form metallic Copper (Cu). Hence, the species being reduced is Cu (specifically the Copper ion in CuO).
• Alternatively, considering Oxidation States, the oxidation number of Copper in CuO is +2, while in its elemental form Cu, it is 0. A decrease in oxidation state from +2 to 0 confirms that reduction has taken place.
• The species H2 gains oxygen to form H2O, which means Hydrogen is undergoing Oxidation. Hence, statement 1 is incorrect.
• Since Cu is the final state of the reduced species and represents the element that underwent the change, it is identified as the species reduced in the context of the options provided. Hence, the correct answer is Cu.

Additional Information

• Oxidizing Agent: This is the substance that provides oxygen or removes electrons from another substance. In this reaction, CuO acts as the oxidizing agent because it oxidizes H2 to H2O.
• Reducing Agent: This is the substance that removes oxygen or provides electrons. Here, H2 acts as the reducing agent as it removes oxygen from CuO.
• Visual Observation: During this experiment, the black colored Copper(II) oxide powder changes into a reddish-brown solid, which is metallic Copper.
• Reaction Type: This is a displacement reaction as well as a redox reaction. Hydrogen is more reactive than Copper under these thermal conditions and displaces it from its oxide.
• Water Formation: The Hydrogen gas acts as a fuel and a reductant, resulting in the formation of water vapor (H2O) as a byproduct of the oxidation process.

Important Points

• Electronic Concept of Reduction: In modern chemistry, reduction is also viewed as the gain of electrons (OIL RIG: Oxidation Is Loss, Reduction Is Gain). The Cu2+ ion gains 2 electrons to become Cu atom.
• Oxidation State Rules: The oxidation state of oxygen is generally -2 in most compounds like CuO and H2O. Therefore, the oxygen itself does not undergo a change in oxidation state in this specific reaction.
• Thermodynamics: The reaction requires Heat because it is endothermic initially to break the Cu-O bonds, though the overall process facilitates the transfer of oxygen to the more stable H-O bond in water.

The correct answer is The optical centre of the lens.

Key Points

• For spherical lenses, we follow a set of sign conventions called the New Cartesian Sign Convention to measure distances.
• According to this convention, the optical centre (denoted by O) of the lens is taken as the origin of the coordinate system.
• The principal axis of the lens is taken as the X-axis (horizontal axis) of the coordinate system.
• All distances are measured from the optical centre of the lens. Distances measured in the direction of incident light (usually to the right of the optical centre) are taken as positive.
• Distances measured in the direction opposite to that of the incident light (usually to the left of the optical centre) are taken as negative.

Additional Information

• Object Distance (u):
• The object is always placed to the left of the lens.
• Since it is measured against the direction of incident light, the object distance (u) is always negative in both convex and concave lenses.
• Image Distance (v):
• For a real image, which is formed on the right side of the lens, the image distance v is positive.
• For a virtual image, which is formed on the left side (the same side as the object), the image distance v is negative.
• Focal Length (f):
• A convex lens (converging lens) has a positive focal length because its principal focus lies on the right side.
• A concave lens (diverging lens) has a negative focal length because its principal focus is considered to be on the left side.
• Height of Object and Image (h and h'):
• Distances measured perpendicularly above the principal axis are taken as positive. Thus, the object height (h) is usually positive.
• Distances measured perpendicularly below the principal axis are taken as negative. Therefore, the height of a real and inverted image is negative, while a virtual and erect image is positive.

Important Points

• Lens Formula: The relationship between the object distance (u), image distance (v), and focal length (f) is given by the formula: 1/v - 1/u = 1/f. This formula is valid for all spherical lenses in all situations.
• Magnification (m): The linear magnification produced by a lens is the ratio of the height of the image to the height of the object. It is expressed as m = h'/h. It is also related to distances as m = v/u.
• Power of a Lens (P): The power of a lens is defined as the reciprocal of its focal length in metres (P = 1/f). The SI unit of power is Dioptre (D). A convex lens has positive power, and a concave lens has negative power.
• Comparison with Mirrors: Unlike lenses where measurements start from the optical centre, in spherical mirrors, all measurements are taken from the pole of the mirror.

The correct answer is Complex permanent tissues are made up of different types of cells performing different tasks.

Key Points

• Complex permanent tissues are defined as plant tissues composed of more than one type of cell that work together as a unit. In these tissues, different cells carry out distinct sub-functions to achieve a collective goal. Hence, statement 1 is correct.
• While the cells in complex tissues coordinate to perform a unified function (like conduction), the individual cell types within the tissue have specialized structures and perform different tasks. For example, some provide support while others transport fluids. Hence, statement 2 is incorrect.
• Complex permanent tissues are not "made of" simple permanent tissues; rather, they are a distinct category of permanent tissues that develop through differentiation from meristematic tissues. They may, however, contain certain types of simple cells like parenchyma as constituent elements. Hence, statement 3 is incorrect.
• Simple permanent tissues (like parenchyma, collenchyma, and sclerenchyma) are made up of similar types of cells, whereas complex permanent tissues are heterogeneous. Hence, statement 4 is incorrect.
• The most prominent examples of complex permanent tissues are Xylem and Phloem, which are together referred to as Vascular Tissues or Conducting Tissues.

Additional Information

• Xylem:
• It is the chief water-conducting tissue of plants, facilitating the unidirectional transport of water and dissolved minerals from the roots to the aerial parts.
• It consists of four main elements: Tracheids, Vessels, Xylem Parenchyma, and Xylem Fibers.
• In Angiosperms, both tracheids and vessels are the primary water-conducting elements, whereas Gymnosperms generally lack vessels.
• Phloem:
• It is responsible for the translocation of organic food (sucrose) manufactured in the leaves to various storage organs and growing regions.
• The transport in phloem is bidirectional (multi-directional), moving from source to sink.
• It consists of Sieve tubes, Companion cells, Phloem Parenchyma, and Phloem Fibers.
• Tissue Differentiation:
• Permanent tissues are formed when the cells produced by apical, lateral, or intercalary meristems stop dividing and take on a permanent shape and function.
• This process is vital for the structural integrity and physiological efficiency of higher plants.

Important Points

• Cell Vitality: In the Xylem, only Xylem Parenchyma is a living cell, while the others are dead at maturity. Conversely, in the Phloem, only Phloem Fibers (bast fibers) are dead cells, while sieve tubes and companion cells are living.
• Lignification: The walls of vessels and tracheids are heavily lignified (thickened with lignin), which provides significant mechanical strength to the plant body, allowing it to stand upright.
• Vascular Bundles: In the stems and roots of plants, xylem and phloem are organized into vascular bundles. Their arrangement varies between Dicotyledons (where they are often arranged in a ring) and Monocotyledons (where they are scattered).
• Companion Cells: These are specialized parenchymatous cells closely associated with sieve tube elements. They help maintain the pressure gradient required for the movement of sugars within the phloem.

The correct answer is They are genetically identical to the parent plant.

Key Points

• Vegetative propagation is a type of asexual reproduction where new plants are produced from vegetative parts of the parent plant, such as roots, stems, and leaves.
• Since only one parent is involved and there is no fusion of gametes, the offspring produced are genetically identical to the parent plant. These identical individuals are often referred to as clones. Hence, statement 4 is correct.
• In this process, the cells divide through mitosis rather than meiosis. Because meiosis and genetic recombination do not occur, there is no genetic diversity among the offspring. Hence, statement 1 is incorrect.
• Vegetative propagation does not involve pollen or ovules, meaning it does not require fertilization to develop new individuals. Hence, statement 3 is incorrect.
• Plants produced through this method generally grow faster and reach maturity earlier than those grown from seeds, as they bypass the delicate seedling stage. Hence, statement 2 is incorrect.

Additional Information

• Natural Methods of Vegetative Propagation:
• Runners (Stolons): Horizontal stems that grow above the ground, such as in Strawberries.
• Rhizomes: Underground stems that grow horizontally and store food, found in Ginger and Turmeric.
• Tubers: Swollen ends of underground branches that store starch, such as the Potato (which has "eyes" that act as buds).
• Bulbs: Short underground stems surrounded by fleshy leaves, as seen in Onions and Garlic.
• Leaves: Some plants like Bryophyllum produce adventitious buds along their leaf margins that drop off to form new plants.
• Artificial Methods (Horticulture):
• Cutting: A part of the stem or root is placed in the soil to grow into a new plant (e.g., Roses, Sugarcane).
• Grafting: Joining parts from two different plants (the scion and the stock) so they grow as one. This is common in Mango and Citrus fruits.
• Layering: A branch is bent to the ground and covered with soil until roots develop (e.g., Jasmine).
• Tissue Culture (Micropropagation): Growing whole plants from small explants (cells or tissues) in a sterile nutrient medium under laboratory conditions.

Important Points

• Advantages:
• It allows for the preservation of desirable traits (quality, taste, size) across generations.
• It is the only way to propagate plants that do not produce viable seeds, such as Bananas, Seedless Grapes, and Oranges.
• The survival rate is higher because the new plant receives sustenance from the parent plant during its initial growth.
• Disadvantages:
• Since offspring are genetically identical, they are all equally susceptible to the same diseases or pests.
• Lack of genetic variation hinders the evolutionary adaptation of the species to changing environmental conditions.
• Overcrowding often occurs near the parent plant due to the lack of a natural dispersal mechanism (unlike seeds).

The correct answer is When the force acts opposite to the direction of displacement.

Key Points

• In physics, work done is defined as the product of the component of the force in the direction of the displacement and the magnitude of this displacement. Mathematically, it is expressed as W = Fs cos(θ), where θ is the angle between the force vector and the displacement vector.
• Negative work occurs when the force acts in the opposite direction to the displacement. In this case, the angle θ between the force and displacement is 180 degrees. Since cos(180°) = -1, the work done becomes W = -Fs. Hence, statement 1 is correct.
• Positive work occurs when the force acts in the same direction as the displacement. Here, the angle θ is 0 degrees, and since cos(0°) = 1, the work done is W = Fs. This happens when you push an object and it moves in the direction of the push. Hence, statement 2 is incorrect.
• Zero work is done when the force applied is zero. If no force is exerted on an object, no energy is transferred via mechanical means, regardless of the motion. Hence, statement 3 is incorrect.
• Work done is also zero when there is no displacement (s = 0), even if a massive force is applied. For example, pushing against a stationary wall results in zero work because the wall does not move. Hence, statement 4 is incorrect.

Additional Information

• Nature of Work: Work is a scalar quantity, meaning it has magnitude but no direction, even though it is derived from two vector quantities (force and displacement). The SI unit of work is the Joule (J), which is equivalent to 1 Newton-meter (N·m).
• The Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its kinetic energy. When negative work is done (such as by friction), the object's kinetic energy decreases, causing it to slow down.
• Examples of Negative Work:
• Frictional Force: When a block slides on a floor, the force of friction acts opposite to the direction of motion, doing negative work and dissipating energy as heat.
• Gravitational Force: When an object is thrown upward, the displacement is up, but gravity pulls downward. Thus, gravity does negative work until the object reaches its peak.
• Air Resistance: A parachute falling through the air experiences upward drag while moving downward, resulting in negative work by the air.

Important Points

• Perpendicular Force: If the force is perpendicular to the displacement (angle θ = 90 degrees), the work done is zero because cos(90°) = 0. A classic example is a coolie carrying a load on his head and walking on a horizontal platform; the force (gravity/normal) is vertical, while displacement is horizontal.
• Centripetal Force: In uniform circular motion, the centripetal force acts toward the center of the circle, while the displacement is tangential (at 90 degrees). Therefore, the work done by centripetal force is always zero.
• Dimensions: The dimensional formula for work is [M1 L2 T-2], which is the same as the dimensions for energy and torque.
• Sign Significance: The positive sign indicates energy is transferred to the object, while a negative sign indicates energy is removed from the object or transferred to the surroundings.

The correct answer is On increasing the number of turns per unit length in a solenoid, the strength of the magnetic field in the solenoid increases.

Key Points

• A solenoid is a long coil containing a large number of close turns of insulated copper wire. When an electric current passes through it, it behaves like a magnet.
• The magnetic field strength (B) inside a long solenoid is given by the formula B = μ₀ n I, where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.
• On increasing the number of turns per unit length (n) in a solenoid, the strength of the magnetic field produced inside it increases proportionally. Hence, statement 3 is correct.
• According to Ohm's Law, current is inversely proportional to resistance (I = V/R). If the resistance of the wire increases, the current (I) decreases, which in turn decreases the magnetic field strength. Hence, statement 1 is incorrect.
• The magnetic field inside a solenoid is directly proportional to the strength of the current flowing through it. If the current increases, the field strength increases. Hence, statement 2 is incorrect.
• The magnetic field lines inside a solenoid are parallel straight lines, indicating that the magnetic field is uniform at all points inside the solenoid. They are not circular inside; they form a pattern similar to that of a bar magnet. Hence, statement 4 is incorrect.

Additional Information

• Structure and Field: A solenoid consists of a cylindrical shape where the wire is wound closely. The field lines outside the solenoid are similar to those of a bar magnet, emerging from the North pole and entering the South pole.
• Uniformity: The fact that field lines are parallel straight lines inside the solenoid is a crucial feature, as it allows for the creation of a constant magnetic field in a specific region, which is essential for many scientific experiments and industrial applications.
• Electromagnets: A strong magnetic field produced inside a solenoid can be used to magnetize a piece of magnetic material, like soft iron, when placed inside the coil. The combination of the coil and the soft iron core is called an electromagnet.
• Polarity: One end of the solenoid acts as a magnetic North pole, while the other behaves as a magnetic South pole. The polarity can be determined using the Clock Rule or the Right-Hand Thumb Rule. If the current flows in a clockwise direction at an end, that end is the South pole.

Important Points

• Factors affecting Magnetic Field:
• Current (I): Directly proportional. Doubling the current doubles the field.
• Number of Turns (N): Directly proportional. More turns result in a stronger field because the field of each turn adds up.
• Core Material: Using a soft iron core can increase the magnetic field strength by thousands of times compared to an air core.
• Permeability (μ₀): The constant μ₀ (permeability of free space) has a value of approximately 4π × 10⁻⁷ T·m/A. This value changes if a material other than air/vacuum is used as the core.
• Comparison: Unlike a permanent bar magnet, the strength of the magnetic field in a solenoid can be controlled by changing the current or the number of turns, and its polarity can be reversed by changing the direction of the current.

The correct answer is Landfilling plastics in soil.

Key Points

• Landfilling plastics in soil: Plastics are non-biodegradable synthetic materials. When they are buried in landfills, they do not decompose for hundreds of years. Instead, they lead to soil degradation, interfere with the natural flow of groundwater, and can leach harmful chemicals like phthalates and bisphenol A (BPA) into the environment. Hence, this method will NOT help in the management of waste and is the correct answer.
• Treating wastes before release: This involves processes like Effluent Treatment for industrial waste and Sewage Treatment for domestic waste. By removing toxic chemicals, pathogens, and heavy metals before releasing waste into water bodies or soil, we prevent pollution and protect ecosystems. Hence, this statement describes a helpful waste management method.
• Vermicomposting: This is a biological process that utilizes certain species of earthworms, such as Red Wigglers (Eisenia fetida), to decompose organic waste into nutrient-rich manure. It effectively manages kitchen and garden waste while producing high-quality organic fertilizer. Hence, this statement describes a helpful waste management method.
• Following 3R’s (Reduce, Reuse, Recycle): This is a globally recognized strategy for Waste Hierarchy. Reducing consumption limits waste generation; Reusing items extends their lifespan; and Recycling processes waste into new raw materials, saving energy and natural resources. Hence, this statement describes a helpful waste management method.

Additional Information

• Plastic Waste Management in India: The Government of India notified the Plastic Waste Management Rules, 2016, which were later amended in 2021 and 2022. These rules include a ban on identified Single-Use Plastic (SUP) items with effect from July 1, 2022, and mandate Extended Producer Responsibility (EPR) for producers and brand owners to ensure the collection and recycling of plastic packaging.
• Classification of Waste:
• Biodegradable Waste: Waste that can be broken down into simpler substances by microorganisms (e.g., peels, paper, wood).
• Non-biodegradable Waste: Waste that cannot be degraded by natural agents (e.g., glass, plastic, metal cans, radioactive waste).
• Sanitary Landfills: Unlike open dumping, a Sanitary Landfill is a scientifically designed facility where waste is deposited in layers, compacted, and covered with soil. It includes systems for Leachate Collection and Methane Extraction to minimize environmental hazards.
• Incineration: This is the process of combusting waste at high temperatures (typically above 850°C) to reduce its volume and convert it into ash, flue gas, and heat. It is commonly used for medical waste and hazardous waste management.
• Waste-to-Energy (WtE): This refers to various technologies that treat non-recyclable waste to generate electricity or heat, thereby reducing the dependency on fossil fuels and minimizing landfill space.

Important Points

• Swachh Bharat Mission (SBM): Launched on October 2, 2014, this national campaign aims to achieve Universal Sanitation Coverage and scientific processing of Solid Waste Management across Urban and Rural India.
• Electronic Waste (E-waste): Rapid urbanization has led to an increase in E-waste, which contains toxic components like Mercury, Lead, and Cadmium. The E-Waste (Management) Rules, 2022 regulate the recycling and disposal of these products to prevent health hazards.
• Global Statistics: According to the World Bank, global waste generation is expected to increase to 3.40 billion tonnes by 2050 if sustainable Circular Economy models are not adopted.

The correct answer is First letter uppercase, second lowercase (in case of two letters).

Key Points

• The International Union of Pure and Applied Chemistry (IUPAC) is the global authority that approves the names of elements and their symbols.
• According to IUPAC nomenclature rules, the symbol of an element is either one or two letters of its name in English or Latin.
• The first letter of a symbol is always written as a capital letter (uppercase) and the second letter is always written as a small letter (lowercase). Hence, the rule stated in option 1 is correct.
• For example, the symbol for Hydrogen is H (single letter, uppercase).
• For elements with two letters, such as Helium, the symbol is He (not HE), and for Aluminum, it is Al (not AL). Similarly, Cobalt is represented as Co.
• If both letters were uppercase, it might lead to confusion with chemical compounds; for instance, Co represents the element Cobalt, whereas CO represents the compound Carbon Monoxide. Hence, statement 2 is incorrect.
• Symbols are never written in numeric codes for standard identification, although atomic numbers are used to organize them in the Periodic Table. Hence, statement 3 is incorrect.
• Using two lowercase letters is against the standard convention as the first letter must distinguish the start of a new element symbol in a formula. Hence, statement 4 is incorrect.

Additional Information

• Origin of Symbols:
• Many symbols are derived from the first letter or the first two letters of the element's English name. Examples include Carbon (C), Oxygen (O), and Neon (Ne).
• Some symbols are derived from Latin, Greek, or German names. For instance, the symbol for Iron is Fe from its Latin name Ferrum; Sodium is Na from Natrium; and Potassium is K from Kalium.
• Other examples of Latin-derived symbols include Copper (Cu from Cuprum), Gold (Au from Aurum), and Silver (Ag from Argentum).
• Historical Context:
• Dalton was the first scientist to use symbols for elements in a specific quantitative sense, using circles with distinct markings.
• Berzelius suggested the modern system where symbols are made from one or two letters of the name of the element, which was later standardized by IUPAC.
• IUPAC (International Union of Pure and Applied Chemistry):
• Founded in 1919, it is an international federation of National Adhering Organizations that represents chemists in individual countries.
• It is best known for its work in standardizing nomenclature in chemistry, ensuring that every element and compound has a unique and universally accepted name.
• For newly discovered heavy elements (atomic number > 100), IUPAC initially assigns systematic three-letter temporary symbols (like Uuo for Ununoctium, now Oganesson - Og) until a permanent name is officially adopted.

The correct answer is Circulatory system.

Key Points

• The circulatory system, also known as the cardiovascular system, is the primary transport system in animals, responsible for moving nutrients, gases, and waste products throughout the body.
• It consists of three main components: the heart (a muscular pumping organ), blood (the transport medium), and blood vessels (the network of tubes).
• The primary function of this system is to deliver oxygen from the respiratory organs to the tissues and to transport carbon dioxide back for elimination.
• It also plays a crucial role in distributing hormones secreted by endocrine glands to their specific target organs and maintaining homeostasis by regulating body temperature and pH levels.
• In multicellular organisms, simple diffusion is insufficient for the transport of materials; hence, a dedicated circulatory system is essential for survival. Hence, the correct answer is Circulatory system.

Additional Information

• Respiratory System:
• This system is primarily responsible for the exchange of gases between the body and the external environment.
• In humans, it includes the lungs, trachea, and bronchi. While it provides oxygen to the blood, it does not transport it to the rest of the body.
• Muscular System:
• The muscular system is composed of skeletal, smooth, and cardiac muscles.
• Its main function is to facilitate locomotion, maintain posture, and support internal movements like peristalsis in the digestive tract.
• Nervous System:
• This system acts as the body's communication network, consisting of the brain, spinal cord, and nerves.
• It processes sensory information and coordinates voluntary and involuntary actions through electrical impulses.
• Types of Circulation:
• Open Circulatory System: Found in arthropods (like insects) and most mollusks, where blood (hemolymph) bathes the organs directly in a hemocoel.
• Closed Circulatory System: Found in annelids (like earthworms) and all vertebrates, where blood is confined within a continuous network of vessels.
• Components of Blood:
• Plasma: The liquid component that carries proteins and electrolytes.
• Red Blood Cells (RBCs): Contain hemoglobin to transport oxygen.
• White Blood Cells (WBCs): Part of the immune system to fight infections.
• Platelets: Small cell fragments essential for blood clotting or coagulation.

The correct answer is Only a and b.

Key Points

• Benzene is a fundamental organic chemical compound and is categorized as the simplest aromatic hydrocarbon. It consists of six carbon atoms arranged in a planar ring. Hence, statement a is correct.
• In its structural representation (known as the Kekulé structure), Benzene is depicted as having alternating three single bonds and three double bonds between the carbon atoms. In modern chemistry, these are understood as delocalized electrons forming a resonance hybrid, where all bond lengths are equal. Hence, statement b is correct.
• The actual molecular formula of Benzene is C6H6, meaning it has six carbon atoms and six hydrogen atoms. The formula C6H14 provided in the statement actually represents Hexane, which is a saturated alkane. Hence, statement c is incorrect.
• Hence, the correct answer is Only a and b.

Additional Information

• Chemical Properties and Discovery:
• Benzene was first isolated and identified by the English scientist Michael Faraday in the year 1825 from the oily residue derived from the production of illuminating gas.
• The cyclic structure of Benzene was famously proposed by August Kekulé in 1865, who suggested the ring of six carbon atoms with alternating bonds.
• It is a colorless and highly flammable liquid with a characteristic sweet odor. It evaporates quickly when exposed to air.
• Structural Features:
• Each carbon atom in the Benzene ring is sp2 hybridized. This hybridization results in a trigonal planar geometry with bond angles of 120 degrees.
• Due to the delocalization of pi-electrons across the ring, Benzene exhibits exceptional stability, a phenomenon known as aromaticity.
• The C-C bond length in Benzene is 1.39 Angstroms, which is intermediate between a standard single bond (1.54 A) and a double bond (1.34 A).
• Industrial Uses and Safety:
• Benzene is a critical industrial chemical used primarily as a precursor to manufacture more complex chemicals like ethylbenzene, cumene, and cyclohexane.
• It is used in the production of plastics, resins, nylon, synthetic fibers, rubbers, lubricants, and detergents.
• From a safety perspective, Benzene is classified as a Group 1 Carcinogen by the International Agency for Research on Cancer (IARC), as long-term exposure is linked to leukemia.

Important Points

• Reactivity: Unlike alkenes, Benzene does not easily undergo addition reactions (like decolorizing bromine water) because that would destroy its aromatic stability. Instead, it typically undergoes Electrophilic Aromatic Substitution (EAS) reactions.
• Comparison: While Benzene (C6H6) is unsaturated and cyclic, Hexane (C6H14) is a straight-chain saturated hydrocarbon, and Cyclohexane (C6H12) is a cyclic saturated hydrocarbon.

The correct answer is Sodium.

Key Points

• A pure substance that consists of only one type of atom is defined as a chemical element.
• Sodium is a chemical element represented by the symbol Na (from the Latin 'natrium') and carries the atomic number 11.
• Since Sodium is composed entirely of atoms that have the same number of protons in their nuclei, it meets the criteria of a substance consisting of only one type of atom. Hence, statement 1 is correct.
• It is categorized as an alkali metal and is located in Group 1 of the periodic table.
• Under standard conditions, it is a soft, silvery-white, and highly reactive metal that cannot be broken down into simpler substances by chemical means.

Additional Information

• Air:
• Air is a homogeneous mixture of several different gases rather than a pure substance.
• It primarily consists of Nitrogen (78%), Oxygen (21%), Argon (0.9%), and Carbon Dioxide (0.04%). Because it contains different types of molecules and atoms, it is not an element. Hence, statement 2 is incorrect.
• Water:
• Water (H2O) is a chemical compound made up of two different elements: hydrogen and oxygen.
• Although it is a pure substance in the sense that it has a fixed chemical composition, it consists of two types of atoms bonded together in a 2:1 ratio. Hence, statement 3 is incorrect.
• Milk:
• Milk is a complex heterogeneous mixture, specifically classified as a colloid.
• It contains various substances such as water, fats, proteins (like casein), lactose (sugar), and minerals. Since it is a mixture of many different compounds, it does not consist of only one type of atom. Hence, statement 4 is incorrect.

Important Points

• Reactivity of Sodium: Sodium is extremely reactive, particularly with water and oxygen. To prevent accidental reactions, it is typically stored in mineral oil or kerosene.
• Atomic Structure: Sodium has 11 protons and 11 electrons. Its electron configuration is [Ne] 3s1, meaning it has one valence electron that it easily loses to form a cation (Na+).
• Natural Occurrence: Due to its high reactivity, Sodium is never found as a free element in nature. It occurs abundantly in compounds, the most common being Sodium Chloride (NaCl), which is table salt found in seawater and rock salt deposits.
• Industrial Uses: Sodium compounds are used in the production of glass, paper, soap, and textiles. It is also vital in biological systems for maintaining fluid balance and nerve signal transmission in the human body.

The correct answer is It involves making food using carbon dioxide and water in the presence of sunlight.

Key Points

• Autotrophic nutrition is a mode of nutrition in which organisms synthesize their own food from simple inorganic substances such as carbon dioxide and water.
• The process primarily utilized by green plants is photosynthesis, which requires the presence of sunlight and the green pigment chlorophyll. Hence, statement 3 is correct.
• Digestion of complex substances outside the body is characteristic of saprotrophic nutrition, common in fungi and certain bacteria. Hence, statement 1 is incorrect.
• Depending on consuming other living organisms for food is known as heterotrophic nutrition, which includes holozoic nutrition found in animals and humans. Hence, statement 2 is incorrect.
• Direct absorption of organic compounds from other organisms is a feature of parasitic nutrition or specific heterotrophic pathways where the organism lives on or inside a host. Hence, statement 4 is incorrect.
• The overall chemical reaction for photosynthesis can be represented as: 6CO2 + 6H2O + Light Energy results in C6H12O6 (Glucose) + 6O2.

Additional Information

• Classification of Autotrophs:
• Photoautotrophs: These organisms use sunlight as their energy source. Examples include green plants, algae, and cyanobacteria.
• Chemoautotrophs: These organisms obtain energy from the oxidation of inorganic chemical compounds like ammonia, nitrites, or sulfur. This is common in certain extremophile bacteria found in deep-sea vents.
• Requirements for Photosynthesis:
• Chlorophyll: A pigment located in the chloroplasts of plant cells that absorbs light energy, specifically in the red and blue wavelengths.
• Carbon Dioxide: Taken from the atmosphere through small pores on leaves called stomata.
• Water: Absorbed from the soil by the roots and transported to the leaves through xylem tissue.
• Energy Storage:
• In autotrophs (plants), the prepared glucose is often converted and stored as starch to serve as an internal energy reserve.
• In heterotrophs (animals), energy is typically stored in the form of glycogen in the liver and muscles.
• Heterotrophic Nutrition Types:
• Holozoic: Organisms take in solid food and break it down inside the body (e.g., humans, amoeba).
• Saprophytic: Organisms feed on dead and decaying organic matter (e.g., mushrooms, bread mould).
• Parasitic: Organisms derive nutrition from a host without killing it immediately (e.g., Cuscuta, ticks, leeches).

Important Points

• The Role of Oxygen: During autotrophic nutrition via photosynthesis, oxygen is released as a byproduct, which is essential for the survival of aerobic organisms on Earth.
• Trophic Levels: Autotrophs are considered producers and form the first trophic level in any food chain or food web, providing the foundational energy for all other consumers.
• Stomata Function: The guard cells regulate the opening and closing of stomata, which controls gas exchange and the rate of transpiration (water loss).
• Photosynthesis Phases: It consists of two stages: the Light-dependent reaction (occurring in the thylakoids) and the Light-independent reaction or Calvin Cycle (occurring in the stroma).

Shortcut Trick

Given Marked Price (MP) = ₹6,370 and Discount% = 50%.

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

SP = 6,370 × (100 − 50) ÷ 100 = 6,370 × 50 ÷ 100.

SP = 6,370 ÷ 2 = 3,185.

∴ The correct answer is ₹3,185.

Alternate Method

Given: Marked Price = ₹6,370, Discount Percentage = 50%.

Formulas: Discount = (Discount% × MP) ÷ 100 | Selling Price (SP) = MP − Discount.

⇒ Discount Amount = 50% of 6,370

⇒ Discount = (50 ÷ 100) × 6,370 = 3,185

⇒ Selling Price = MP − Discount = 6,370 − 3,185

⇒ Selling Price = 3,185

∴ The correct answer is ₹3,185.

Additional Information

Marked Price & Discount

Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP unless specified otherwise. Formula: SP = MP × (100 − D)%.

50% Discount Rule

A 50% discount implies the customer pays exactly half of the Marked Price. It is equivalent to a 1:1 ratio between the discount and the selling price.

Shortcut Trick

Total Capacity (LCM of 15 and 40) = 120 units

Efficiency of P + Q = 120 ÷ 15 = 8 units/hr

Efficiency of P = 120 ÷ 40 = 3 units/hr

Efficiency of Q = 8 − 3 = 5 units/hr

Time for Q to fill 1/4th = (120 × 1/4) ÷ 5 = 30 ÷ 5 = 6 hours

∴ The correct answer is 6 hours.

Alternate Method

Given: P + Q can fill in 15 hours. P alone can fill in 40 hours.

Formula Used: Total Work = Efficiency × Time | 1/Q = 1/(P+Q) − 1/P

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

⇒ Efficiency of Q = 8 − 3 = 5 units/hr

⇒ Total capacity of cistern = 120 units

⇒ One-fourth of the cistern = 120 ÷ 4 = 30 units

⇒ Time taken by Q = 30 ÷ 5 = 6 hours

∴ The correct answer is 6 hours.

Additional Information

Efficiency Concept

Efficiency is inversely proportional to the time taken to complete a work. Total Work = LCM of individual time durations.

Combined Work Formula

If two pipes A and B fill a tank in x and y hours, then together they fill it in (xy) ÷ (x + y) hours.

Pipes and Cisterns

Inlet pipes have positive efficiency while outlet/leakage pipes have negative efficiency.

Shortcut Trick

Use the algebraic identity: if x + y + z = 0, then x³ + y³ + z³ = 3xyz.

In the numerator, (a² − b²) + (b² − c²) + (c² − a²) = 0, so it becomes 3(a² − b²)(b² − c²)(c² − a²).

In the denominator, (a − b) + (b − c) + (c − a) = 0, so it becomes 3(a − b)(b − c)(c − a).

Dividing these gives [(a−b)(a+b)(b−c)(b+c)(c−a)(c+a)] ÷ [(a−b)(b−c)(c−a)].

Canceling the common terms leaves (a + b)(b + c)(c + a).

∴ The correct answer is (a + b)(b + c)(c + a).

Alternate Method

Given: ((a² − b²)³ + (b² − c²)³ + (c² − a²)³) ÷ ((a − b)³ + (b − c)³ + (c − a)³)

Formula Used: If x + y + z = 0, then x³ + y³ + z³ = 3xyz; and x² − y² = (x − y)(x + y).

⇒ Let X = a² − b², Y = b² − c², and Z = c² − a².

⇒ X + Y + Z = a² − b² + b² − c² + c² − a² = 0.

⇒ Therefore, Numerator = 3(a² − b²)(b² − c²)(c² − a²).

⇒ Now, let p = a − b, q = b − c, and r = c − a.

⇒ p + q + r = a − b + b − c + c − a = 0.

⇒ Therefore, Denominator = 3(a − b)(b − c)(c − a).

⇒ Expression = [3(a² − b²)(b² − c²)(c² − a²)] ÷ [3(a − b)(b − c)(c − a)]

⇒ Expression = [3(a − b)(a + b)(b − c)(b + c)(c − a)(c + a)] ÷ [3(a − b)(b − c)(c − a)]

⇒ Expression = (a + b)(b + c)(c + a).

∴ The correct answer is (a + b)(b + c)(c + a).

Additional Information

Conditional Identity

The condition a + b + c = 0 implies a³ + b³ + c³ = 3abc, which is a specific case of the expansion of a³ + b³ + c³ − 3abc.

Sum of Three Cubes Formula

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

Difference of Squares

The property a² − b² = (a − b)(a + b) allows for the factorization and cancellation of terms in complex rational expressions.

Shortcut Trick

Solve the brackets using the BODMAS rule directly.

Inner bracket: (−8 − 20 − 18) = −46

Square bracket: [1 × {10 + (−2) × (−3)}] = [1 × {10 + 6}] = 16

Remaining expression: 86 − (−16) × (−46) ÷ 16

⇒ 86 − (736 ÷ 16) = 86 − 46 = 40

∴ The correct answer is 40.

Alternate Method

Given:

Expression: 86 − (−16) × (−8 − 20 − 18) ÷ [1 × {10 + (−2) × (−3)}]

Method Used:

BODMAS Rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Calculations:

⇒ 86 − (−16) × (−8 − 20 − 18) ÷ [1 × {10 + (−2) × (−3)}]

⇒ 86 − (−16) × (−46) ÷ [1 × {10 + 6}]

⇒ 86 − (−16) × (−46) ÷ [1 × 16]

⇒ 86 − (−16) × (−46) ÷ 16

⇒ 86 − 736 ÷ 16

⇒ 86 − 46

⇒ 40

∴ The correct answer is 40.

Additional Information

BODMAS Rule

It stands for Bracket, Of, Division, Multiplication, Addition, and Subtraction. Expressions are simplified in this strict sequential order.

Order of Brackets

Always solve brackets in the order: Line bracket (bar), Simple brackets (), Curly brackets {}, and finally Square brackets [].

Sign Multiplication Rules

When multiplying numbers with the same signs, the result is positive (− × − = +). When multiplying numbers with different signs, the result is negative (+ × − = −).

Shortcut Trick

To find 31³, focus on the unit digit. Since 31 ends in 1, its cube must end in 1 (1 × 1 × 1 = 1).

Check the given options: (1) ends in 9, (2) ends in 8, (3) ends in 1, and (4) ends in 5.

Since only 29791 ends in 1, it is the correct answer.

∴ The correct answer is 29791 m³.

Alternate Method

Given: The edge of the cube (a) = 31 m.

Formula Used: Volume of a cube = Side³ = a³

⇒ Volume = (31 m)³

⇒ Volume = 31 × 31 × 31

⇒ Volume = 961 × 31

⇒ Volume = 29791 m³

∴ The correct answer is 29791 m³.

Additional Information

Total Surface Area of Cube

The total surface area is the sum of the areas of all six faces: TSA = 6a².

Lateral Surface Area of Cube

The area of the four side faces (excluding top and bottom): LSA = 4a².

Diagonal of a Cube

The longest distance between two opposite corners: d = a√3.

Shortcut Trick

Ratio of Profit = Ratio of Investment (when time is constant).

Sunita : Javed : Rani = 139 : 47 : 154.

Total ratio parts = 139 + 47 + 154 = 340 units.

340 units = ₹4,50,500 ⇒ 1 unit = 4,50,500 ÷ 340 = ₹1,325.

Rani's share (154 units) = 154 × 1,325 = ₹2,04,050.

∴ The correct answer is ₹2,04,050.

Alternate Method

Given → Sunita's Investment = ₹13,90,000 | Javed's Investment = ₹4,70,000 | Rani's Investment = ₹15,40,000 | Total Profit = ₹4,50,500. Formulas: Profit Share = (Individual Investment ÷ Total Investment) × Total Profit.

⇒ Ratio of investments = 13,90,000 : 4,70,000 : 1540,000 = 139 : 47 : 154. ⇒ Sum of ratios = 139 + 47 + 154 = 340. ⇒ Rani's share = (Rani's ratio ÷ Total ratio) × Total Profit. ⇒ Rani's share = (154 ÷ 340) × 4,50,500. ⇒ Rani's share = 154 × 1,325 = 2,04,050.

∴ The correct answer is ₹2,04,050.

Additional Information

Partnership Concept

When multiple people invest in a business, the profit is distributed in the ratio of (Investment × Time Period).

Simple vs. Compound Partnership

In a simple partnership, all partners invest for the same duration. In a compound partnership, durations differ.

Shortcut Trick

When the distance for both journeys is the same, use the harmonic mean formula.

Average Speed = (2 × S₁ × S₂) ÷ (S₁ + S₂)

⇒ (2 × 131 × 24) ÷ (131 + 24) = 6288 ÷ 155 ≈ 40.57 km/hr

∴ The correct answer is 40.57 km/hr.

Alternate Method

Given:

Distance (A to B) = 663 km

Speed from A to B (S₁) = 131 km/hr

Speed from B to A (S₂) = 24 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 663 + 663 = 1326 km

⇒ Time taken (A to B) = 663 ÷ 131 ≈ 5.061 hours

⇒ Time taken (B to A) = 663 ÷ 24 = 27.625 hours

⇒ Total Time = 5.061 + 27.625 = 32.686 hours

⇒ Average Speed = 1326 ÷ 32.686 ≈ 40.57 km/hr

∴ The correct answer is 40.57 km/hr.

Additional Information

Average Speed (Different Distances)

If a person covers distances d₁, d₂, d₃ with speeds s₁, s₂, s₃, then Average Speed = (d₁ + d₂ + d₃) ÷ (d₁/s₁ + d₂/s₂ + d₃/s₃).

Relative Speed

When two objects move in opposite directions, relative speed is (u + v). When they move in the same direction, relative speed is |u − v|.

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

Larger number = (Sum + Difference) ÷ 2 = (74 + 12) ÷ 2 = 43

Smaller number = (Sum − Difference) ÷ 2 = (74 − 12) ÷ 2 = 31

Product = 43 × 31 = 1333

∴ The correct answer is 1333.

Alternate Method

Given: Sum of two numbers = 74 and Difference of two numbers = 12.

Formula Used: Product (xy) = [(x + y)² − (x − y)²] ÷ 4

⇒ Let the numbers be x and y.

⇒ x + y = 74 --- (1)

⇒ x − y = 12 --- (2)

⇒ Adding (1) and (2): 2x = 86 ⇒ x = 43

⇒ Substituting x in (1): 43 + y = 74 ⇒ y = 31

⇒ Product = 43 × 31

⇒ Product = 1333

∴ The correct answer is 1333.

Additional Information

Algebraic Identity for Product

The relationship between sum, difference, and product is given by: 4xy = (x + y)² − (x − y)². This allows finding the product without calculating individual numbers.

Finding Individual Numbers

If S is the sum and D is the difference, then the numbers are (S + D) ÷ 2 and (S − D) ÷ 2.

Sum and Product Relationship

For a quadratic equation t² − (sum)t + (product) = 0, the roots are the two numbers themselves.

Shortcut Trick

For any quadrilateral EFGH, the angle formed by the angle bisectors of two adjacent angles (say, ∠G and ∠H) meeting at a point X is equal to half the sum of the other two angles (∠E and ∠F):

∠HXG = (∠E + ∠F) ÷ 2

Given: ∠E = 52° and ∠F = 37°

⇒ ∠HXG = (52° + 37°) ÷ 2

⇒ ∠HXG = 89° ÷ 2 = 44.5°

∴ The correct answer is 44.5°.

Alternate Method

Given:

In quadrilateral EFGH:

∠E = 52°

∠F = 37°

The bisectors of ∠G and ∠H meet at X.

Formula Used:

1. Sum of interior angles of a quadrilateral = 360°
2. Sum of angles of a triangle = 180°

Calculations:

In quadrilateral EFGH:

⇒ ∠E + ∠F + ∠G + ∠H = 360°

⇒ 52° + 37° + ∠G + ∠H = 360°

⇒ 89° + ∠G + ∠H = 360°

⇒ ∠G + ∠H = 360° − 89° = 271°

Since GX and HX are the angle bisectors of ∠G and ∠H respectively:

⇒ ∠XGH = ∠G ÷ 2

⇒ ∠XHG = ∠H ÷ 2

In ΔHXG:

⇒ ∠HXG + ∠XGH + ∠XHG = 180°

⇒ ∠HXG + (∠G ÷ 2) + (∠H ÷ 2) = 180°

⇒ ∠HXG + (∠G + ∠H) ÷ 2 = 180°

⇒ ∠HXG + 271° ÷ 2 = 180°

⇒ ∠HXG + 135.5° = 180°

⇒ ∠HXG = 180° − 135.5° = 44.5°

∴ The correct answer is 44.5°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

The angle formed by the bisectors of two adjacent angles of a quadrilateral is always equal to half the sum of the other two remaining angles.

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 360°.

Angle Bisectors of Opposite Angles

If the bisectors of opposite angles of a quadrilateral meet, the angle between them is equal to half the difference of the other two angles.

Shortcut Trick

Use the direct relationship formula: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

MP ÷ 1,600 = (100 + 20) ÷ (100 − 25) = 120 ÷ 75.

MP = 1,600 × (120 ÷ 75) = 1,600 × (8 ÷ 5).

MP = 320 × 8 = 2,560.

∴ The correct answer is ₹2,560.

Alternate Method

Given: Cost Price (CP) = ₹1,600, Profit = 20%, Discount = 25%. Formulas: SP = CP × (100 + P%) ÷ 100 | MP = (SP × 100) ÷ (100 − D%)

⇒ Step 1: Calculate Selling Price (SP) = 1,600 × (120 ÷ 100) = ₹1,920.

⇒ Step 2: Marked Price (MP) × (100 − 25) ÷ 100 = 1,920.

⇒ Step 3: MP × (75 ÷ 100) = 1,920.

⇒ Step 4: MP = 1,920 × (4 ÷ 3) = 640 × 4 = ₹2,560.

∴ The correct answer is ₹2,560.

Additional Information

Marked Price & Discount

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

CP-MP Relationship

For problems involving both Profit% and Discount%, the ratio MP/CP = (100 + P%)/(100 − D%) is the most efficient solving method.

Shortcut Trick Given: Selling Price (SP) of 10 watches = Cost Price (CP) of 8 watches. Using the ratio method: 10 × SP = 8 × CP. Ratio of SP/CP = 8/10 = 4/5. Loss = CP − SP = 5 − 4 = 1 unit. Loss% = (Loss ÷ CP) × 100 = (1 ÷ 5) × 100 = 20%. ∴ The correct answer is 20%.

Alternate Method Given: Total SP of 10 watches = ₹10,500. CP of 8 watches = ₹10,500. Formulas: CP of 1 watch = Total CP ÷ Quantity | SP of 1 watch = Total SP ÷ Quantity | Loss% = ((CP − SP) ÷ CP) × 100.

⇒ Cost Price (CP) of 8 watches = ₹10,500. ⇒ CP of 1 watch = 10,500 ÷ 8 = ₹1312.5. ⇒ Selling Price (SP) of 10 watches = ₹10,500. ⇒ SP of 1 watch = 10,500 ÷ 10 = ₹1050. ⇒ Loss per watch = 1312.5 − 1050 = ₹262.5. ⇒ Loss% = (262.5 ÷ 1312.5) × 100 = 20%. ∴ The correct answer is 20%.

Additional Information Loss and CP Relationship When the SP of 'n' articles equals the CP of 'm' articles, if n > m, there is always a loss. Loss% = ((n − m) ÷ n) × 100. Percentage Calculation Loss and Profit percentages are always calculated on the Cost Price (CP) unless stated otherwise in the problem.

Shortcut Trick

Required percentage = (Value ÷ Base Value) × 100

Required percentage = (1/30 ÷ 2/20) × 100

⇒ (1/30 × 20/2 = 10/1) × 100 = 1/3 × 100

⇒ 33.33%

∴ The correct answer is 33.33%.

Alternate Method

Given:

First fraction (Part) = 1/30

Second fraction (Base) = 2/20 = 1/10

Formula Used:

Percentage = (Part ÷ Base) × 100

Calculations:

⇒ Percentage = ((1/30) / (1/10)) × 100

⇒ Percentage = (1/30 × 10/1) × 100

⇒ Percentage = (1/3) × 100

⇒ Percentage = 33.3333...

⇒ Percentage ≈ 33.33%

∴ The correct answer is 33.33%.

Additional Information

Fraction to Percentage

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

Percentage Change

Percentage change = (Difference ÷ Original Value) × 100.

Reciprocal Percentages

1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16.67%.

Shortcut Trick

To find the Mode, identify the observation that repeats most frequently in the given data.

Scanning the dataset: 53 appears 5 times, whereas 40 appears 3 times.

Values 50 and 51 appear only 2 times each, and others appear once.

Since 53 has the highest Frequency, it is the mode.

∴ The correct answer is 53.

Alternate Method

Given: Data = 53, 53, 40, 49, 50, 48, 53, 50, 41, 40, 53, 45, 40, 47, 43, 52, 51, 42, 51, 53

Formula Used: Mode is the value with the Maximum Frequency in a distribution.

Calculations:

⇒ Count the frequency of each distinct number in the set:

⇒ Frequency of 53 = 5

⇒ Frequency of 40 = 3

⇒ Frequency of 50 = 2

⇒ Frequency of 51 = 2

⇒ Frequency of {40, 41, 42, 43, 45, 47, 48, 49, 52} = 1 each

⇒ The value with the highest frequency (5) is 53.

∴ The correct answer is 53.

Additional Information

Mean

The Mean is the average of all observations, calculated as Σx ÷ n.

Median

The Median is the middle value when data is arranged in ascending or descending order. For even n, it is the average of the two middle terms.

Empirical Relationship

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

Bimodal and Multimodal

If two values have the same maximum frequency, the data is Bimodal. If more than two, it is Multimodal.

Shortcut Trick

Total work (LCM of 25 and 30) = 150 units.

Efficiency of Pipe F = 150 ÷ 25 = 6 units/min; Efficiency of Pipe M = 150 ÷ 30 = 5 units/min.

Pipe M runs for the full 15 minutes: Work by M = 15 × 5 = 75 units.

Remaining work for F = 150 − 75 = 75 units.

Time for pipe F = 75 ÷ 6 = 12.5 minutes.

∴ The correct answer is 12.5 min.

Alternate Method

Given: Time taken by Pipe F = 25 min, Time taken by Pipe M = 30 min, Total time to fill = 15 min.

Formula Used: Work = Efficiency × Time; Total Work = LCM of individual times.

⇒ Let Pipe F be closed after t minutes.

⇒ Pipe F works for t minutes and Pipe M works for the entire 15 minutes.

⇒ (Efficiency of F × t) + (Efficiency of M × 15) = Total Work

⇒ (6 × t) + (5 × 15) = 150

⇒ 6t + 75 = 150

⇒ 6t = 75

⇒ t = 75 ÷ 6 = 12.5 minutes

∴ The correct answer is 12.5 min.

Additional Information

Pipes and Cisterns Efficiency

Efficiency is inversely proportional to the time taken to fill the tank. If a pipe fills a tank in x hours, its efficiency is 1/x part per hour.

Negative Work (Inlet vs Outlet)

Inlet pipes do positive work (filling), while outlet pipes (leaks) do negative work (emptying). Net work = Σ(Inlet efficiencies) − Σ(Outlet efficiencies).

Combined Work Formula

If two pipes fill a tank in x and y mins respectively, then together they fill it in (x × y) ÷ (x + y) mins.

Shortcut Trick

For an angle of elevation of 45°, the height of the tower is always equal to the distance from the foot.

Since the distance is 29.6 m, the height must also be 29.6 m.

∴ The correct answer is 29.6 m.

Alternate Method

Given:

Distance from the foot of the tower (BC) = 29.6 m

Angle of elevation (θ) = 45°

Formula Used:

tan θ = Perpendicular / Base = AB / BC

Calculations:

⇒ tan 45° = Height (h) / 29.6

⇒ 1 = h / 29.6

⇒ h = 29.6 × 1

⇒ h = 29.6 m

∴ The correct answer is 29.6 m.

Additional Information

Trigonometric Ratios for 45°

In a right-angled triangle with a 45° angle, the Perpendicular and Base are equal because tan 45° = 1.

Special Right Triangle (45-45-90)

The sides of a 45°−45°−90° triangle are in the ratio 1 : 1 : √2. If height is 'a', the hypotenuse is a√2.

Angle of Elevation

It is the angle formed by the line of sight with the horizontal when looking up at an object.

Shortcut Trick

For a number to be divisible by 9, the sum of its digits must be a multiple of 9.

Sum of digits of A6B59 = A + 6 + B + 5 + 9 = 20 + A + B.

Next multiples of 9 after 20 are 27, 36... ⇒ A + B = 7 (for sum 27) or 16 (for sum 36).

To minimize $(A^2+2B)$ with A + B = 7, test pairs: (1,6) → 1+12=13; (2,5) → 4+10=14; (3,4) → 9+8=17.

∴ The correct answer is 13.

Alternate Method

Given: The number A6B59 is divisible by 9. We need to find the least value of $(A^2+2B)$.

Divisibility Rule of 9: Sum of digits is divisible by 9.

⇒ A + 6 + B + 5 + 9 = 20 + A + B

⇒ 20 + A + B must be 27, 36, ...

Case 1: A + B = 7 (Since 20 + 7 = 27)

⇒ Possible (A, B) pairs: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0). (Note: A cannot be 0 as it is the first digit).

⇒ If A = 1, B = 6: $1^2 + 2(6)$ = 1 + 12 = 13

⇒ If A = 2, B = 5: $2^2 + 2(5)$ = 4 + 10 = 14

⇒ If A = 3, B = 4: $3^2 + 2(4)$ = 9 + 8 = 17

⇒ If A = 7, B = 0: $7^2 + 2(0)$ = 49

Case 2: A + B = 16 (Since 20 + 16 = 36)

⇒ Min A is 7, B = 9: $7^2 + 2(9)$ = 49 + 18 = 67 (Higher than Case 1 values)

⇒ Comparing all results, the minimum value is 13.

∴ The correct answer is 13.

Additional Information

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility Rule of 11

Difference between the sum of digits at odd places and even places must be 0 or a multiple of 11.

Minimization Logic

To minimize a sum of squares and linear terms like $A^2 + 2B$, generally, the variable with the higher power (A) should be as small as possible.

Shortcut Trick

Original Volume = Final Volume ÷ 0.8 = 10,752 ÷ 0.8 = 13,440 cm³

Volume Formula: 4/3 × 3.14 × R³ = 13,440

R³ = 13,440 × 3 ÷ 12.56 ≈ 3210.19

Since 14³ = 2744 and 15³ = 3375, R ≈ 14.75 cm.

Rounding to the nearest integer, R = 15 cm.

∴ The correct answer is 15 cm.

Alternate Method

Given:

⇒ Total volume of 64 small spheres = 10,752 cm³

⇒ Metal waste percentage = 20%

⇒ Value of π = 3.14

Formula Used:

⇒ Volume of a sphere = 4/3 × π × R³

Calculations:

⇒ Let the radius of the original sphere be R.

⇒ Effective metal percentage after loss = 100% − 20% = 80%

⇒ Total volume of original sphere × 80% = Total volume of 64 spheres

⇒ Original Volume × 0.8 = 10,752

⇒ Original Volume = 10,752 ÷ 0.8 = 13,440 cm³

⇒ 4/3 × 3.14 × R³ = 13,440

⇒ 12.56 ÷ 3 × R³ = 13,440

⇒ R³ = (13,440 × 3) ÷ 12.56

⇒ R³ = 40,320 ÷ 12.56 = 3210.19

⇒ R = √³3210.19 ≈ 14.75 cm

⇒ Nearest integer to 14.75 is 15.

∴ The correct answer is 15 cm.

Additional Information

Volume Conservation in Recasting

When a solid is melted and recast into other shapes, the total volume remains constant if there is no loss. If loss occurs, Final Volume = Initial Volume − Waste.

Sphere Properties

Volume = (4/3)πr³ and Surface Area = 4πr². If the radius is scaled by factor k, the volume is scaled by k³.

Approximate Cube Roots

Memorizing basic cubes like 10³ = 1000, 15³ = 3375, and 20³ = 8000 helps in estimating the radius during exams without full calculation.

Shortcut Trick

Incomes ratio = 5 : 9 and Expenditures ratio = 1 : 3.

Since savings are equal (��51,000), the difference between income and expenditure parts must be equal.

Difference in income parts (9 − 5) = 4; Difference in expenditure parts (3 − 1) = 2.

Make differences equal: Multiply expenditure ratio by 2 → 2 : 6.

Now, Savings = 5 − 2 = 3 units. ⇒ 3 units = ₹51,000 ⇒ 1 unit = ₹17,000.

Chetan monthly income = 5 units = 5 × ₹17,000 = ₹85,000.

∴ The correct answer is ₹85,000.

Alternate Method

Given: Ratio of Monthly Incomes = 5 : 9 | Monthly Savings = ₹51,000 each | Ratio of Expenditures = 1 : 3.

Formula: Income − Savings = Expenditure

Let the incomes be 5x and 9x.

⇒ (5x − 51,000) / (9x − 51,000) = 1 / 3

⇒ 3 × (5x − 51,000) = 1 × (9x − 51,000)

⇒ 15x − 1,53,000 = 9x − 51,000

⇒ 6x = 1,02,000

⇒ x = 17,000

⇒ Chetan monthly income = 5x = 5 × 17,000 = 85,000

∴ The correct answer is ₹85,000.

Additional Information

Income, Expenditure, and Saving Relation

The basic governing formula for these problems is Income = Expenditure + Savings.

Ratio Equalization Method

When a constant value is added or subtracted from both terms of a ratio, the difference between the ratio parts should be made equal to maintain proportionality.

Cross Multiplication Method

For two ratios a:b and c:d with a constant change k: x = |(bk − dk) / (ad − bc)| helps in finding the variable value quickly.

Shortcut Trick

Ratio of numbers = 12 : 14 : 4

25% of 1st number (12 units) = 3 units

Given: 3 units = 18 ⇒ 1 unit = 6

Difference between 2nd and 3rd = 14 − 4 = 10 units = 60

30% of 60 = 18

∴ The correct answer is 18.

Alternate Method

Given: Ratio of numbers = 12 : 14 : 4, 25% of the first number = 18.

Formula Used: Percentage = (Value/Total) × 100

⇒ Let the numbers be 12x, 14x, and 4x.

⇒ According to question: 25% of 12x = 18

⇒ (25/100) × 12x = 18

⇒ 3x = 18 ⇒ x = 6

⇒ Second number = 14 × 6 = 84

⇒ Third number = 4 × 6 = 24

⇒ Difference = 84 − 24 = 60

⇒ Required value = 30% of 60 = (30/100) × 60 = 18

∴ The correct answer is 18.

Additional Information

Ratio and Proportion

A ratio compares two quantities; if a:b = c:d, then product of extremes equals product of means (a × d = b × c).

Percentage Conversion

To find x% of a number N, use the formula: (x/100) × N. Common fractions: 25% = 1/4, 50% = 1/2.

Shortcut Trick

Sum of ages after 20 years = 92 years.

Present sum of ages = 92 − (2 × 20) = 52 years.

Ratio of Wife : Husband = 6 : 7 (Total parts = 13).

Value of 1 part = 52 ÷ 13 = 4 years.

Present ages = 6 × 4 and 7 × 4 = 24 and 28 years.

∴ The correct answer is 24, 28 years.

Alternate Method

Given: Present ratio = 6 : 7; Sum after 20 years = 92.

Formula: Let Present Age of Wife = 6x and Husband = 7x.

⇒ Age after 20 years: (6x + 20) and (7x + 20)

⇒ Sum: (6x + 20) + (7x + 20) = 92

⇒ 13x + 40 = 92

⇒ 13x = 52

⇒ x = 4

⇒ Wife's present age = 6 × 4 = 24 years

⇒ Husband's present age = 7 × 4 = 28 years

∴ The correct answer is 24, 28 years.

Additional Information

Age Sum Property

If the sum of ages of 'n' persons is 'S' now, then after 't' years, the sum will be S + (n × t).

Ratio & Proportion

If ages are in ratio a:b and sum is S, individual ages are S × [a/(a+b)] and S × [b/(a+b)].

Age Difference Property

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

Shortcut Trick

Direct formula for Median: Median = (Mode + 2 × Mean) ÷ 3

Substitute values: Median = (88.7 + 2 × 20) ÷ 3

Median = (88.7 + 40) ÷ 3 = 128.7 ÷ 3

Median = 42.9

∴ The correct answer is 42.9.

Alternate Method

Given:

Mean of data = 20

Mode of data = 88.7

Formula Used:

The empirical relationship between central tendencies is:

Mode = 3 × Median − 2 × Mean

Calculations:

⇒ 88.7 = 3 × Median − 2 × 20

⇒ 88.7 = 3 × Median − 40

⇒ 3 × Median = 88.7 + 40

⇒ 3 × Median = 128.7

⇒ Median = 128.7 ÷ 3

⇒ Median = 42.9

∴ The correct answer is 42.9.

Additional Information

Empirical Relationship

This formula is applicable for moderately asymmetrical distributions. It was suggested by Karl Pearson.

Symmetrical Distribution

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

Skewness Property

If Mode > Median > Mean, the distribution is negatively skewed (skewed to the left). If Mean > Median > Mode, it is positively skewed.

Shortcut Trick

Given HCF = 23 and Ratio = 1 : 2 : 3.

Numbers = Ratio Parts × HCF.

First number = 1 × 23 = 23.

Second number = 2 × 23 = 46.

Third number = 3 × 23 = 69.

∴ The correct answer is 23, 46, 69.

Alternate Method

Given: Ratio of three numbers = 1 : 2 : 3 and Highest Common Factor (HCF) = 23.

Let the three numbers be 1x, 2x, and 3x.

The HCF of 1x, 2x, and 3x is the common variable x.

⇒ x = 23

⇒ First number = 1 × 23 = 23

⇒ Second number = 2 × 23 = 46

⇒ Third number = 3 × 23 = 69

∴ The correct answer is 23, 46, 69.

Additional Information

HCF and LCM Relation

For two numbers a and b, Product of numbers = HCF × LCM.

Ratio and Multiples

If numbers are in ratio a:b:c, then the actual numbers are a×H, b×H, and c×H where H is the HCF.

LCM Formula

LCM of numbers in ratio a:b:c is equal to (a × b × c × HCF) if a, b, c are co-prime.

Shortcut Trick

Successive rate for 2 years at 20% = 20 + 20 + (20 × 20)/100 = 44%.

Total Amount (%) = 100% (Principal) + 44% (Interest) = 144%.

Amount = 144% of ₹5,200 = 1.44 × 5200 = ₹7,488.

∴ The correct answer is ₹7,488.

Alternate Method

Given:

Principal (P) = ₹5,200

Rate of Interest (R) = 20% per annum

Time (n) = 2 years (Compounded Annually)

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 5200(1 + 20/100)²

⇒ A = 5200(1 + 1/5)²

⇒ A = 5200 × (6/5) × (6/5)

⇒ A = 5200 × 36 / 25

⇒ A = 208 × 36

⇒ A = ₹7,488

∴ The correct answer is ₹7,488.

Additional Information

Effective Rate for Compound Interest

For 2 years, the effective rate is calculated as R + R + (R × R)/100.

Compound Interest (CI)

CI is the difference between the total Amount and the Principal: CI = A − P.

Difference between CI and SI

For a period of 2 years, the difference (D) between Compound Interest and Simple Interest is D = P × (R/100)².

Shortcut Trick

Total distance = Length of A + Length of B = 150 + 100 = 250 m.

Relative speed (opposite direction) = Speed of A + Speed of B = 72 + 54 = 126 km/h.

Convert speed to m/s: 126 × (5/18) = 35 m/s.

Time = Distance ÷ Relative Speed = 250 ÷ 35 ≈ 7.14 sec.

∴ The correct answer is 7.14 sec.

Alternate Method

Given:

Length of Train A (L₁) = 150 m | Length of Train B (L₂) = 100 m

Speed of Train A (S₁) = 72 km/h | Speed of Train B (S₂) = 54 km/h

Formula Used:

Time taken to cross = (L₁ + L₂) ÷ (S₁ + S₂)

Calculations:

⇒ Total Distance = 150 + 100 = 250 m

⇒ Relative Speed = 72 + 54 = 126 km/h

⇒ Relative Speed in m/s = 126 × (5/18) = 7 × 5 = 35 m/s

⇒ Time taken = 250 ÷ 35

⇒ Time taken = 50 ÷ 7 ≈ 7.1428... sec

∴ The correct answer is 7.14 sec.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). If they move in the same direction, the speeds are subtracted (|S₁ − S₂|).

Unit Conversion

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

Crossing Distance

When a train crosses another train or a bridge/platform, the total distance covered is always the sum of the lengths of both objects.

Shortcut Trick

Using the Multiplying Factor method: Let original price be 100.

New Price = 100 × 0.85 × 0.70 × 0.40 × 1.60

New Price = 100 × 0.595 × 0.64 = 38.08

Net change = 100 − 38.08 = 61.92% decrease.

∴ The correct answer is Decreases by 61.92%.

Alternate Method

Given: Successive changes are −15%, −30%, −60% and +60%.

Formula Used: Final Price = Initial Price × (1 ± r₁/100) × (1 ± r₂/100) × ...

⇒ Let original price = 100

⇒ Price after 1st month = 100 − 15% of 100 = 85

⇒ Price after 2nd month = 85 − 30% of 85 = 85 − 25.5 = 59.5

⇒ Price after 3rd month = 59.5 − 60% of 59.5 = 59.5 − 35.7 = 23.8

⇒ Price after 4th month = 23.8 + 60% of 23.8 = 23.8 + 14.28 = 38.08

⇒ Total Decrease = 100 − 38.08 = 61.92

⇒ Percentage Decrease = (61.92 / 100) × 100 = 61.92%

∴ The correct answer is Decreases by 61.92%.

Additional Information

Successive Percentage Change Formula

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

Base Value Stability

In successive changes, the base for the next percentage calculation is the resultant value of the previous calculation.

Multiplying Factor (MF)

For x% increase, MF = (1 + x/100); for x% decrease, MF = (1 − x/100). Net Result = Initial × MF₁ × MF₂...

Given:

‘ELMS’ is coded as ‘4892’

‘ALES’ is coded as ‘7428’

Common letter analysis:

Therefore, ‘A’ is coded as ‘7’.

Hence, the correct answer is "Option 3".

Logic followed here is: The series follows an addition pattern where the difference between consecutive numbers increases by 5 at each step.

For the given series progression:

 

 

Therefore, the code for the question mark (?) is 245.

Hence, the correct answer is "Option 2".

Logic followed here is: Each letter in the word follows a specific sequence of addition or subtraction based on its position in the English alphabetical order.

 

Therefore, the missing term in the series is HSD.

Hence, the correct answer is "Option 1".

Given:

Seven boxes A, B, C, D, P, Q and R are kept one above the other.

• D is kept immediately above Q.
• R is kept immediately above C.
• Only P is kept above A.
• Only two boxes are kept above D.
• R is not at the third position from the bottom.

Find the number of boxes kept between B and P.

Final Arrangement (Top to Bottom)

 

P is at Position 1 and B is at Position 5.

Boxes between them are: A, D and Q

Number of boxes between B and P = 3.

Hence, the correct answer is Option 4 (Three).

The logic followed here is: Logic: In each pair, the second letter-cluster is obtained by subtracting 2 from the positional values of the letters of the first letter-cluster (Cluster 1 Letter - 2 = Cluster 2 Letter).

(1) IN - GL →

(2) MR - KP →

 

(3) GL - EJ →

 

(4) OT - MS →

 

In all pairs except 'OT - MS', the pattern followed is a subtraction of 2 from both letters. In 'OT - MS', the second letter is obtained by subtracting 1.

Therefore, the odd letter-cluster pair is OT - MS.

Henvce, the correct answer is "Option 4".

Shortcut Trick

Interchange the signs as per the logic: (÷ → ×), (+ → −), (× → ÷), (− → +).

New expression: 58 × 11 − 624 ÷ 13 + 471

Step 1: 58 × 11 = 638 and 624 ÷ 13 = 48

Step 2: 638 − 48 + 471 = 590 + 471 = 1061

∴ The correct answer is 1061.

Alternate Method

Given:

Expression: 58 ÷ 11 + 624 × 13 − 471

Conditions: + ↔ − and × ↔ ÷

Calculations:

After interchanging the signs, the new equation is:

⇒ 58 × 11 − 624 ÷ 13 + 471

Applying BODMAS rule (Division first):

⇒ 58 × 11 − 48 + 471

Applying BODMAS rule (Multiplication):

⇒ 638 − 48 + 471

Applying BODMAS rule (Addition and Subtraction):

⇒ 590 + 471

⇒ 1061

∴ The correct answer is 1061.

Additional Information

BODMAS Rule

It defines the order of operations: Brackets, Of (orders/powers), Division, Multiplication, Addition, and Subtraction.

Mathematical Sign Interchange

In these problems, always rewrite the full expression with new signs before starting any calculation to avoid errors.

Multiplication Trick for 11

To multiply a two-digit number AB by 11, the result is A | (A+B) | B. For 58 × 11: 5 | (5+8) | 8 → 5 | 13 | 8 → 638.

Logic followed here is: Three clusters follow the pattern (+4, +2, +4), while one does not.

1. HLNR →

2. INOS →

3. SWYC →

 

4. VZBF →

  

 

 

Therefore, the letter-cluster 'INOS' does not belong to the group as it follows a different pattern.

Hence, the correct answer is "Option 2".

Given:

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

We need to find even numbers that are:

→ Immediately preceded by an odd number

→ Immediately followed by an odd number

Pattern:

Odd Number → Even Number → Odd Number

Checking the series:

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

The required even numbers are:

4 in (9 4 1)

6 in (1 6 7)

8 in (7 8 3)

2 in (7 2 1)

Therefore, there are 4 such even numbers.

Hence, the correct answer is "Option 3".

Shortcut Trick

The series follows a multiplication pattern with factors increasing by 0.5 each time.

Pattern: ×0.5, ×1, ×1.5, ×2, ×2.5

Missing term = 1.5 × 2 = 3

Check: 3 × 2.5 = 7.5 (Matches the last term)

∴ The correct answer is 3.

Alternate Method

Given Series: 2, 1, 1, 1.5, ?, 7.5

2 ×0.5 → 1 ×1.0 → 1 ×1.5 → 1.5 ×2.0 → 3 ×2.5 → 7.5

The logic of the series is multiplying each term by an arithmetic progression (0.5, 1.0, 1.5, 2.0, 2.5...).

⇒ 2 × 0.5 = 1

⇒ 1 × 1.0 = 1

⇒ 1 × 1.5 = 1.5

⇒ 1.5 × 2.0 = 3

⇒ 3 × 2.5 = 7.5

∴ The correct answer is 3.

Additional Information

Multiplication Series

In this type, each term is multiplied by a fixed or a varying number to get the next term. Common varying factors include AP (0.5, 1, 1.5...) or squares/cubes.

Decimal Multiplication Pattern

Often used in exams, this pattern usually starts with ×0.5 or ×1.5, resulting in numbers decreasing once then increasing rapidly.

Arithmetic-Geometric Series

A complex series where both addition and multiplication (or division) occur in a single step, such as (n × 1) + 1, (n × 2) + 2, etc.