RRB Group D Official Paper (Held On: 21 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is option4. The detailed solution will be updated soon.

p data-end="57" data-start="0">The correct answer is Same mass number but different atomic numbers.

Key Points

• Isobars are defined as atoms of different chemical elements that possess the same mass number but have different atomic numbers.
• The mass number (represented as A) is the total sum of protons and neutrons in the nucleus. In isobars, this sum remains constant across different elements.
• Because they have different atomic numbers (represented as Z), isobars belong to different places in the Periodic Table and represent entirely different chemical elements. Hence, statement 3 is correct.
• A classic example of isobars includes Argon (Atomic Number 18), Potassium (Atomic Number 19), and Calcium (Atomic Number 20). All three can have a mass number of 40.
• In these elements, the number of nucleons (protons + neutrons) is 40, even though the number of protons and electrons varies (18, 19, and 20 respectively).

Additional Information

• Isotopes:
• Isotopes are atoms of the same element that have the same atomic number but different mass numbers. This occurs due to a difference in the number of neutrons.
• Example: Protium, Deuterium, and Tritium are isotopes of Hydrogen with mass numbers 1, 2, and 3 respectively. Hence, statement 4 is incorrect.
• Isotones:
• Isotones are atoms of different elements that have the same number of neutrons but different atomic numbers and mass numbers.
• Example: Carbon-14 (8 neutrons) and Oxygen-16 (8 neutrons). Hence, statement 2 is incorrect.
• Isoelectronic Species:
• These are atoms, ions, or molecules that have the same number of electrons.
• Example: The Neon atom, Sodium ion (Na+), and Magnesium ion (Mg2+) all contain 10 electrons. Hence, statement 1 is incorrect.

Important Points

• Chemical and Physical Properties:
• Isobars have different chemical properties because their electronic configurations are different (owing to different atomic numbers).
• They generally have different physical properties as well, though they may share similarities in properties that depend heavily on atomic mass.
• Nuclear Physics Context:
• In nuclear physics, the study of isobars is vital for understanding Beta Decay. During beta-minus decay, a neutron turns into a proton, increasing the atomic number by 1 while the mass number remains the same, thus creating an isobar.
• Mattauch's isobar rule states that if two adjacent elements on the periodic table have isotopes of the same mass number, one of them must be radioactive.

The correct answer is decreases.

Key Points

• Pressure is defined as the physical force exerted on an object per unit area. It is mathematically expressed as Pressure = Thrust / Area.
• Thrust is the total force acting perpendicularly on a surface. When the thrust remains constant, the pressure depends solely on the surface area over which the force is distributed.
• Since Pressure is inversely proportional to the Area (P ∝ 1/A), any increase in the surface area will lead to a decrease in the pressure experienced by that surface.
• Conversely, if the surface area is decreased while the thrust remains the same, the pressure on the surface will increase significantly.
• The SI unit of pressure is the Pascal (Pa), which is equivalent to one Newton per square meter (1 N/m²). This unit is named after the French scientist Blaise Pascal.

Additional Information

• Applications of Increasing Area:
• Foundations of buildings are made wide so that the weight of the structure (thrust) is distributed over a larger area, thereby reducing the pressure on the ground and preventing the building from sinking.
• School bags are provided with broad straps rather than thin strings. This ensures the weight of the bag is spread over a larger surface area of the shoulder, making it less painful to carry by reducing pressure.
• Tractors and heavy trucks are equipped with wide tyres to ensure the heavy load does not exert enough pressure to sink the vehicle into soft mud or sand.
• Applications of Decreasing Area:
• A sharp knife has a very small surface area at its cutting edge. This allows a small force to produce a very high pressure, making it easy to cut through materials.
• Nails and pins have pointed tips (extremely small area) so that even a gentle hammer strike produces enough pressure to penetrate a wooden plank or a wall.
• Liquid and Gas Pressure:
• In fluids (liquids and gases), pressure is exerted in all directions. The pressure in a liquid increases with depth due to the weight of the overhead liquid column.
• Atmospheric pressure is the pressure exerted by the weight of the atmosphere. At sea level, its value is approximately 101,325 Pa or 1 atmosphere (atm).

Important Points

• The relationship between force and pressure is a fundamental principle in physics and engineering used to design safety equipment and heavy machinery.
• Scalars and Vectors: While force is a vector quantity (having both magnitude and direction), pressure is a scalar quantity because it relates the magnitude of the normal force to the area.
• Dimensions: The dimensional formula for pressure is [ML⁻¹T⁻²].
• The concept of thrust and pressure explains why a person can walk on snow more easily using skis (large area, low pressure) than wearing normal shoes (small area, high pressure).

The correct answer is Because both gametes get halved before fertilisation to form the zygote.

Key Points

• In sexually reproducing organisms, the gametes (sperm and egg) are produced through meiosis.
• Meiosis reduces the chromosome number and DNA content to half, producing haploid (n) gametes from diploid (2n) cells.
• During fertilisation, the male and female gametes fuse to form a diploid (2n) zygote.
• Thus, the halving of the chromosome number during gamete formation prevents the DNA/chromosome number from doubling in every generation.
• Hence, the correct answer is Option 3 – Because both gametes get halved before fertilisation to form the zygote.

Additional Information

• Diploid cells (2n): Contain two sets of chromosomes, one set from each parent.
• Haploid gametes (n): Contain only one set of chromosomes.
• Meiosis: Reduces the chromosome number from 2n to n.
• Fertilisation: Combines two haploid gametes (n + n) to restore the diploid condition (2n).

Important Points

• Before meiosis: Diploid cell = 2n.
• After meiosis: Gametes = n.
• Fertilisation: n + n = 2n (zygote).
• Purpose of meiosis: To maintain a constant chromosome number across generations in sexually reproducing organisms.
• Exam Tip: Remember the sequence: 2n → Meiosis → n → Fertilisation → 2n.

The correct answer is By storing them in vacuoles or excreting through falling leaves and resins.

Key Points

• Excretion in plants is significantly different from animals because plants do not possess a specialized excretory system or specific organs like kidneys.
• One of the primary methods plants use to manage metabolic waste is storage in cellular vacuoles. These vacuoles are membrane-bound sacs within the cell that sequester toxic or unnecessary substances, keeping them away from the vital cytoplasm.
• Plants often accumulate waste products in older leaves. When these leaves reach a certain concentration of waste, they turn yellow and fall off (a process known as abscission), thereby removing the waste from the plant's body. Hence, statement 4 is correct.
• Another common strategy involves the deposition of waste products as resins and gums. These are often found in old xylem tissues that are no longer active in water transport.
• Plants also eliminate certain waste substances into the soil directly through their root systems. However, this is just one of many methods, not the exclusive way. Hence, statement 1 is incorrect.
• While plants exchange oxygen and carbon dioxide, they do not "convert all wastes" into these gases. Solid and liquid metabolic byproducts require different disposal routes. Hence, statement 2 is incorrect.
• Nephrons are the microscopic functional units of kidneys found in vertebrates. Plants completely lack these complex multi-cellular excretory organs. Hence, statement 3 is incorrect.

Additional Information

• Gaseous Waste Management:
• During the day, plants produce oxygen as a byproduct of photosynthesis. Some of this is used for respiration, and the excess is released through stomata.
• At night, carbon dioxide produced during respiration is the main gaseous waste expelled.
• Small openings on the woody stems called lenticels also facilitate the exchange of these gases.
• Water Excretion:
• Plants remove excess water through the process of transpiration, where water evaporates from the stomata on the leaf surface.
• In certain conditions, plants may also release liquid water droplets through specialized structures called hydathodes, a process known as guttation.
• Secondary Metabolites:
• Many plant wastes are "useful" wastes known as secondary metabolites, such as alkaloids (e.g., quinine, morphine), essential oils, and tannins.
• Latex, a milky fluid found in plants like rubber trees, is also a form of plant byproduct stored in specialized cells called laticifers.

Important Points

• Calcium Oxalate: Some plants convert liquid waste into solid crystals of calcium oxalate, known as raphides. These are often needle-shaped and are stored in vacuoles to prevent them from interfering with metabolic processes.
• Heartwood: In large trees, the central, non-functional part of the trunk called heartwood acts as a permanent storage site for tannins, resins, and oils. This not only disposes of waste but also makes the wood more durable and resistant to fungal and insect attacks.
• Bark Shedding: Certain trees deposit waste materials in their bark. As the tree grows, the outer bark peels or sheds, taking the accumulated waste products with it.

The correct answer is Selfish exploitation causes environmental harm and misery.

Key Points

• The core realization for sustainable resource management is that unregulated selfish exploitation of natural resources inevitably leads to ecological degradation and human suffering. This is often described in environmental economics as the Tragedy of the Commons.
• When individuals or entities prioritize short-term profits over long-term stability, they deplete renewable resources (like forests and fisheries) and non-renewable resources (like minerals and fossil fuels) beyond their carrying capacity.
• This exploitation results in environmental harm, such as biodiversity loss, soil erosion, and climate change, which directly causes socio-economic misery by destroying livelihoods and reducing the quality of life for present and future populations.
• Recognizing the interconnectedness of ecosystems helps society understand that sustainable use is not just an ethical choice but a biological necessity for survival.

Additional Information

• Sustainable Development: This concept was famously defined in the Brundtland Report (also known as Our Common Future) in 1987 as development that meets the needs of the present without compromising the ability of future generations to meet their own needs.
• The Three Pillars: Sustainability is built on three interconnected pillars: Environmental Protection, Social Equity, and Economic Viability. Focusing solely on unlimited growth (Option 2) ignores the biophysical limits of Earth.
• Intergenerational Equity: This legal and ethical principle suggests that natural resources belong to all generations. While future generations must benefit (Option 3), the realization must also encompass the current generation's responsible consumption to avoid immediate misery.
• The Role of Regulation: While rules and legislation are essential for conservation (Option 4), they are often insufficient without a collective realization and cultural shift toward stewardship and voluntary compliance.

Important Points

• Resource Depletion Data: According to the Global Footprint Network, humanity is currently using nature 1.75 times faster than our planet’s ecosystems can regenerate, leading to ecological overshoot.
• United Nations SDGs: The Sustainable Development Goals (SDGs), specifically Goal 12 (Responsible Consumption and Production) and Goal 15 (Life on Land), were adopted in 2015 to provide a global blueprint for mitigating selfish exploitation.
• Economic Impact: The Dasgupta Review (2021) on the Economics of Biodiversity highlighted that our global economy is embedded within nature, not external to it, meaning environmental harm eventually results in economic collapse.

The correct answer is – 7.5 m/s².

Key Points

• Acceleration is defined as the rate of change of velocity of an object with respect to time. It is a vector quantity, meaning it possesses both magnitude and direction.
• The mathematical formula to calculate average acceleration (a) is given by: a = (v - u) / t, where v is the final velocity, u is the initial velocity, and t is the time interval.
• In the given problem, the initial velocity (u) is +20 m/s, the final velocity (v) is -10 m/s, and the time taken (t) is 4 seconds.
• Substituting the values into the formula: a = (-10 m/s - 20 m/s) / 4 s. This simplifies to a = -30 / 4, which equals -7.5 m/s².
• The negative sign indicates that the acceleration is acting in the opposite direction to the initial motion, or that the body is experiencing retardation relative to the positive direction. Hence, the correct answer is – 7.5 m/s².

Additional Information

• SI Unit of Acceleration: The standard unit for measuring acceleration in the International System of Units (SI) is the meter per second squared (m/s²).
• Vector Nature: Since velocity is a vector, any change in either the speed or the direction of motion results in acceleration. In this case, the body not only slowed down but also reversed its direction (moving from positive to negative velocity).
• Uniform vs. Non-Uniform Acceleration: Uniform acceleration occurs when the velocity changes by equal amounts in equal intervals of time. Non-uniform acceleration occurs when the rate of change of velocity varies over time.
• Retardation/Deceleration: When the acceleration of a body is in the direction opposite to its velocity, it is often referred to as deceleration or retardation. It results in a decrease in the magnitude of velocity (speed) until the object stops or reverses.

Important Points

• Sign Convention: In physics, direction is often indicated by positive (+) and negative (-) signs. A change from +20 m/s to -10 m/s implies a total change in velocity of 30 m/s in the negative direction.
• Instantaneous vs. Average: While the formula (v-u)/t provides the average acceleration over the 4-second interval, instantaneous acceleration refers to the acceleration at a specific moment in time, calculated using derivatives (dv/dt).
• Graphical Representation: On a velocity-time graph, the slope of the line represents the acceleration. A downward-sloping line indicates negative acceleration.

The correct answer is The components of brass are uniformly distributed and cannot be distinguished visually.

Key Points

• Brass is categorized as a homogeneous mixture because it is a solid solution where the solute (zinc) and solvent (copper) are mixed uniformly at the atomic level.
• In a homogeneous mixture, the composition is consistent throughout the entire sample, and there are no visible boundaries of separation between the constituent elements. Hence, statement 1 is correct.
• Unlike chemical compounds, the elements in an alloy are not chemically bonded in fixed proportions. They are physically blended in a molten state and retain their individual chemical identities. Hence, statement 2 is incorrect.
• Brass is an alloy, which by definition is a mixture of two or more elements (at least one being a metal). It consists of copper and zinc atoms, rather than being made of a single type of atom. Hence, statement 3 is incorrect.
• A compound is a substance formed when two or more elements chemically react to form a new substance with entirely different properties. Brass is a physical mixture of metals that exhibits variable composition. Hence, statement 4 is incorrect.
• The uniformity of brass ensures that any small sample taken from the musical instrument will have the same ratio of copper to zinc as any other sample from the same mold.

Additional Information

• Composition of Brass: It typically consists of approximately 60% to 70% Copper (Cu) and 30% to 40% Zinc (Zn). The exact percentage can be adjusted to alter the hardness, color, and ductility of the metal.
• Properties: Brass is highly prized for its malleability (easier to shape than bronze or zinc) and its acoustic properties, which make it ideal for musical instruments like trumpets, horns, and bells.
• Corrosion Resistance: Due to the presence of Copper, brass is highly resistant to corrosion, making it useful for plumbing fixtures, marine hardware, and electrical terminals.
• Substitutional Alloy: Brass is specifically a substitutional alloy, meaning the Zinc atoms replace Copper atoms within the metallic crystal lattice.
• Applications: Beyond music, it is used in ammunition casings, decorative items, and low-friction applications like locks, gears, and bearings.

Important Points

• Alloys vs. Mixtures: Alloys are considered mixtures because they show the properties of their constituents and can have variable compositions (unlike compounds which have a fixed formula).
• Separation: While homogeneous mixtures cannot be separated by simple physical methods like filtration, the components of brass can be separated using metallurgical processes that exploit different melting points or chemical reactivities.
• History: Brass has been known to humans since ancient times (roughly 500 BC), though it was often produced by smelting copper ores rich in zinc before pure zinc was identified as an element.
• Identification: To the naked eye or under a standard microscope, a homogeneous mixture like brass appears as a single pure substance, which is a key characteristic of solid solutions.

The correct answer is a cathode.

Key Points

• In the process of electrolytic refining, the pure metal always serves as the cathode, which is the negative electrode in the electrolytic cell.
• A thin strip or rod of pure metal is placed at the cathode to act as a starter sheet upon which the refined metal will deposit.
• The impure metal (the metal to be refined) is made the anode, which is the positive electrode.
• When an electric current is passed through the circuit, the metal atoms from the impure anode lose electrons and enter the electrolyte as positive ions.
• These positive metal ions are then attracted to the negative cathode, where they gain electrons (undergo reduction) and deposit as pure metal atoms.
• The concentration of the electrolyte remains constant throughout the process because the rate at which metal dissolves from the anode is equal to the rate at which it deposits on the cathode.

Additional Information

• The Electrolyte:
• The electrolyte used in this process is a soluble salt solution of the same metal that is being refined.
• For example, in the electrolytic refining of copper, the electrolyte is usually a solution of acidified copper sulphate (CuSO4).
• Commonly Refined Metals:
• This method is widely utilized for the purification of various non-ferrous metals such as Copper (Cu), Zinc (Zn), Tin (Sn), Nickel (Ni), Silver (Ag), and Gold (Au).
• It is particularly effective for obtaining metals with a very high degree of purity, often exceeding 99.9%.
• Mechanism at Electrodes:
• At the Anode: The reaction involves oxidation, where the metal M becomes M(n+) + ne(-).
• At the Cathode: The reaction involves reduction, where the metal ions M(n+) + ne(-) become the pure metal M.
• Treatment of Impurities:
• Soluble impurities go into the electrolyte solution and remain there.
• Insoluble impurities, which do not dissolve, fall to the bottom of the tank just below the anode.

Important Points

• Anode Mud: This refers to the collection of insoluble impurities that settle under the anode. In many cases, anode mud contains valuable elements such as antimony, selenium, tellurium, silver, gold, and platinum. The recovery of these precious metals often covers the cost of the refining process.
• Copper Refining: In the specific case of copper, the impure anode is often referred to as blister copper. As the process proceeds, the anode gradually gets thinner while the cathode becomes thicker and heavier with pure copper.
• Voltage and Current: The process requires a carefully controlled Direct Current (DC) and a specific voltage to ensure that only the desired metal ions are reduced at the cathode, preventing the deposition of unwanted impurities.

The correct answer is 20 m.

Key Points

• The motion of the block can be described using the Equations of Motion formulated by Sir Isaac Newton. These equations relate displacement, velocity, acceleration, and time for an object moving with constant acceleration.
• In this problem, the block is "dropped," which implies that its initial velocity (u) is 0 m/s. The height from which it is dropped is X, and it hits the ground with a final velocity (v) of 20 m/s.
• The acceleration due to gravity (g) acting on the block is given as 10 m/s^2. Since the motion is downward, the acceleration is positive in the direction of displacement.
• To find the height X, we use the Third Equation of Motion: v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement (height X).
• Substituting the values: (20)^2 = (0)^2 + 2 * 10 * X. This simplifies to 400 = 20 * X. Dividing 400 by 20, we find that X = 20 meters. Hence, the correct answer is 20 m.

Additional Information

• Conservation of Energy:
• This problem can also be solved using the principle of Conservation of Mechanical Energy. At the top of the building, the block has Potential Energy (PE) equal to mgh.
• As the block falls, its potential energy is converted into Kinetic Energy (KE), which is 1/2 * m * v^2 at the moment of impact.
• Equating the two (mgh = 1/2 * m * v^2), the mass (m) cancels out from both sides, proving that the height is independent of the mass of the object.
• Calculation: 10 * X = 0.5 * (20)^2 yields 10X = 200, resulting in X = 20 m.
• Role of Mass in Free Fall:
• The question specifies a mass of 1 kg. In the absence of air resistance, all objects fall at the same rate regardless of their mass. This was famously demonstrated by Galileo Galilei at the Leaning Tower of Pisa.
• Whether the block was 1 kg or 100 kg, the height required to reach a velocity of 20 m/s would remain 20 m.
• Acceleration due to Gravity:
• The standard value of g on Earth is approximately 9.81 m/s^2, but for competitive exams and simplified calculations, it is often rounded to 10 m/s^2.
• Gravity is a vector quantity that always acts toward the center of the Earth.

Important Points

• Units and Measurements:
• It is essential to ensure all units are in the SI system (Standard International). Here, velocity is in m/s, acceleration is in m/s^2, and height is in meters (m).
• If the velocity was given in km/h, it would need to be converted by multiplying by 5/18 before using the kinematic equations.
• Kinematic Equations:
• First Equation: v = u + at (used to find time or velocity).
• Second Equation: s = ut + 1/2 * at^2 (used to find displacement when time is known).
• Third Equation: v^2 = u^2 + 2as (used when time is unknown, as in this specific question).
• Assumptions:
• These calculations assume a vacuum or negligible air resistance. In a real-world scenario with significant air friction, the block would hit the ground with a lower velocity than 20 m/s when dropped from 20 m.

The correct answer is Micelles.

Key Points

• Soap molecules are sodium or potassium salts of long-chain carboxylic acids (fatty acids). They possess a unique structure consisting of a hydrophilic (water-loving) ionic head and a hydrophobic (water-fearing) hydrocarbon tail.
• When soap is dissolved in water, the molecules orient themselves in a specific cluster called a Micelle. In this arrangement, the hydrophobic tails aggregate together at the interior of the cluster to stay away from water, while the ionic heads face outward, interacting with the water molecules.
• The cleaning action occurs because most dirt and stains are oily or greasy in nature. The hydrophobic end of the soap molecule attaches itself to the oil or dirt particle, while the hydrophilic end remains attached to the water.
• This results in the formation of an emulsion where the oily dirt is trapped at the center of the micelle. Since the surfaces of these micelles are similarly charged (usually negative), they repel each other and remain suspended in the water as a colloid, preventing the dirt from re-depositing on the clothes. Hence, the correct answer is Micelles.

Additional Information

• Chemical Composition of Soap:
• Common soaps are made from fats and oils reacting with Sodium Hydroxide (NaOH) or Potassium Hydroxide (KOH). This chemical reaction is known as Saponification.
• Common fatty acids used include stearic acid, palmitic acid, and oleic acid. Sodium stearate (C17H35COONa) is a typical example of a soap molecule.
• Hard Water and Scum Formation:
• In hard water, which contains high concentrations of Calcium (Ca2+) and Magnesium (Mg2+) ions, soap reacts with these ions to form an insoluble, greyish-white precipitate called scum.
• Because of this precipitation, a large amount of soap is wasted before it can start forming micelles for cleaning.
• Detergents vs. Soap:
• Detergents are generally ammonium or sulphonate salts of long-chain carboxylic acids.
• Unlike soaps, the charged ends of detergents do not form insoluble precipitates with the calcium and magnesium ions in hard water. Therefore, detergents are effective for cleaning even in hard water conditions.
• Mechanism of Action:
• The cleansing process involves agitation (scrubbing or rotating in a washing machine), which helps pull the micelles (containing the trapped dirt) away from the surface of the fabric into the bulk of the water.
• Soap lowers the surface tension of water, allowing it to wet surfaces more effectively and penetrate deeper into the fibers of the cloth.

Important Points

• Hydrophobic Effect: This is the primary driving force behind micelle formation. The non-polar hydrocarbon tails are "pushed" together by water molecules that prefer to bond with each other and the polar heads of the soap.
• Tyndall Effect: Because a soap solution containing micelles is a colloidal system, it is capable of scattering a beam of light, a phenomenon known as the Tyndall effect.
• Shape: While often represented as spheres, micelles can take various shapes depending on the concentration of soap and the temperature, including ellipsoidal or cylindrical forms.
• Critical Micelle Concentration (CMC): This is the specific concentration of surfactants above which micelles form and all additional surfactants added to the system go to micelles. Below this concentration, soap molecules exist individually on the surface or in the bulk of the liquid.

The correct answer is One element displaces another from a compound.

Key Points

• A displacement reaction is a type of chemical reaction in which a more reactive element removes or displaces a less reactive element from its compound. It is often represented by the general formula: A + BC → AC + B.
• In this reaction, the element A must be more chemically reactive than the element B for the displacement to occur. If A is less reactive than B, no reaction will take place.
• Statement 1: "Two reactants combine to generate one product" describes a Combination Reaction (also known as a Synthesis Reaction), such as the formation of water from hydrogen and oxygen. Hence, statement 1 is incorrect.
• Statement 2: "Compounds disintegrate into two or more components" describes a Decomposition Reaction, where a single reactant breaks down, often requiring energy in the form of heat, light, or electricity. Hence, statement 2 is incorrect.
• Statement 3: "Ion exchange occurs between two substances" refers to a Double Displacement Reaction (or Metathesis), where parts of two ionic compounds are exchanged to make two new compounds. Hence, statement 3 is incorrect.
• Statement 4: "One element displaces another from a compound" is the fundamental definition of a Single Displacement Reaction. This process is governed by the Electrochemical Series or Reactivity Series. Hence, statement 4 is correct.

Additional Information

• The Reactivity Series: This is a vertical arrangement of metals in the decreasing order of their reactivity. Potassium (K) and Sodium (Na) are at the top (most reactive), while Gold (Au) and Platinum (Pt) are at the bottom (least reactive).
• Example of Metal Displacement: When an Iron (Fe) nail is placed in a Copper Sulphate (CuSO4) solution, the blue color of the solution fades and turns light green. This happens because Iron is more reactive than Copper and displaces it to form Iron Sulphate (FeSO4).
• Hydrogen Displacement: Many metals like Zinc (Zn) or Magnesium (Mg) react with dilute acids (such as HCl) to displace Hydrogen. For instance, Zn + 2HCl → ZnCl2 + H2.
• Applications: Displacement reactions are widely used in metallurgy for the extraction of metals, in Thermite welding for repairing railway tracks, and in batteries to generate electricity.
• Halogen Displacement: Non-metals also undergo these reactions. For example, Chlorine (Cl2) can displace Bromine (Br2) from a solution of Potassium Bromide (KBr) because Chlorine is more electronegative and reactive.

Important Points

• Thermodynamics: These reactions are generally exothermic, meaning they release heat energy to the surroundings as the bonds in the products are more stable than the reactants.
• Redox Nature: Most displacement reactions are Redox Reactions, involving the oxidation of the more reactive element (losing electrons) and the reduction of the less reactive element (gaining electrons).
• Reaction Speed: The speed or rate of reaction often depends on the distance between the two metals in the reactivity series; the further apart they are, the more vigorous the reaction tends to be.

The correct answer is Object is at the centre of curvature.

Key Points

• The magnification (m) of a spherical mirror is defined as the ratio of the height of the image (h') to the height of the object (h). It is also expressed in terms of object distance (u) and image distance (v) as m = -v/u.
• In this question, the magnification is -1. The negative sign indicates that the image formed is real and inverted. Hence, the image is formed in front of the mirror.
• The numerical value 1 indicates that the size of the image is exactly equal to the size of the object (|v| = |u|).
• For a concave mirror, the image is of the same size as the object only when the object is placed at the centre of curvature (C). In this specific case, the image is also formed at the centre of curvature, is real, and is inverted. Hence, the statement "Object is at the centre of curvature" is correct.
• If the object is beyond the centre of curvature, the image is formed between the focus (F) and the centre of curvature (C) and is diminished (|m| < 1). Hence, option 1 is incorrect.
• If the object is between the focus (F) and the pole (P), the image formed is virtual, erect, and magnified, meaning the magnification would be positive and greater than 1. Hence, option 2 is incorrect.
• If the object is at the focus (F), the image is formed at infinity and is highly magnified. Hence, option 3 is incorrect.

Additional Information

• Concave Mirror: It is a spherical mirror whose reflecting surface is curved inwards, towards the centre of the sphere. It is also known as a converging mirror because it converges light rays that fall on it.
• Mirror Formula: The relationship between the focal length (f), object distance (u), and image distance (v) is given by the formula: 1/f = 1/v + 1/u.
• Sign Convention: According to the New Cartesian Sign Convention, the pole (P) of the mirror is taken as the origin. The object distance (u) is always negative. For a concave mirror, the focal length (f) and the radius of curvature (R) are always negative.
• Radius of Curvature (R): The distance between the pole and the centre of curvature is called the radius of curvature. It is related to the focal length by the equation R = 2f.
• Uses of Concave Mirrors: They are commonly used in shaving mirrors to see a larger image of the face, in torches and searchlights to get powerful parallel beams of light, and by dentists to see large images of teeth.

Important Points

• If m is positive, the image is virtual and erect.
• If m is negative, the image is real and inverted.
• If |m| > 1, the image is enlarged.
• If |m| < 1, the image is diminished.
• If |m| = 1, the image is of the same size as the object.
• A plane mirror always has a magnification of +1, meaning the image is virtual, erect, and of the same size as the object.
• A convex mirror always produces a virtual, erect, and diminished image, so its magnification is always positive and less than 1.

The correct answer is atomic, mass.

Key Points

• Isobars are defined as atoms of different chemical elements that have the same mass number (total number of nucleons) but different atomic numbers (number of protons).
• The term isobar was suggested by Alfred Walter Stewart in the year 1918. It is derived from the Greek words "isos" meaning equal and "baros" meaning weight.
• Because isobars have different atomic numbers, they occupy different positions in the periodic table and possess different chemical properties.
• Examples of isobars include Argon (atomic number 18), Potassium (atomic number 19), and Calcium (atomic number 20), all of which can have a mass number of 40.
• Since their mass numbers are the same, their physical properties, which depend on the mass of the atom, may be quite similar, though their electronic configurations differ significantly.

Additional Information

• Atomic Number (Z):
• It represents the number of protons found in the nucleus of every atom of that element.
• It uniquely identifies a chemical element. For instance, any atom with 6 protons is an atom of Carbon.
• Mass Number (A):
• It is the sum of the total number of protons and neutrons in an atomic nucleus.
• While the atomic number is usually constant for an element, the mass number can vary, leading to the formation of isotopes.
• Isotopes:
• These are atoms of the same element that have the same atomic number but different mass numbers.
• Example: Hydrogen has three isotopes: Protium (1 proton, 0 neutrons), Deuterium (1 proton, 1 neutron), and Tritium (1 proton, 2 neutrons).
• Isotones:
• These are atoms of different elements that have the same number of neutrons but different atomic numbers and mass numbers.
• Example: Carbon-14 (6 protons, 8 neutrons) and Oxygen-16 (8 protons, 8 neutrons) are isotones.

Important Points

• Nucleons is the collective term used for protons and neutrons, both of which reside in the nucleus of an atom.
• The number of electrons in a neutral atom is equal to its atomic number. Since isobars have different atomic numbers, they have a different number of electrons, leading to their distinct chemical reactivity.
• In nuclear physics, isobars are relevant in beta decay. In beta minus decay, a neutron turns into a proton, meaning the atomic number increases by 1 while the mass number remains constant, effectively creating an isobar of the parent atom.
• The atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes and is measured in atomic mass units (amu).

The correct answer is Focal length (in metres).

Key Points

• The power of a lens is defined as the ability of the lens to converge or diverge light rays falling on it. It is mathematically expressed as the reciprocal of its focal length.
• The standard formula used to calculate the power (P) is P = 1/f, where f must be measured in metres.
• The SI unit of power of a lens is the Dioptre, which is denoted by the symbol D. One Dioptre is equal to 1 m⁻¹.
• A convex lens has a positive focal length, and therefore, its power is always positive. This is because it converges light rays at a real focus.
• A concave lens has a negative focal length, resulting in a negative power value. This lens is known for diverging light rays.
• The power of a lens is directly related to its refractive index and the radii of curvature of its surfaces, as described by the lens maker's formula.

Additional Information

• Combination of Lenses: When two or more thin lenses are placed in contact, the total power (P) of the combination is the algebraic sum of the individual powers of the lenses (P = P1 + P2 + P3...). This principle is widely used in designing camera lenses and microscopes.
• Focal Length and Power Relationship: A lens with a short focal length has more power because it bends light rays more sharply, leading to a higher degree of convergence or divergence. Conversely, a lens with a long focal length has less power.
• Optical Instruments: Opticians prescribe corrective lenses using power instead of focal length. For example, a prescription of +2.0 D indicates a convex lens with a focal length of 0.5 metres (50 cm).
• Refractive Index Influence: The power of a lens also depends on the medium in which it is placed. If a lens is immersed in a liquid with a higher refractive index than air, its focal length increases, and its power decreases.

Important Points

• Myopia (Short-sightedness): To correct this vision defect, a concave lens with negative power is used. This helps in diverging the incoming light so the image forms correctly on the retina.
• Hypermetropia (Long-sightedness): This condition is corrected using a convex lens with positive power, which increases the converging power of the eye's optical system.
• Calculation Tip: Always remember to convert the focal length from centimetres to metres before using the power formula. For instance, if f = 25 cm, then f = 0.25 m, and P = 1/0.25 = 4 D.
• Physical Dimensions: While thickness and diameter (aperture) affect the intensity of light and aberrations, they do not directly define the numerical "power" of the lens in the same way the focal length does.

The correct answer is Receptor → Spinal cord → Motor neuron → Muscle.

Key Points

• A reflex arc is a neural pathway that controls a reflex action, which is an immediate, involuntary response to a stimulus.
• The process begins at the Receptor (like the skin or eyes), which perceives the stimulus (such as heat or pain).
• From the receptor, the impulse is carried by a sensory neuron (also known as an afferent neuron) to the Central Nervous System (CNS), specifically the Spinal cord.
• In the Spinal cord, the interneuron processes the signal and immediately relays it to a Motor neuron (also known as an efferent neuron) without waiting for the brain to provide a conscious command.
• The Motor neuron then transmits the impulse to the Effector, which is usually a Muscle or a gland, to perform the necessary action.
• The sequence Receptor → Spinal cord → Motor neuron → Muscle correctly identifies the bypass of the brain to ensure speed. Hence, option 1 is correct.
• The sequence Receptor → Spinal cord → Brain → Muscle implies that the brain must process the reflex before the muscle acts, which would be too slow for a reflex. Hence, option 2 is incorrect.
• The sequence Receptor → Brain → Spinal cord → Muscle describes a conscious voluntary action rather than an involuntary reflex. Hence, option 3 is incorrect.
• The sequence Receptor → Motor neuron → Spinal cord → Muscle is anatomically incorrect because motor neurons carry impulses away from the spinal cord, not toward it. Hence, option 4 is incorrect.

Additional Information

• Sensory Neurons (Afferent): These specialized nerve cells are responsible for converting external stimuli from the environment into internal electrical impulses. They carry signals from the peripheral nervous system to the CNS.
• Motor Neurons (Efferent): These are neurons located in the CNS that project their axons outside the CNS to directly or indirectly control muscles and glands.
• Reflex Action: This is a rapid, automatic response to a stimulus. Examples include the knee-jerk reflex, blinking when an object approaches the eye, or withdrawing a hand from a hot surface.
• Evolutionary Advantage: The primary function of the reflex arc is survival. By bypassing the conscious brain, the body can react to dangerous stimuli in milliseconds, significantly reducing the risk of tissue damage.
• Interneurons: These are found within the Spinal cord and act as the "middleman," connecting sensory and motor neurons to allow for a quick circuit.

Important Points

• Types of Reflexes:
• Monosynaptic Reflex: A simple reflex arc consisting of only one synapse between the sensory and motor neuron (e.g., the Patellar reflex).
• Polysynaptic Reflex: A reflex involving one or more interneurons between the sensory and motor components (e.g., the withdrawal reflex from pain).
• Anatomy of the Spinal Cord: The grey matter of the spinal cord is where the synapses of the reflex arc occur. It contains the cell bodies of motor neurons and interneurons.
• Synapse: This is the microscopic gap between two neurons where the electrical impulse is converted into a chemical signal (using neurotransmitters) to cross to the next cell.
• Stimulus: Any change in the environment (internal or external) that is capable of eliciting a nerve impulse.

The correct answer is Butane (C4H10).

Key Points

• Butane (C4H10) is an alkane with four carbon atoms. It is the first hydrocarbon in the series that can form a branched chain structure known as isobutane (or 2-methylpropane), where one carbon atom is attached to the center carbon of a three-carbon chain. Hence, option 1 is correct.
• Ethane (C2H6) consists of two carbon atoms connected by a single bond. Because it only has two carbons, they can only be arranged in a straight line, making branching impossible. Hence, option 2 is incorrect.
• Cyclohexane (C6H12) is a cycloalkane where six carbon atoms are joined in a closed-ring structure. While it contains multiple carbons, its primary arrangement is cyclic rather than a branched open chain. Hence, option 3 is incorrect.
• Methane (CH4) is the simplest hydrocarbon, containing only one carbon atom bonded to four hydrogen atoms. Since there is only one carbon, there is no possibility of any chain formation or branching. Hence, option 4 is incorrect.
• In organic chemistry, branching occurs when at least one carbon atom is bonded to three or four other carbon atoms, a phenomenon that requires a minimum of four carbon atoms in an open-chain alkane.

Additional Information

• Structural Isomerism:
• This is a form of isomerism where compounds have the same molecular formula but different structural arrangements of atoms.
• Butane has two structural isomers: n-Butane (straight chain) and isobutane (branched chain).
• As the number of carbon atoms increases, the number of possible isomers increases significantly. For example, Pentane (C5H12) has three isomers, while Hexane (C6H14) has five.
• Hydrocarbon Classification:
• Alkanes: Saturated hydrocarbons with the general formula CnH2n+2. They contain only single covalent bonds.
• Alkenes and Alkynes: Unsaturated hydrocarbons containing double or triple bonds, respectively.
• Cycloalkanes: Saturated hydrocarbons where the carbon atoms are arranged in a ring, following the formula CnH2n.
• Physical Properties of Branched Chains:
• Branching generally lowers the boiling point of a compound compared to its straight-chain isomer.
• This is because a branched-chain molecule has a more spherical shape, which reduces the surface area and weakens the Van der Waals forces between molecules.
• For instance, n-butane has a boiling point of approximately -0.5 degrees Celsius, while isobutane boils at -11.7 degrees Celsius.

Important Points

• Carbon's Tetravalency: Carbon has four valence electrons, allowing it to form four covalent bonds. This property, known as tetravalency, combined with catenation (the ability to link with other carbon atoms), allows the formation of complex straight, branched, and cyclic structures.
• Nomenclature: According to IUPAC rules, the branched isomer of butane is named 2-methylpropane because the longest continuous chain has three carbons (propane) and a methyl group is attached to the second carbon.
• Natural Occurrence: Butane is a highly flammable, colorless gas that is easily liquefied. It is commonly used as a fuel in LPG (Liquefied Petroleum Gas) cylinders and cigarette lighters.

The correct answer is Golgi apparatus.

Key Points

• The Golgi apparatus, also known as the Golgi complex or Golgi body, is a membrane-bound organelle found in eukaryotic cells. It consists of a series of flattened, stacked pouches called cisternae.
• It was first discovered by the Italian biologist Camillo Golgi in 1898 while he was examining nerve cells. The cisternae are typically 0.5µm to 1.0µm in diameter and are stacked parallel to each other.
• The primary function of the Golgi apparatus is the processing, packaging, and secretion of proteins and lipids. It acts as the shipping center of the cell, modifying products received from the Endoplasmic Reticulum (ER).
• The Golgi apparatus is directly involved in the formation of lysosomes. Lysosomes are formed by the budding of vesicles from the trans-face (shipping side) of the Golgi complex, containing concentrated hydrolytic enzymes.
• The organelle possesses a distinct polarity, with a cis face (forming face) that receives materials from the ER and a trans face (maturing face) where modified materials are dispatched in secretory vesicles.

Additional Information

• Chromoplast:
• Chromoplasts are a type of plastid found in plant cells that contain pigments other than chlorophyll.
• They are responsible for the yellow, orange, and red colors seen in flowers, fruits, and aging leaves due to the presence of carotenoid pigments.
• Unlike the Golgi apparatus, they do not contain cisternae and are not involved in lysosome formation.
• Leucoplast:
• Leucoplasts are colorless plastids primarily used for the storage of nutrients in plants.
• They are categorized into Amyloplasts (store starch), Elaioplasts (store oils/fats), and Aleuroplasts (store proteins).
• These are typically found in non-photosynthetic tissues like roots, bulbs, and seeds.
• Mitochondria:
• Known as the "Powerhouse of the cell," mitochondria are responsible for cellular respiration and the production of ATP (Adenosine Triphosphate).
• They have a double-membrane structure. The inner membrane is folded into cristae (not cisternae) to increase surface area for oxidative phosphorylation.
• Mitochondria contain their own DNA (mtDNA) and ribosomes, allowing them to be semi-autonomous.
• Lysosomes:
• These are suicide bags of the cell, containing digestive enzymes that break down waste materials, cellular debris, and foreign particles.
• The acidic pH within lysosomes is maintained to optimize the activity of acid hydrolases.

Important Points

• The term "cisternae" is specifically used for the flattened sacs in the Golgi apparatus and the Endoplasmic Reticulum, whereas "cristae" refers to the folds of the inner mitochondrial membrane.
• The Endomembrane System includes the nuclear envelope, ER, Golgi apparatus, lysosomes, vacuoles, and the plasma membrane, all working together to transport and process cellular materials.
• In plant cells, the Golgi apparatus is often referred to as Dictyosomes, which are usually more scattered and smaller than those found in animal cells.

The correct answer is Rhinoceros.

Key Points

• Infrasound refers to sound waves with frequencies below the lower limit of human audibility, which is typically less than 20 Hz.
• Rhinoceroses are well-known in the animal kingdom for their ability to communicate using infrasound. They use these low-frequency sounds to maintain contact with other individuals and coordinate social behavior, especially in environments where visibility is low.
• Research indicates that Rhinoceros vocalizations can reach frequencies as low as 5 Hz to 10 Hz, allowing the sound to travel over long distances and through dense vegetation. Hence, the correct answer is Rhinoceros.
• In contrast, animals like Dolphins, Bats, and Rats are primarily associated with ultrasound, which consists of frequencies higher than 20,000 Hz (the upper limit of human hearing).
• Bats and Dolphins use high-frequency ultrasonic pulses for echolocation, enabling them to navigate and hunt prey with extreme precision. Hence, the other options are incorrect in the context of infrasound.

Additional Information

• Infrasound Characteristics:
• Infrasonic waves have long wavelengths, which allows them to bypass obstacles and travel vast distances without significant loss of energy.
• Natural phenomena such as earthquakes, volcanic eruptions, and lightning also produce infrasound.
• Apart from Rhinoceroses, other large mammals like Elephants, Whales (specifically baleen whales), Giraffes, and Hippopotamuses are known to communicate using infrasonic rumbles.
• Ultrasound Applications:
• Ultrasound (frequencies above 20 kHz) is widely used by Microbats for hunting insects. They emit pulses and listen to the echoes to determine the size, location, and speed of their prey.
• Dolphins use ultrasound for underwater navigation and social bonding, with some clicks reaching frequencies up to 150 kHz.
• In human technology, ultrasound is utilized in medical imaging (sonography) and industrial non-destructive testing.
• Audible Range:
• The standard audible range for a healthy young human is 20 Hz to 20,000 Hz. As humans age, the sensitivity to higher frequencies typically declines.
• Many animals have evolved specialized hearing to occupy niches outside this range to avoid competition or to facilitate specialized survival strategies.

Important Points

• Elephants: They use infrasound to communicate between herds that are several kilometers apart. These "secret" conversations help them find water sources or mates.
• Pigeons: Surprisingly, Pigeons are capable of hearing very low-frequency infrasound (down to 0.5 Hz), which may help them in navigation by detecting the "acoustic signature" of distant mountains or oceans.
• Rats: While often associated with ultrasound for social chirping (at 22 kHz or 50 kHz), they do not use infrasound for primary communication.
• Scientific Studies: Studies conducted on Sumatran Rhinoceroses in captivity have confirmed their use of infrasonic signals, which share similarities with the vocalizations of Humpback Whales.

The correct answer is Stamen includes both anther and filament.

Key Points

• The stamen is the male reproductive organ of a flower. It consists of a long, slender stalk called the filament and a sac-like structure at the tip called the anther. Hence, statement 1 is correct.
• The ovary is the basal, swollen part of the pistil (or carpel), which is the female reproductive part. It is not located within the stamen. Hence, statement 2 is incorrect.
• The pistil (also known as the carpel) functions as the female reproductive part of the flower, consisting of the stigma, style, and ovary. Hence, statement 3 is incorrect.
• Pollen grains are produced within the anther (the male part) through a process called microsporogenesis. The ovule is the structure inside the ovary that contains the female gamete and develops into a seed after fertilization. Hence, statement 4 is incorrect.

Additional Information

• Structure of the Pistil:
• Stigma: The receptive sticky tip of the pistil that serves as a landing platform for pollen grains during pollination.
• Style: An elongated tube that connects the stigma to the ovary, through which the pollen tube grows.
• Ovary: The enlarged basal portion containing ovules. After fertilization, the ovary typically matures into a fruit.
• Structure of the Stamen:
• Anther: Usually bilobed in angiosperms, it contains pollen sacs where pollen grains are stored and released.
• Filament: The stalk that supports the anther, positioning it to effectively distribute pollen to pollinators like bees, butterflies, or the wind.
• Flower Classification:
• Bisexual Flowers: Flowers that contain both stamens and pistils (e.g., Hibiscus, Lily, and Rose).
• Unisexual Flowers: Flowers that contain either only stamens or only pistils (e.g., Papaya, Watermelon, and Corn).

Important Points

• Pollination: This is the transfer of pollen grains from the anther to the stigma. It can be self-pollination (within the same flower or plant) or cross-pollination (between different plants of the same species).
• Fertilization: Once pollen lands on the stigma, it germinates to form a pollen tube that carries male gametes to the ovule inside the ovary. The fusion of male and female gametes forms a zygote.
• Double Fertilization: A unique characteristic of angiosperms (flowering plants) where one male gamete fuses with the egg to form a zygote, and another fuses with polar nuclei to form the endosperm (nutritive tissue for the embryo).
• Non-Reproductive Parts: Petals (corolla) are often brightly colored to attract pollinators, while sepals (calyx) protect the flower in its bud stage.

The correct answer is Positive - negative ions.

Key Points

• A double displacement reaction, also known as a metathesis reaction, is a chemical process involving the exchange of ions between two compounds to form new compounds.
• In these reactions, the positive ions (cations) and negative ions (anions) of the reactants switch places. Specifically, the cation of the first reactant combines with the anion of the second reactant, while the cation of the second reactant pairs with the anion of the first.
• The general chemical equation for this reaction is AB + CD → AD + CB. In this formula, A and C represent positively charged ions, while B and D represent negatively charged ions.
• The reaction occurs primarily in aqueous solutions where the ions are free to move and interact. The driving force for such a reaction is typically the formation of a precipitate, a gas, or a stable molecular compound like water.
• Because the fundamental nature of the bond formation involves a cation attracting an anion to maintain electrical neutrality, the exchange is characterized as occurring between positive and negative ions.

Additional Information

• Precipitation Reactions: This is a common subtype where two soluble salts react to form an insoluble solid. For example, when Silver Nitrate (AgNO3) is mixed with Sodium Chloride (NaCl), the Silver (Ag+) ions and Chloride (Cl-) ions bond to form Silver Chloride (AgCl), which appears as a white precipitate.
• Neutralization Reactions: These occur between an acid and a base. The Hydrogen (H+) ion from the acid and the Hydroxyl (OH-) ion from the base exchange to form Water (H2O), while the remaining positive and negative ions form a salt. A classic example is HCl + NaOH → NaCl + H2O.
• Gas Formation: Some double displacement reactions result in the production of a gas. For instance, the reaction between Sodium Sulfide (Na2S) and Hydrochloric Acid (HCl) produces Hydrogen Sulfide (H2S) gas and Sodium Chloride.
• Applications: Double displacement is used extensively in industrial chemistry, qualitative analysis to detect specific ions, and in water treatment. Ion exchange resins used in water softeners operate on this principle to remove Calcium (Ca2+) and Magnesium (Mg2+) ions.

Important Points

• Oxidation State: Unlike Redox reactions, the oxidation state of the atoms typically does not change during a double displacement reaction. The atoms simply rearrange their ionic partnerships.
• Solubility Rules: To predict if a double displacement will occur, chemists use solubility rules. If both predicted products are soluble (stay as ions in the water), no net reaction has actually taken place; this is often called "no reaction."
• Contrast with Single Displacement: In a single displacement reaction, one element replaces another in a compound (e.g., Zn + CuSO4 → ZnSO4 + Cu). In double displacement, there is a mutual exchange between two compounds.
• Ionic Equations: These reactions are best represented by Net Ionic Equations, which highlight only the species that undergo a chemical change, excluding spectator ions that remain dissolved in the solution.

The correct answer is n^2B.

Key Points

• The magnetic field (B) at the center of a circular coil carrying a current (I) is directly proportional to the number of turns (N) and inversely proportional to the radius (R) of the coil.
• The mathematical formula for the magnetic field at the center of a circular loop is B = (mu_0 * N * I) / (2 * R), where mu_0 is the permeability of free space.
• In the initial case, for a single-turn coil (N = 1) with radius R, the field is B = (mu_0 * I) / (2 * R). Hence, this represents the baseline magnetic field strength.
• In the second case, the same wire is wound into a coil of n turns. This means the new number of turns N' = n.
• The new radius (R') is given as R/n. This is because the total length of the wire remains constant, so increasing the turns must reduce the radius proportionally. Hence, the change in radius is a direct consequence of the change in turns.
• The new magnetic field B' is calculated as: B' = (mu_0 * n * I) / (2 * (R/n)). Simplifying this expression gives B' = n^2 * [(mu_0 * I) / (2 * R)].
• Comparing the two equations, we find that B' = n^2 * B. Hence, the correct answer is n^2B.

Additional Information

• Magnetic Field Induction: This phenomenon is a part of electromagnetism, specifically explained by the Biot-Savart Law, which relates magnetic fields to the magnitude, direction, length, and proximity of the electric current.
• Dependence on Current: If the current (I) were doubled while keeping the turns and radius constant, the magnetic field would also double, showing a linear relationship.
• Permeability (mu_0): The constant mu_0 has a standard value of 4 * pi * 10^-7 T*m/A. It represents how easily magnetic flux can be established in a vacuum.
• Right-Hand Thumb Rule: This rule is used to determine the direction of the magnetic field. If the fingers of the right hand curl in the direction of the current in the loop, the thumb points in the direction of the magnetic field at the center.
• Applications: The principle of increasing turns to strengthen a magnetic field is widely used in transformers, electric motors, generators, and electromagnets found in industrial machinery.

Important Points

• Length of Wire Constraint: In such problems, it is crucial to remember that the length of the wire (L) is L = 2 * pi * R * N. If N is multiplied by n, R must be divided by n to keep L constant.
• Squared Relationship: The field strength at the center increases with the square of the number of turns (n^2) when the same length of wire is used because both the increase in turns and the decrease in radius contribute equally to the field enhancement.
• Units: The SI unit for the magnetic field is the Tesla (T), named after Nikola Tesla. Another commonly used unit is the Gauss (G), where 1 Tesla = 10,000 Gauss.
• Center vs. Axis: The formula used here is specifically for the center of the coil. The field at a distance x along the axis of the coil would follow a different formula involving (R^2 + x^2)^3/2 in the denominator.

The correct answer is living and loosely arranged with intercellular spaces.

Key Points

• Parenchyma is the most fundamental and versatile type of simple permanent tissue found in plants. It consists of living cells that possess a functional nucleus and cytoplasm.
• The cells of parenchyma are typically loosely arranged, which results in the presence of prominent intercellular spaces between individual cells. Hence, statement 3 is correct.
• The cell walls of parenchyma are thin and primarily composed of cellulose. These cells are usually isodiametric, meaning they are roughly equal in all dimensions, appearing spherical, oval, or polygonal.
• Parenchyma cells serve as the "filler" tissue in the soft parts of plants, such as the cortex of roots, the pith of stems, and the mesophyll of leaves.
• Unlike certain other tissues, parenchyma cells are not dead at maturity; they remain metabolically active to perform vital functions like secretion and regeneration. Hence, statement 1 and 2 are incorrect as they describe dead tissues.

Additional Information

• Classification of Simple Tissues:
• Collenchyma: These are living cells that provide mechanical support and flexibility. They are characterized by being thickened at the corners due to the deposition of cellulose and pectin, often leaving no intercellular spaces. Hence, statement 4 is incorrect.
• Sclerenchyma: This tissue consists of dead cells at maturity. The walls are heavily thickened with lignin, a chemical substance that acts as cement and hardens them. These cells are compactly arranged and provide structural rigidity to parts like the husk of a coconut.
• Specialized Forms of Parenchyma:
• Chlorenchyma: When parenchyma cells contain chlorophyll, they are known as chlorenchyma. These are responsible for performing photosynthesis in green leaves and stems.
• Aerenchyma: In aquatic plants, parenchyma cells develop large air cavities to provide buoyancy, helping the plants float on water.
• Functions of Parenchyma:
• Storage: One of the primary roles is the storage of food (starch), water, and waste products (tannins, resins).
• Photosynthesis: Through chlorenchyma, it helps in the synthesis of organic compounds.
• Wound Healing: Because these cells retain the ability to divide (meristematic potential) even at maturity, they play a critical role in healing plant injuries.

Important Points

• Comparison of Cell Walls: While Parenchyma has thin cellulosic walls, Collenchyma has unevenly thickened walls (at corners), and Sclerenchyma has uniformly thick lignified walls.
• Location: Parenchyma is found in almost all organs of the plant, including roots, stems, leaves, flowers, and fruits.
• Protoplasm: Since they are living cells, parenchyma cells contain a dense protoplasm and a large central vacuole.
• Shape: Though often isodiametric, they can also be elongated in specific areas like the palisade parenchyma of a leaf.

The correct answer is 2as = v2 + u2.

Key Points

• The equations of motion describe the basic concepts of a moving object, such as its velocity, displacement, and acceleration, relative to time. These are specifically valid for motion with uniform acceleration (constant acceleration).
• The equation v = u + at is known as the first equation of motion. It shows the relationship between final velocity (v), initial velocity (u), acceleration (a), and time (t). Hence, statement 2 is correct.
• The equation s = ut + 1/2at2 is the second equation of motion. It is used to calculate the displacement (s) of an object over a period of time when the initial velocity and acceleration are constant. Hence, statement 3 is correct.
• The equation v2 - u2 = 2as is the third equation of motion. It relates final velocity, initial velocity, acceleration, and displacement without involving the variable of time. Hence, statement 4 is correct.
• The expression 2as = v2 + u2 does not correctly represent a standard kinematic relationship. In the correct formula, the initial velocity squared (u2) must be subtracted from the final velocity squared (v2) to equal 2as. Hence, statement 1 is incorrect.
• These equations were famously summarized in Sir Isaac Newton's work, Philosophiæ Naturalis Principia Mathematica, published in 1687, which laid the foundation for classical mechanics.

Additional Information

• Definitions of Variables:
• Initial Velocity (u): The velocity of the object at time t = 0. It is measured in meters per second (m/s).
• Final Velocity (v): The velocity of the object after time t has elapsed, measured in m/s.
• Acceleration (a): The rate of change of velocity. For these equations to hold true, a must be constant. It is measured in meters per second squared (m/s2).
• Displacement (s): The shortest distance between the starting and ending points, a vector quantity measured in meters (m).
• Graphical Representation:
• In a Velocity-Time (v-t) graph, the slope represents the acceleration of the object.
• The area under the curve of a Velocity-Time graph represents the total displacement covered by the object.
• A straight line in a v-t graph indicates uniform acceleration, whereas a horizontal line indicates zero acceleration (constant velocity).
• Motion Under Gravity:
• A common application of these equations is free fall. In this case, acceleration (a) is replaced by acceleration due to gravity (g), which is approximately 9.8 m/s2 on Earth.
• When an object is thrown upward, the value of g is taken as negative because it acts in the opposite direction of motion.

Important Points

• Conditions for Validity: These equations are only applicable when the object moves in a straight line and the acceleration remains uniform. They cannot be used for circular motion or variable acceleration without calculus.
• Retardation/Deceleration: If the final velocity is less than the initial velocity, the acceleration is negative, often referred to as retardation or deceleration.
• Special Cases:
• If an object starts from rest, the initial velocity (u) is taken as 0.
• If an object comes to a stop (brakes applied), the final velocity (v) is taken as 0.
• Vector Nature: Since displacement, velocity, and acceleration are vectors, their direction (positive or negative) must be consistently maintained throughout the calculation.

The correct answer is Length, area of cross-section, material of the conductor, and temperature.

Key Points

• The resistance of a conductor is a property that opposes the flow of electric current through it. It is mathematically expressed by the formula R = ρ(L/A), where ρ is the resistivity, L is the length, and A is the area of cross-section.
• Length (L): The resistance of a conductor is directly proportional to its length. This means if the length of the wire is doubled, the resistance also doubles because electrons have to travel a longer distance and face more collisions. Hence, the statement regarding length is correct.
• Area of Cross-section (A): Resistance is inversely proportional to the area of cross-section of the conductor. A thicker wire (larger area) provides a wider path for electrons, resulting in lower resistance, whereas a thinner wire has higher resistance. Hence, the statement regarding the area of cross-section is correct.
• Material of the Conductor: Resistance depends on the nature of the material used. Different materials have different resistivities (ρ) due to their unique atomic structures. For example, Silver and Copper have very low resistivity and are excellent conductors, while Nichrome has high resistivity. Hence, the statement regarding the material is correct.
• Temperature: For metallic conductors, the resistance increases with an increase in temperature. As temperature rises, the metal ions vibrate more vigorously, increasing the frequency of collisions between flowing electrons and the ions. Hence, the statement regarding temperature is correct.
• While Ohm's Law states V = IR, the resistance itself does not depend on the voltage applied or the current flowing through it; these are external circuit variables. Resistance is an intrinsic physical property of the conductor. Hence, the correct answer is Length, area of cross-section, material of the conductor, and temperature.

Additional Information

• Resistivity (ρ): This is also known as specific resistance. It is the resistance offered by a conductor of unit length and unit area of cross-section. The SI unit of resistivity is the Ohm-meter (Ωm).
• Conductors vs. Insulators: Materials with very low resistivity are called conductors (like Aluminum and Gold), while those with very high resistivity are insulators (like Glass and Rubber).
• Temperature Coefficient: Metals generally have a positive temperature coefficient of resistance, meaning resistance increases with heat. Conversely, semiconductors like Silicon and Germanium have a negative temperature coefficient, where resistance decreases as temperature increases.
• Superconductivity: This is a phenomenon where certain materials exhibit zero electrical resistance when cooled below a specific critical temperature. This was first discovered by Heike Kamerlingh Onnes in 1911.

Important Points

• SI Unit: The SI unit of resistance is the Ohm, symbolized by the Greek letter Ω (Omega). It is named after the German physicist Georg Simon Ohm.
• Conductance: The reciprocal of resistance is known as conductance (G). Its SI unit is Siemens (S) or mho. A material with high resistance has low conductance.
• Ohmic and Non-Ohmic Conductors: Conductors that obey Ohm's Law (where V is proportional to I) are called Ohmic conductors (like most metals). Those that do not, such as diodes or transistors, are called non-Ohmic.
• Effect of Stretching: If a wire is stretched to double its length, its area of cross-section decreases proportionally (since volume remains constant), causing the resistance to increase by four times (n squared).

Shortcut Trick

Savings percentage = 100% − 75% = 25%.

Given: 25% units = ₹4,827.

Expenses = 75% units = 3 × 25% units.

Monthly Expenses = 3 × 4,827 = ₹14,481.

∴ The correct answer is ₹14,481.

Alternate Method

Given:

Spending = 75% of Monthly Income

Savings = ₹4,827

Formula Used:

Income = Expenditure + Savings

Calculations:

⇒ Let Total Income be x.

⇒ Expenditure = 75% of x = 0.75x

⇒ Savings = x − 0.75x = 0.25x

⇒ 0.25x = 4,827

⇒ x = 4,827 ÷ 0.25 = 4,827 × 4 = 19,308

⇒ Monthly Expenses = 75% of 19,308

⇒ Expenses = 0.75 × 19,308 = ₹14,481

∴ The correct answer is ₹14,481.

Additional Information

Percentage Change

Percentage increase or decrease is calculated as (Change ÷ Original Value) × 100.

Income-Expenditure Relation

If income increases by a% and expenditure increases by b%, the percentage change in savings depends on the initial ratio of expenditure to savings.

Base Change Rule

To find the total value when a part is known: Total = (Part Value ÷ Part Percentage) × 100.

Shortcut Trick

Simplify ratio: 4/8 : 11/16 : 18/21 → 1/2 : 11/16 : 6/7.

Multiply by LCM(2, 16, 7) = 112 → 56 : 77 : 96.

Difference between greatest (96) and smallest (56) = 40 units.

Given 40 units = 40, so 1 unit = 1. Largest number = 96 units = 96.

∴ The correct answer is 96.

Alternate Method

Given: Ratio of three numbers = 4/8 : 11/16 : 18/21

Step 1: Simplify fractions → 1/2 : 11/16 : 6/7

Step 2: To remove denominators, multiply by LCM of 2, 16, and 7, which is 112.

⇒ Number 1 = 1/2 × 112 = 56

⇒ Number 2 = 11/16 × 112 = 11 × 7 = 77

⇒ Number 3 = 6/7 × 112 = 6 × 16 = 96

Step 3: Identify greatest and smallest.

⇒ Greatest = 96x, Smallest = 56x

⇒ Difference = 96x − 56x = 40x

Step 4: Solve for x using given difference.

⇒ 40x = 40 ⇒ x = 1

⇒ Largest number = 96 × 1 = 96

∴ The correct answer is 96.

Additional Information

Ratio Simplification

To simplify a ratio involving fractions, multiply each term by the Least Common Multiple (LCM) of all the denominators to convert them into integers.

Proportional Parts

If a total value V is divided in ratio a:b:c, the largest part is calculated as [Max(a,b,c) / (a+b+c)] × V.

Difference Rule

When the difference between two parts is given, 1 unit value = (Actual Difference) ÷ (Difference in Ratio parts).

Shortcut Trick

For fourth proportional: P : 25 :: 8 : 4

Product of extremes = Product of means

P × 4 = 25 × 8

P = 25 × 2 = 50

∴ The correct answer is 50.

Alternate Method

Given:

First term = P, Second term = 25, Third term = 8, Fourth term = 4

Formula Used:

If a, b, c, d are in proportion, then a/b = c/d

Calculations:

⇒ P / 25 = 8 / 4

⇒ P / 25 = 2

⇒ P = 2 × 25

⇒ P = 50

∴ The correct answer is 50.

Additional Information

Fourth Proportional

If four quantities a, b, c, and d are in proportion, then d is called the fourth proportional. Formula: d = (b × c) / a.

Third Proportional

If a, b, and c are in continued proportion (a:b = b:c), then c is the third proportional. Formula: c = b² / a.

Mean Proportional

The mean proportional between two numbers a and b is √(a × b).

Shortcut Trick

The Mode is defined as the observation that occurs most frequently in the given data set.

By scanning the data, we observe the frequency of each number:

Value 48 occurs 5 times (highest frequency).

Value 42 occurs 3 times, while others occur 2 times or once.

∴ The correct answer is 48.

Alternate Method

Given Data: 41, 48, 54, 50, 54, 48, 42, 43, 48, 42, 53, 45, 44, 50, 49, 45, 43, 48, 42, 48

Formula Used: Mode = Value with the Maximum Frequency

To find the Mode, we construct a frequency distribution table:

⇒ Frequency of 41 = 1

⇒ Frequency of 42 = 3

⇒ Frequency of 43 = 2

⇒ Frequency of 44 = 1

⇒ Frequency of 45 = 2

⇒ Frequency of 48 = 5

⇒ Frequency of 49 = 1

⇒ Frequency of 50 = 2

⇒ Frequency of 53 = 1

⇒ Frequency of 54 = 2

⇒ Since 48 has the maximum frequency of 5.

∴ The correct answer is 48.

Additional Information

Empirical Relationship

For a moderately skewed distribution, the relationship between measures of central tendency is: Mode = 3 Median − 2 Mean.

Types of Mode

A data set can be Unimodal (one mode), Bimodal (two modes), or Multimodal (more than two modes). If all values occur with the same frequency, there is no mode.

Mean and Median

Mean is the sum of observations divided by their count. Median is the middle value when data is arranged in ascending or descending order.

Shortcut Trick

If x + 1/x = k, then the value of x² + 1/x² = k² − 2.

Here, the given value k = 17.

Required value = 17² − 2

Required value = 289 − 2 = 287.

∴ The correct answer is 287.

Alternate Method

Given: x + 1/x = 17

Formula Used: (a + b)² = a² + b² + 2ab

On squaring both sides of the given equation:

⇒ (x + 1/x)² = (17)²

⇒ x² + (1/x)² + 2 × x × (1/x) = 289

⇒ x² + 1/x² + 2 = 289

⇒ x² + 1/x² = 289 − 2

⇒ x² + 1/x² = 287

∴ The correct answer is 287.

Additional Information

Negative Sign Identity

If x − 1/x = k, then x² + 1/x² = k² + 2.

Cubic Algebraic Identity

If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Power of Four Identity

If x + 1/x = k, then x⁴ + 1/x⁴ = (k² − 2)² − 2.

Shortcut Trick

Assume the smaller number is x.

The other number is 2x + 5.

Sum: x + (2x + 5) = 32 → 3x = 27.

Solving gives x = 9.

∴ The correct answer is 9.

Alternate Method

Given: Sum of two numbers = 32. Relation: Larger = 2 × Smaller + 5.

Let the smaller number be x and the larger number be y.

⇒ x + y = 32 ... (i)

⇒ y = 2x + 5 ... (ii)

Substitute y from (ii) into (i):

⇒ x + (2x + 5) = 32

⇒ 3x + 5 = 32

⇒ 3x = 32 − 5

⇒ 3x = 27

⇒ x = 27 ÷ 3 = 9

∴ The correct answer is 9.

Additional Information

Linear Equations in One Variable

A linear equation involves a single variable with a power of 1, typically solved by isolating the variable on one side.

Substitution Method

This involves expressing one variable in terms of another and substituting it into a second equation to find the value.

Sum and Difference Problems

If the sum (S) and difference (D) are known, larger number = (S+D)/2 and smaller number = (S−D)/2.

Shortcut Trick

The difference in ages remains constant (2 years).

Ratio after 18 years = 21 : 22. Difference in ratio parts = 22 − 21 = 1 unit.

1 unit = 2 years → Gopal = 21 × 2 = 42 years, Manoj = 22 × 2 = 44 years (after 18 years).

Sum 10 years ago = (42 + 44) − 28 (for 18 years back to now + 10 years back) = 86 − 56 = 30 years.

∴ The correct answer is 30 years.

Alternate Method

Let the present ages of Gopal and Manoj be G and M respectively.

Given: M − G = 2 years (Since Manoj is older in the ratio 21:22).

After 18 years: (G + 18) / (M + 18) = 21 / 22

⇒ 22(G + 18) = 21(M + 18)

⇒ 22G + 396 = 21M + 378

⇒ 21M − 22G = 18

Substitute M = G + 2:

⇒ 21(G + 2) − 22G = 18 ⇒ 21G + 42 − 22G = 18

⇒ −G = 18 − 42 ⇒ G = 24 years, M = 26 years.

Sum of ages 10 years ago = (24 − 10) + (26 − 10) = 14 + 16 = 30 years.

∴ The correct answer is 30 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two individuals remains the same at any point in time (past, present, or future).

Ratio Balancing

If the age difference is 'd' and the ratio is a:b, then (b − a) units = d years. This helps in finding ages without complex algebra.

Shortcut Trick

Identify the Highest Common Factor (HCF) first to reduce the numbers.

HCF of 780, 2080, and 2600 is 260.

Divide numbers by 260: 780/260 = 3, 2080/260 = 8, 2600/260 = 10.

LCM of 3, 8, and 10 is 120.

Final LCM = 260 × 120 = 31200.

∴ The correct answer is 31200.

Alternate Method

Given: Numbers are 780, 2080, and 2600.

Formula Used: Prime Factorization Method for finding LCM.

Prime factorization of each number:

⇒ 780 = 2 × 2 × 3 × 5 × 13 = 2² × 3¹ × 5¹ × 13¹

⇒ 2080 = 2 × 2 × 2 × 2 × 2 × 5 × 13 = 2⁵ × 5¹ × 13¹

⇒ 2600 = 2 × 2 × 2 × 5 × 5 × 13 = 2³ × 5² × 13¹

⇒ LCM = Product of highest powers of all prime factors

⇒ LCM = 2⁵ × 3¹ × 5² × 13¹

⇒ LCM = 32 × 3 × 25 × 13

⇒ LCM = 96 × 325 = 31200

∴ The correct answer is 31200.

Additional Information

LCM (Least Common Multiple)

The smallest positive integer that is divisible by each of the given numbers. Found using division method or prime factorization.

HCF and LCM Relationship

For two numbers A and B: Product of A × B = HCF(A,B) × LCM(A,B).

LCM of Fractions

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

Total Distance = Sum of lengths = 248 + 142 = 390 m

Relative Speed = Sum of speeds (opposite direction) = 34 + 74 = 108 kmph

Convert kmph to m/s: 108 × (5/18) = 30 m/s

Time = Distance ÷ Relative Speed = 390 ÷ 30 = 13 sec

∴ The correct answer is 13 sec.

Alternate Method

Given:

Length of first train (L₁) = 248 m

Length of second train (L₂) = 142 m

Speed of first train (S₁) = 34 km/hr

Speed of second train (S₂) = 74 km/hr

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Time = (L₁ + L₂) ÷ Relative Speed

Calculations:

⇒ Total distance = 248 + 142 = 390 m

⇒ Relative speed = 34 + 74 = 108 km/hr

⇒ Speed in m/s = 108 × (5/18) = 6 × 5 = 30 m/s

⇒ Time taken = 390 ÷ 30

⇒ Time taken = 13 seconds

∴ The correct answer is 13 sec.

Additional Information

Relative Speed Concepts

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

Unit Conversion

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

Train Crossing a Point Object

If a train crosses a pole or a stationary man, the distance traveled is equal to the length of the train itself.

Shortcut Trick

Speed ratio (New : Normal) = 3 : 4

Since distance is constant, Time ratio (New : Normal) = 4 : 3

Difference in time ratio parts = 4 − 3 = 1 unit

Given, 1 unit = 19 minutes

Usual time = 3 units = 3 × 19 minutes = 57 minutes

∴ The correct answer is 57 min.

Alternate Method

Given:

New Speed = 3/4 × Normal Speed

Late time = 19 minutes

Formula Used:

Distance = Speed × Time

Calculations:

⇒ Let normal speed be S and usual time be T.

⇒ Normal Distance = S × T

⇒ New Speed = 3S/4

⇒ New Time = T + 19

⇒ Since distance is same: S × T = (3S/4) × (T + 19)

⇒ T = (3/4) × (T + 19)

⇒ 4T = 3T + 57

⇒ T = 57 minutes

∴ The correct answer is 57 min.

Additional Information

Inverse Proportion Rule

When distance is constant, speed is inversely proportional to time (S ∝ 1/T). If speed becomes a/b of its original, time becomes b/a of the original.

General Formula for Late/Early

Usual Time = (Late time × Numerator) ÷ (Difference between Denominator and Numerator) when speed is a/b of usual speed.

Average Speed

If a person covers equal distances at speeds x and y, then Average Speed = 2xy/(x + y).

Shortcut Trick

ഒരു സംഖ്യ 2 കൊണ്ടും 5 കൊണ്ടും കൃത്യമായി ഹരിക്കണമെങ്കിൽ, അത് 10 കൊണ്ട് ഹരിക്കാൻ കഴിയുന്നതായിരിക്കണം (2 × 5 = 10).

10-ന്റെ ഗുണിതങ്ങളുടെ അവസാന അക്കം എപ്പോഴും 0 ആയിരിക്കും.

നൽകിയിരിക്കുന്ന ഓപ്ഷനുകളിൽ 130 എന്ന സംഖ്യയുടെ അവസാന അക്കം 0 ആണ്.

∴ The correct answer is 130.

Alternate Method

നൽകിയിരിക്കുന്നവ: സംഖ്യകൾ 134, 130, 135, 133.

ഉപയോഗിച്ചിരിക്കുന്ന നിയമങ്ങൾ:

1. 2-ന്റെ ഭാജ്യതാനിയമം: അവസാന അക്കം ഇരട്ടസംഖ്യയോ 0-ഓ ആയിരിക്കണം (0, 2, 4, 6, 8).
2. 5-ന്റെ ഭാജ്യതാനിയമം: അവസാന അക്കം 0 അല്ലെങ്കിൽ 5 ആയിരിക്കണം.

⇒ ഓപ്ഷൻ (1) 134: 2 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 4), എന്നാൽ 5 കൊണ്ട് ഹരിക്കാൻ കഴിയില്ല.

⇒ ഓപ്ഷൻ (2) 130: 2 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 0), 5 കൊണ്ടും ഹരിക്കാം (അവസാന അക്കം 0).

⇒ ഓപ്ഷൻ (3) 135: 5 കൊണ്ട് ഹരിക്കാം (അവസാന അക്കം 5), എന്നാൽ 2 കൊണ്ട് ഹരിക്കാൻ കഴിയില്ല (ഒറ്റസംഖ്യ).

⇒ ഓപ്ഷൻ (4) 133: 2 കൊണ്ടോ 5 കൊണ്ടോ ഹരിക്കാൻ കഴിയില്ല.

∴ The correct answer is 130.

Additional Information

10-ന്റെ ഭാജ്യതാനിയമം

ഒരു സംഖ്യയുടെ അവസാന അക്കം 0 ആണെങ്കിൽ അത് 10 കൊണ്ട് കൃത്യമായി ഹരിക്കാം. ഇത് 2, 5 എന്നീ സംഖ്യകളുടെ സംയുക്ത നിയമമാണ്.

3-ന്റെ ഭാജ്യതാനിയമം

ഒരു സംഖ്യയിലെ അക്കങ്ങളുടെ തുക 3-ന്റെ ഗുണിതമാണെങ്കിൽ ആ സംഖ്യയെ 3 കൊണ്ട് ഹരിക്കാം.

ഇരട്ട സംഖ്യകളും ഒറ്റ സംഖ്യകളും

2 കൊണ്ട് ഹരിക്കാൻ കഴിയുന്ന സംഖ്യകളെ ഇരട്ട സംഖ്യകൾ എന്നും അല്ലാത്തവയെ ഒറ്റ സംഖ്യകൾ എന്നും വിളിക്കുന്നു.

Shortcut Trick

Ratio of Income (R : S) = 5 : 9 (Difference = 4 units)

Ratio of Expenditure (R : S) = 1 : 3 (Difference = 2 units)

To equate savings, make differences equal: Multiply Expenditure ratio by 2 → 2 : 6

Difference in units (Income − Expenditure) = 5 − 2 = 3 units = ₹39,000

1 unit = ₹13,000 ∴ Rakesh income (5 units) = 5 × 13,000 = ₹65,000

∴ The correct answer is ₹65,000.

Alternate Method

Given: Ratio of incomes = 5 : 9 | Ratio of expenditures = 1 : 3 | Savings = ₹39,000 each.

Formula Used: Income − Savings = Expenditure

⇒ Let the incomes of Rakesh and Shivam be 5x and 9x respectively.

⇒ According to the question: (5x − 39000) / (9x − 39000) = 1 / 3

⇒ Cross-multiplying: 3(5x − 39000) = 1(9x − 39000)

⇒ 15x − 1,17,000 = 9x − 39,000

⇒ 15x − 9x = 1,17,000 − 39,000

⇒ 6x = 78,000

⇒ x = 13,000

⇒ Rakesh monthly income = 5x = 5 × 13,000 = ₹65,000

∴ The correct answer is ₹65,000.

Additional Information

Income, Expenditure, and Savings Relationship

The basic accounting identity used in such problems is Income = Expenditure + Savings.

Cross-Multiplication Method for Ratios

When the same amount is added/subtracted from both terms of a ratio, the unit difference method is the fastest way to find the initial values.

Ratio Balancing

To balance two ratios with a constant difference, multiply each ratio by the difference of the other ratio.

Shortcut Trick

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

CP : MP = (100 − 12) : (100 + 10) = 88 : 110.

Simplify the ratio: 88 : 110 = 4 : 5.

Given 4 units (CP) = ₹ 7660, then 1 unit = 7660 ÷ 4 = 1915.

Marked Price (5 units) = 5 × 1915 = ₹ 9575.

∴ The correct answer is ₹ 9575.

Alternate Method

Given: Cost Price (CP) = ₹ 7660, Profit = 10%, Discount = 12%.

Formulas: SP = CP × (100 + P%) ÷ 100 | MP = (SP × 100) ÷ (100 − D%).

⇒ Step 1: Find Selling Price (SP) = 7660 × (110 ÷ 100) = 766 × 11 = ₹ 8426.

⇒ Step 2: Since SP is obtained after a 12% discount on MP, SP = 88% of MP.

⇒ Step 3: Marked Price (MP) = (8426 ÷ 88) × 100.

⇒ Step 4: MP = 95.75 × 100 = ₹ 9575.

∴ The correct answer is ₹ 9575.

Additional Information

Marked Price & Discount

Marked Price is the price printed on the label. Discount is always calculated on the Marked Price unless stated otherwise.

Relation between CP and MP

The direct formula linking CP, MP, Profit, and Discount is: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Shortcut Trick

Apply the BODMAS rule directly: compute 'Of', followed by Division, Multiplication, Addition, and Subtraction.

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − 12 of 4 + 9 × 7

⇒ 21 + 9 × 5 − 48 + 63

⇒ 21 + 45 − 48 + 63

⇒ 129 − 48 = 81

∴ The correct answer is 81.

Alternate Method

Given:

189 ÷ 9 + 117 ÷ 13 × 5 − 12 of 4 + 9 × 7

Concept Used:

BODMAS Rule: First solve Brackets, then Of, Division, Multiplication, Addition, and lastly Subtraction.

Calculation:

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − (12 × 4) + 9 × 7

⇒ 189 ÷ 9 + 117 ÷ 13 × 5 − 48 + 9 × 7

⇒ 21 + 9 × 5 − 48 + 9 × 7

⇒ 21 + 45 − 48 + 63

⇒ 66 − 48 + 63

⇒ 18 + 63

⇒ 81

∴ The correct answer is 81.

Additional Information

BODMAS Rule

The fundamental rule that dictates the order of mathematical operations: Brackets first, then Of (multiplication), Division, Multiplication, Addition, and Subtraction.

VBODMAS Extension

'V' stands for Vinculum (or bar). If an expression contains a horizontal bar over some numbers, that specific part must be evaluated first before applying regular BODMAS.

Unit Digit Method

A quick cross-checking tool where you compute only the last digits of each term. In this expression: 1 + 5 − 8 + 3 = 1, matching the unit digit of the correct answer 81.

Shortcut Trick The Empirical relationship between Mean, Median, and Mode is given by: Median = (Mode + 2 × Mean) ÷ 3 Substituting the values: Median = (14.1 + 2 × 53.1) ÷ 3 Median = (14.1 + 106.2) ÷ 3 = 120.3 ÷ 3 = 40.1 ∴ The correct answer is 40.1.

Alternate Method Given: ⇒ Mean (x̄) = 53.1 ⇒ Mode (Z) = 14.1 Formula Used: ⇒ Mode = 3 × Median − 2 × Mean

Calculations: ⇒ 14.1 = 3 × Median − 2 × 53.1 ⇒ 14.1 = 3 × Median − 106.2 ⇒ 3 × Median = 14.1 + 106.2 ⇒ 3 × Median = 120.3 ⇒ Median = 120.3 ÷ 3 ⇒ Median = 40.1 ∴ The correct answer is 40.1.

Additional Information Empirical Formula Applicability This relationship, Mode = 3 Median − 2 Mean, is valid for moderately skewed distributions. It is also known as Karl Pearson's formula. Symmetrical Distribution In a perfectly symmetrical distribution (like the Normal Distribution), the Mean, Median, and Mode are all equal: Mean = Median = Mode. Positively Skewed Distribution When the tail on the right side is longer, the distribution is positively skewed. In this case: Mean > Median > Mode. Negatively Skewed Distribution When the tail on the left side is longer, the distribution is negatively skewed. In this case: Mean < Median < Mode.

Shortcut Trick

Total Cost Price (CP) of 9 articles = ₹16.

Total Selling Price (SP) of 9 articles = 9 × 2.8 = ₹25.2.

Profit = SP − CP = 25.2 − 16 = ₹9.2.

Profit% = (Profit ÷ CP) × 100 = (9.2 ÷ 16) × 100.

Profit% = 0.575 × 100 = 57.5%.

∴ The correct answer is 57.5%.

Alternate Method

Given: Total CP of 9 articles = ₹16, SP of 1 article = ₹2.8.

Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ Total Cost Price (CP) = ₹16.

⇒ Total Selling Price (SP) = 9 × 2.8 = ₹25.2.

⇒ Total Profit = 25.2 − 16 = ₹9.2.

⇒ Profit Percentage = (9.2 ÷ 16) × 100 = 57.5%.

∴ The correct answer is 57.5%.

Additional Information

Profit and Loss Basics

Profit occurs when the Selling Price (SP) is greater than the Cost Price (CP). Formula: Profit = SP − CP.

Profit Percentage Calculation

Profit percentage is always calculated on the Cost Price (CP) unless stated otherwise. Formula: Profit% = (Profit ÷ CP) × 100.

Shortcut Trick

In a parallelogram, the area remains constant: Base₁ × Height₁ = Base₂ × Height₂

Let longer side be b₁ = 11.9 cm and shorter side be b₂ = a cm.

⇒ 11.9 × 3.9 = a × 5.1

⇒ a = (11.9 × 3.9) ÷ 5.1 = 9.1 cm

Perimeter = 2 × (Sum of adjacent sides) = 2 × (11.9 + 9.1) = 42 cm

∴ The correct answer is 42 cm.

Alternate Method

Given:

Longer side (L) = 11.9 cm

Shorter side (S) = a cm

Distance between longer sides (h_L) = 3.9 cm

Distance between shorter sides (h_S) = 5.1 cm

Formula Used:

Area of Parallelogram = Base × Perpendicular Height

Perimeter of Parallelogram = 2 × (L + S)

Calculations:

⇒ Area = L × h_L = S × h_S

⇒ 11.9 × 3.9 = a × 5.1

⇒ a = (11.9 × 3.9) / 5.1

⇒ a = (11.9 × 39) / 51

⇒ a = (11.9 × 13) / 17

⇒ a = 0.7 × 13

⇒ a = 9.1 cm

Now, finding the Perimeter:

⇒ Perimeter = 2 × (11.9 + 9.1)

⇒ Perimeter = 2 × 21.0

⇒ Perimeter = 42 cm

∴ The correct answer is 42 cm.

Additional Information

Properties of a Parallelogram

Opposite sides are equal and parallel. Opposite angles are equal. Adjacent angles are supplementary (sum = 180°).

Area and Height Relationship

Area = b₁h₁ = b₂h₂. This implies that the ratio of sides is inversely proportional to the ratio of their corresponding heights: b₁/b₂ = h₂/h₁.

Diagonals of a Parallelogram

The diagonals bisect each other but are not necessarily equal in length or perpendicular. The sum of the squares of the diagonals equals the sum of the squares of the four sides: d₁² + d₂² = 2(a² + b²).

Shortcut Trick

If a tap fills a cistern in x hours and a leak empties it in y hours, the time taken to fill it is (xy) ÷ (y − x).

Given x = 7 hours and y = 8 hours.

Time = (7 × 8) ÷ (8 − 7) = 56 ÷ 1 = 56 hours.

∴ The correct answer is 56 hr.

Alternate Method

Given:

Filling Tap time = 7 hours

Emptying Hole time = 8 hours

Calculations:

⇒ Total Work (Capacity of Cistern) = LCM of (7, 8) = 56 units

⇒ Efficiency of Filling Tap = 56 ÷ 7 = +8 units/hr

⇒ Efficiency of Emptying Hole = 56 ÷ 8 = −7 units/hr

⇒ Combined Efficiency (Net flow) = 8 − 7 = +1 unit/hr

⇒ Time taken to fill = Total Work ÷ Combined Efficiency

⇒ Time = 56 ÷ 1 = 56 hours

∴ The correct answer is 56 hr.

Additional Information

Net Efficiency Rule

When pipes/taps work together, their efficiencies are added (positive for filling, negative for emptying) to find the total work per unit time.

Pipes and Cisterns Concept

This topic is an application of Time and Work where filling pipes represent positive work and leaks/outlets represent negative work.

Shortcut Trick

In right-angled ΔBDC, the angle at C is 90° − 71° = 19°.

For ΔAEC, ∠AEB is the exterior angle at vertex E.

By the exterior angle property, ∠AEB = ∠CAE + ∠ACE.

∠AEB = 55° + 19° = 74°.

∴ The correct answer is 74°.

Alternate Method

Given: BD ⊥ AC at D, ∠DBC = 71°, and ∠CAE = 55°.

Formula Used: Angle Sum Property of a triangle and the Exterior Angle Theorem.

⇒ In ΔBDC, ∠BDC = 90° (since BD ⊥ AC).

⇒ ∠BCD = 180° − (∠BDC + ∠DBC)

⇒ ∠BCD = 180° − (90° + 71°) = 180° − 161° = 19°.

⇒ In ΔAEC, ∠AEB is the exterior angle at vertex E because B, E, C are collinear.

⇒ ∠AEB = ∠CAE + ∠ACE

⇒ ∠AEB = 55° + 19° = 74°.

∴ The correct answer is 74°.

Additional Information

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Angle Sum Property

The sum of all interior angles of any triangle is always 180°.

Complementary Angles

In a right-angled triangle, the two acute angles are complementary, meaning their sum is exactly 90°.

Shortcut Trick

Relation between Volume (V) and Curved Surface Area (CSA): V = (CSA × r) ÷ 2

First find Radius (r): 2 × (22/7) × r × 6 = 264 ⇒ r = 7 cm

Direct Calculation: Volume = (264 × 7) ÷ 2 = 132 × 7 = 924 cm³

∴ The correct answer is 924 cm³.

Alternate Method

Given:

Height (h) of the cylinder = 6 cm

Curved Surface Area (CSA) = 264 cm²

Formula Used:

Curved Surface Area (CSA) = 2πrh

Volume (V) = πr²h

Calculations:

⇒ 2 × (22/7) × r × 6 = 264

⇒ (264/7) × r = 264

⇒ r = 7 cm

Now, substitute the value of Radius (r) in the Volume formula:

⇒ Volume = (22/7) × 7 × 7 × 6

⇒ Volume = 22 × 7 × 6

⇒ Volume = 154 × 6

⇒ Volume = 924 cm³

∴ The correct answer is 924 cm³.

Additional Information

Total Surface Area (TSA) of Cylinder

The total surface area includes the curved surface and the area of two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Hollow Cylinder

If the cylinder has an outer radius R and inner radius r, the volume is V = π(R² − r²)h.

Relation between Cone and Cylinder

For the same radius and height, the volume of a cylinder is exactly three times the volume of a cone (V_cylinder = 3 × V_cone).

Shortcut Trick

Net Profit/Loss% = a + b + (a × b) ÷ 100, where a is Mark-up% and b is Discount% (negative).

Given: a = +37 and b = −16.

Net change = 37 − 16 + (37 × (−16)) ÷ 100.

Net change = 21 − (592 ÷ 100) = 21 − 5.92 = 15.08%.

Positive result indicates a gain.

∴ The correct answer is 15.08% gain.

Alternate Method

Given → Mark-up = 37% above CP, Discount = 16% on MP.

Formulas → MP = CP × (100 + Mark-up%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Gain% = (SP − CP) ÷ CP × 100.

⇒ Let Cost Price (CP) = 100.

⇒ Marked Price (MP) = 100 + 37% of 100 = 137.

⇒ Selling Price (SP) = 137 − 16% of 137.

⇒ SP = 137 − (16 × 137 ÷ 100) = 137 − 21.92 = 115.08.

⇒ Gain = SP − CP = 115.08 − 100 = 15.08.

⇒ Gain% = (15.08 ÷ 100) × 100 = 15.08%.

∴ The correct answer is 15.08% gain.

Additional Information

Relation between CP and MP

The ratio of Marked Price to Cost Price is given by CP ÷ MP = (100 − D%) ÷ (100 + P%).

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net effect is (x − y − xy ÷ 100)%.

Shortcut Trick

Calculate the total number of days: Feb (25) + March (31) + April (17) = 73 days.

Convert days to years: T = 73 ÷ 365 = 1/5 year.

Apply direct SI formula: SI = (2000 × 7 × 1) ÷ (100 × 5).

SI = 20 × 7 ÷ 5 = 4 × 7 = 28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate of Interest (R) = 7% per annum

Time period = 3 February 2023 to 17 April 2023

Formula Used:

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

Calculations:

⇒ Days in February (Non-leap year) = 28 − 3 = 25 days

⇒ Days in March = 31 days

⇒ Days in April = 17 days

⇒ Total days (T) = 25 + 31 + 17 = 73 days

⇒ Time in years = 73 ÷ 365 = 1/5 year

⇒ SI = (2000 × 7 × (1/5)) ÷ 100

⇒ SI = (2000 × 7) ÷ 500

⇒ SI = 14000 ÷ 500 = 28

∴ The correct answer is ₹28.

Additional Information

Time Calculation Rule

In interest calculation, the day on which money is deposited is generally excluded, and the day on which money is withdrawn is included.

Standard Day Fractions

73 days is a very common period in exams as it equals exactly 1/5 of a 365-day year. Similarly, 146 days = 2/5 year, and 219 days = 3/5 year.

Leap Year Check

Always check if the year is a leap year (divisible by 4). If it is, February has 29 days and the total year has 366 days.

Shortcut Trick

Net rate of filling = 1/6 − 1/15 = (5 − 2)/30 = 3/30 = 1/10 per minute.

Time to fill full tank = 10 minutes.

Time to fill one-half of the tank = 10 ÷ 2 = 5 minutes.

∴ The correct answer is 5 minutes.

Alternate Method

Given: Filling time = 6 min, Emptying time = 15 min.

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

⇒ Net Efficiency (A + B) = 5 − 2 = 3 units/min.

⇒ Target Work = 1/2 × Total Capacity = 1/2 × 30 = 15 units.

⇒ Time required = Target Work ÷ Net Efficiency = 15 ÷ 3 = 5 minutes.

∴ The correct answer is 5 minutes.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency) while outlet pipes remove water (negative efficiency) from the tank.

Net Rate Formula

If an inlet fills in 'x' and an outlet empties in 'y', net work done in 1 hour = (1/x − 1/y).

Pipes and Cisterns vs Time and Work

The concept is identical to Time and Work, where emptying a tank is equivalent to doing negative work.

Shortcut Trick

Let Tower Height be h = 50 m.

Using standard 30°-60°-90° triangle ratios: Distance to car = h × √3 = 50√3 m.

Relative height of balloon from tower top = Distance × tan(60°) = 50√3 × √3 = 150 m.

Total height of balloon = Height of tower + Relative height = 50 + 150 = 200 m.

∴ The correct answer is 200 metres.

Alternate Method

Given: Height of Tower (AB) = 50 m, Angle of depression = 30°, Angle of elevation = 60°.

Formula Used: tan(θ) = Perpendicular / Base

⇒ Let distance from tower to car be x.

⇒ In ΔABC: tan(30°) = 50 / x

⇒ 1/√3 = 50 / x ⇒ x = 50√3 m.

⇒ Let height of balloon above the tower top level be H.

⇒ In the upper triangle: tan(60°) = H / x

⇒ √3 = H / 50√3

⇒ H = 50√3 × √3 = 150 m.

⇒ Total Height of Balloon from ground = Tower Height + H = 50 + 150 = 200 m.

∴ The correct answer is 200 metres.

Additional Information

Angle of Depression and Elevation

The Angle of Depression from an observer to an object below is equal to the Angle of Elevation from the object to the observer due to alternate interior angles.

Trigonometric Ratios for Standard Angles

For heights and distances, remember: tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) = √3.

Shortcut Trick

Let original price = 100

Net multiplier = (1 − 0.45) × (1 − 0.55) × (1 − 0.50) × (1 + 0.40)

Net multiplier = 0.55 × 0.45 × 0.50 × 1.4 = 0.17325

Net % change = (0.17325 − 1) × 100 = −82.675%

∴ The correct answer is Decreases by 82.68%.

Alternate Method

Given: Successive decreases of 45%, 55%, 50% and an increase of 40%.

Step-by-step Calculation:

⇒ Let the initial price of the fuel be 100 units.

⇒ Price after 1^st month decrease of 45% = 100 × (1 − 0.45) = 55

⇒ Price after 2^nd month decrease of 55% = 55 × (1 − 0.55) = 55 × 0.45 = 24.75

⇒ Price after 3^rd month decrease of 50% = 24.75 × (1 − 0.50) = 12.375

⇒ Price after 4^th month increase of 40% = 12.375 × (1 + 0.40) = 12.375 × 1.4 = 17.325

⇒ Net Change = Final Price − Original Price = 17.325 − 100 = −82.675

⇒ Net Percentage Decrease = (82.675 / 100) × 100 = 82.675% ≈ 82.68%

∴ The correct answer is Decreases by 82.68%.

Additional Information

Successive Percentage Formula

For two successive changes of a% and b%, the net change is (a + b + ab/100)%. Negative values are used for decrease.

Multiplying Factor Method

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ... This is the most efficient method for multiple successive changes.

Base Change Property

Successive percentage changes are cumulative, meaning each subsequent percentage is calculated on the updated value, not the original base.

Given:

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

We need to find the even digits which are:

Immediately preceded by an even digit and immediately followed by an even digit.

Required pattern:

Even digit → Even digit → Even digit

Checking the series:

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

The even digits satisfying the condition are:

• 6 in (4 6 2)

• 4 in (8 4 2)

Therefore, there are 2 such even digits.

Hence, the correct answer is "Option 1".

Given:

FTBJ → OCKS

PDLT → YMUC

Observe the pattern:

For, FTBJ → OCKS

For, PDLT → YMUC

So, each letter is moved 9 positions forward in cyclic alphabetical order.

Now apply the same logic to QEMU:

Therefore, QEMU → ZNVD

Hence, the correct answer is "Option 2".

Given:

‘A + B’ means ‘A is the mother of B’

‘A − B’ means ‘A is the sister of B’

‘A × B’ means ‘A is the son of B’

‘A ÷ B’ means ‘A is the brother of B’

Expression:

D + E − F ÷ G × K

Decode the expression from left to right:

D + E → D is the mother of E.

E − F → E is the sister of F.

F ÷ G → F is the brother of G.

G × K → G is the son of K.

Form the relationship chain:

Therefore, D is the wife of K.

Hence, the correct answer is "Option 1".

Given series:

5, 9, 16, 26, 39, ?

Hence, the correct answer is "Option 1".

The given equation is:

70 B 4 D (50 A 5) B 11 C (24 A 2) B 4 C 10 = ? →

After replacing the letters with their respective arithmetic symbols:

70 × 4 − (50 ÷ 5) × 11 + (24 ÷ 2) × 4 + 10

Step-by-step solution using BODMAS:

⇒ 70 × 4 − (50 ÷ 5) × 11 + (24 ÷ 2) × 4 + 10

⇒ 280 - 10 × 11 + 12 × 4 + 10

⇒ 280 - 110 + 48 + 10

⇒ 338 - 110

⇒ 228

Therefore, the value that will come in place of the question mark is 228.

Hence, the correct answeris "Option 2".

Logic followed here is: Identify the highest and lowest numbers from the given sequence, then add the second digit of the highest number to the second digit of the lowest number.

For Highest Number

791 is the highest number among (783, 791, 531, 563, 594) →

Second digit of the highest number = 9

For Lowest Number

531 is the lowest number among (783, 791, 531, 563, 594) →

Second digit of the lowest number = 3

The pattern observed is: (Second digit of highest number) + (Second digit of lowest number).

Resultant:

Resultant = 9 + 3 = 12

Hence, the correct answer is "Option 4".

Given Series:

DRT UIK LZB CQS ?

The pattern followed here is:

Hence, the correct answer is "Option 1".

Logic followed here is:

In each triad, the first letters of the pairs decrease by 2, while the second letters follow a pattern of decreasing by 2 and then increasing by 9.

For

Triad 1: ME-KC-IL follows the pattern →

Triad 2: PH-NF-LO follows the pattern →

The pattern observed is: (1st letters: -2, -2 sequence) and (2nd letters: -2, +9 sequence).

Similarly, let's find the pattern for WO-UM-SV:

Therefore, the triad that follows the same pattern is 'WO-UM-SV'.

Hence, the correct answer is "Option 2".

Given:

Six people A, B, C, D, E and F are sitting in a row facing north.

A sits third to the left of E.

E sits second to the left of F.

D sits to the immediate right of E.

B sits to the immediate left of C.

People between C and F are:

E and D

Therefore, 2 people sit between C and F.

Hence, the correct answer is "Option 3".

Given letter-clusters:

XZW

MNL

RSQ

IJH

Observe the pattern:

In three clusters, the letters follow the pattern:

1st letter → +1 → 2nd letter → −1 → 3rd letter

Check each option:

Option 2: MNL

Option 3: RSQ

Option 4: IJH

Option 1: XZW

This does not follow the same pattern.

Therefore, the odd one out is: XZW

Hence, the correct answer is "Option 1".

Logic followed here is substituting the given letters with their corresponding mathematical operators and then solving the equation using the BODMAS rule.

The mapping of letters to symbols is as follows:

Given equation:

(80 A 8) B 5 C 14 B 2 D 11 B 3 C 72 A 4 = ?

Step 1: Replace the letters with their respective symbols:

(80 ÷ 8) × 5 + 14 × 2 - 11 × 3 + 72 ÷ 4 = ?