RRB Group D Official Paper (Held On: 16 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is becomes four times.

Key Points

• Kinetic Energy (K.E.) is the energy possessed by an object by virtue of its motion. Every moving object, whether it is a falling grain of sand or a speeding planet, possesses some amount of kinetic energy.
• The standard mathematical formula for kinetic energy is K.E. = 1/2 mv², where m represents the mass of the object and v represents its velocity or speed.
• In this formula, the velocity term is squared. This implies that the kinetic energy of an object is directly proportional to the square of its speed.
• When the speed of an object is doubled (v becomes 2v), the new kinetic energy calculation becomes: K.E_new = 1/2 m(2v)² = 1/2 m(4v²) = 4 * (1/2 mv²).
• As a result, doubling the speed results in the kinetic energy increasing by a factor of four. Hence, the statement that it becomes four times is correct.

Additional Information

• Work-Energy Theorem: This fundamental principle of physics states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle. If work is done on an object, its kinetic energy increases.
• Scalar Nature: Kinetic energy is a scalar quantity, meaning it has only magnitude and no specific direction. Unlike velocity or momentum, kinetic energy does not change if only the direction of motion changes while the speed remains constant.
• Dependency on Mass: While kinetic energy depends quadratically on speed, it is linearly proportional to the mass. If the mass of an object is doubled while the speed remains the same, the kinetic energy will only double.
• Relationship with Momentum: Kinetic energy and linear momentum (p) are closely related. The relationship is expressed as K.E. = p² / 2m. This shows that if the momentum of an object is doubled, its kinetic energy also becomes four times the original value.

Important Points

• Units and Dimensions: The SI unit of kinetic energy is the Joule (J), which is defined as 1 kg·m²/s². Its dimensional formula is [ML²T⁻²].
• Potential Energy Comparison: Unlike kinetic energy, which is energy of motion, Potential Energy is the energy stored in an object due to its position (like a height in a gravitational field) or configuration (like a stretched spring).
• Braking Distance and Safety: The fact that kinetic energy increases with the square of the speed is a critical factor in road safety. Because a car traveling at 60 km/h has four times the energy of a car at 30 km/h, it requires much more work (and distance) from the brakes to bring it to a complete stop.
• Relativistic Limit: At speeds approaching the speed of light (c), the classical formula 1/2 mv² becomes inaccurate. In such cases, Einstein's theory of relativity provides the corrected formula for relativistic kinetic energy.
• Zero Kinetic Energy: An object at rest (velocity = 0) has zero kinetic energy, regardless of how much mass it possesses.

The correct answer is acetic acid (CH3COOH) and sodium hydroxide (NaOH).

Key Points

• An acid-base pair consists of one substance that acts as an acid (donates protons) and another that acts as a base (accepts protons or provides hydroxyl ions).
• Sulphuric acid (H2SO4) and Nitric acid (HNO3) are both strong mineral acids. Since both substances in this pair are acids, they do not form an acid-base category. Hence, statement 1 is incorrect.
• Acetic acid (CH3COOH) is a weak organic acid, while Sodium hydroxide (NaOH) is a strong base. Together, they represent a valid acid-base pair. Hence, statement 2 is correct.
• Magnesium hydroxide [Mg(OH)2] and Ammonium hydroxide (NH4OH) are both classified as bases. Because both chemicals possess alkaline properties, they do not constitute an acid-base pair. Hence, statement 3 is incorrect.
• Calcium hydroxide [Ca(OH)2] and Potassium hydroxide (KOH) are both strong bases (metal hydroxides). This grouping consists of two bases rather than an acid and a base. Hence, statement 4 is incorrect.
• In chemistry, identifying these pairs is fundamental for understanding neutralization reactions and buffer solutions.

Additional Information

• Acids: These are substances that release Hydrogen ions (H+) when dissolved in water. They are known for their sour taste and their ability to turn blue litmus paper red. Common examples include Hydrochloric acid (HCl) and Citric acid.
• Bases: These substances release Hydroxyl ions (OH-) in aqueous solutions. They typically have a bitter taste, a soapy/slippery feel, and turn red litmus paper blue. Examples include NaOH and Calcium carbonate.
• pH Scale: The acidity or alkalinity of a solution is measured on the pH scale, which ranges from 0 to 14. A pH of 7 is neutral (like pure water), below 7 is acidic, and above 7 is basic or alkaline.
• Neutralization: This is a chemical reaction where an acid and a base react with each other to produce salt and water. For example, CH3COOH reacting with NaOH produces Sodium Acetate (CH3COONa) and H2O.
• Strength of Acids/Bases: Strong acids (like H2SO4) ionize completely in water, whereas weak acids (like CH3COOH) ionize only partially. Similarly, NaOH is a strong base compared to the weak base NH4OH.

Important Points

• Acetic Acid (CH3COOH): Also known as ethanoic acid, it is the primary component of vinegar, usually making up about 4% to 8% of the volume. It is widely used in the food industry as a preservative.
• Sodium Hydroxide (NaOH): Commonly referred to as caustic soda, it is a highly corrosive base used in the manufacturing of soaps, paper, and detergents.
• Arrhenius Theory: This theory, proposed in 1884, defines acids as substances that increase H+ concentration and bases as substances that increase OH- concentration in water.
• Indicators: Chemical substances like Phenolphthalein and Methyl Orange are used to detect the presence of acids and bases. Phenolphthalein turns pink in a basic medium and remains colorless in an acidic medium.
• Magnesium hydroxide [Mg(OH)2]: Known as Milk of Magnesia, it is used medicinally as an antacid to treat indigestion by neutralizing excess stomach acid.

The correct answer is Sexual reproduction involves the fusion of male and female gametes (sex cells).

Key Points

• Sexual Reproduction is a biological process that involves the fusion of male and female gametes to form a zygote, which eventually develops into a new organism. This process ensures genetic diversity within a species. Hence, statement 2 is correct.
• In animals, fertilization is the process where the sperm (male gamete) meets the egg (female gamete). This can occur either inside the female body (Internal Fertilization) or outside in the environment (External Fertilization), such as in water for many fish and amphibians. It is not "always" outside. Hence, statement 1 is incorrect.
• Animals do not reproduce throughout their entire lifespan. They go through different phases: the juvenile phase (period of growth before reaching maturity), the reproductive phase (when they can reproduce), and the senescent phase (old age). Hence, statement 3 is incorrect.
• Sexual reproduction typically involves two parents (male and female) and results in offspring that are genetically different from the parents due to the mixing of DNA. The creation of identical offspring from a single parent is a characteristic of asexual reproduction. Hence, statement 4 is incorrect.

Additional Information

• Gametes and Zygote:
• The male gamete is called sperm, which is usually small and motile.
• The female gamete is called the ovum or egg, which is typically larger and contains food reserves for the developing embryo.
• The fusion of these two haploid cells results in a diploid zygote, restoring the chromosome number of the species.
• Types of Fertilization:
• Internal Fertilization: Common in mammals, birds, and reptiles. It provides better protection to the developing embryo.
• External Fertilization: Observed in frogs, starfish, and most fish. Large numbers of gametes are released into the water to increase the chances of survival.
• Significance of Sexual Reproduction:
• It promotes evolution by producing variations. These variations help organisms adapt to changing environmental conditions over thousands of years.

Important Points

• Asexual vs. Sexual Reproduction:
• Asexual Reproduction involves a single parent and produces clones (genetically identical copies). Examples include Binary Fission in Amoeba and Budding in Hydra.
• Sexual Reproduction involves meiosis (cell division that reduces chromosome number) and syngamy (fusion of gametes).
• Puberty: This is the age or stage at which the reproductive organs become functional, marking the beginning of the reproductive phase in animals and humans.

The correct answer is Ionic compound.

Key Points

• An Ionic compound is formed when Sodium (Na) reacts with Chlorine (Cl). This reaction results in the formation of Sodium Chloride (NaCl), commonly known as table salt.
• Sodium is a metal belonging to Group 1 of the periodic table with an atomic number of 11. It has one electron in its outermost shell and seeks to lose it to achieve a stable octet configuration.
• Chlorine is a non-metal belonging to Group 17 (Halogens) with an atomic number of 17. It has seven electrons in its valence shell and requires one additional electron to complete its octet.
• During the reaction, Sodium transfers its single valence electron to Chlorine. This complete transfer of electrons creates a Sodium cation (Na+) and a Chloride anion (Cl-).
• The strong electrostatic force of attraction between these oppositely charged ions holds them together, forming an Ionic Bond. Hence, option 1 is correct.

Additional Information

• Covalent Compounds:
• These are formed by the sharing of electrons between two or more non-metals.
• Examples include Water (H2O), Methane (CH4), and Carbon Dioxide (CO2). Since Sodium is a metal and Chlorine is a non-metal, they do not form a covalent bond. Hence, option 2 is incorrect.
• Coordinate (Dative) Compounds:
• This is a type of covalent bond where both electrons in the shared pair come from the same atom.
• These are common in complex ions and ammonium ions (NH4+). Sodium and Chlorine involve a complete transfer rather than a one-sided sharing. Hence, option 3 is incorrect.
• Metallic Compounds:
• These are formed through metallic bonding, which occurs between metal atoms where valence electrons are delocalized in a "sea of electrons."
• Examples include Pure Iron (Fe) or Copper (Cu). Sodium Chloride is a salt, not a metallic alloy or pure metal. Hence, option 4 is incorrect.

Important Points

• Properties of NaCl: Sodium Chloride forms a crystalline lattice structure. It has a high melting point of approximately 801 degrees Celsius and a high boiling point of 1413 degrees Celsius.
• Electronegativity: The Pauling electronegativity of Sodium is 0.93, while Chlorine is 3.16. The large difference of 2.23 (greater than 1.7) confirms that the bond is predominantly ionic.
• Conductivity: As an ionic compound, NaCl does not conduct electricity in its solid state but is an excellent conductor when molten or dissolved in water due to the mobility of ions.
• Solubility: It is highly soluble in polar solvents like Water because the polar water molecules can stabilize the Na+ and Cl- ions through ion-dipole interactions.

The correct answer is Ligaments connect bones to bones and are very elastic with little matrix.

Key Points

• Ligaments are specialized types of connective tissue that serve the primary function of connecting two bones together at a joint.
• A key physical property of ligaments is that they are highly elastic and possess great strength, allowing for the necessary range of motion in the skeletal system while maintaining structural integrity. Hence, statement 3 is correct.
• Unlike some other connective tissues, ligaments contain very little matrix, as they are primarily composed of densely packed fibrous proteins.
• Ligaments are flexible enough to allow movement but are strong enough to prevent dislocation of the joints. Hence, statement 1 is incorrect as it claims they are not flexible.
• By definition, their main role is inter-bone connectivity. Hence, statement 2 is incorrect because it denies this connection.
• The task of connecting muscles to bones is performed by tendons, not ligaments. Hence, statement 4 is incorrect.

Additional Information

• Composition of Ligaments:
• Ligaments are primarily made of collagen fibers and elastin fibers. While collagen provides tensile strength to withstand pressure, elastin provides the flexibility needed for movement.
• The cells found within ligaments are known as fibroblasts, which are responsible for the synthesis of the extracellular matrix and collagen.
• Comparison with Tendons:
• While both are fibrous connective tissues, tendons are tough and inelastic compared to ligaments.
• Tendons anchor skeletal muscles to bones, facilitating the transmission of mechanical force required for movement.
• Classification:
• Ligaments are categorized under Dense Regular Connective Tissue because the collagen fibers are arranged in a parallel pattern to provide maximal resistance to pulling forces.

Important Points

• Clinical Significance:
• An injury involving the overstretching or tearing of a ligament is medically termed a sprain. This is different from a strain, which involves muscles or tendons.
• Common examples of ligaments include the Anterior Cruciate Ligament (ACL) in the knee, which is vital for stabilizing the knee joint during rotational movements.
• Functionality:
• Ligaments act as mechanical checks; they limit the range of motion of a joint to protect it from moving in abnormal directions that could cause injury.
• They also contain sensory nerve endings (proprioceptors) that provide the brain with information regarding the position and movement of the joints.

The correct answer is Electron.

Key Points

• The discovery of the electron by Sir J.J. Thomson in 1897 was the primary catalyst for the development of the first modern atomic model.
• Thomson conducted experiments using a Cathode Ray Tube (CRT), where he observed that rays were deflected away from a negatively charged plate and toward a positively charged plate, indicating they consisted of negatively charged particles.
• Following this discovery, Thomson proposed the Plum Pudding Model (also known as the Watermelon Model) in 1904 to explain the structure of the atom.
• In this model, the atom was envisioned as a positively charged sphere in which negatively charged electrons were embedded, similar to seeds in a watermelon or raisins in a pudding.
• This discovery was revolutionary because it proved that the atom was not an indivisible particle, as John Dalton had previously proposed in his Atomic Theory, but instead contained smaller subatomic particles.
• The model also established the concept of electrical neutrality, suggesting that the total positive charge of the sphere was exactly balanced by the total negative charge of the embedded electrons.

Additional Information

• Proton:
• While Eugen Goldstein observed canal rays (positive ions) in 1886, the proton was identified as a fundamental subatomic particle by Ernest Rutherford.
• In 1919, Rutherford proved that the nucleus of a hydrogen atom was present in other nuclei, leading to the naming of the proton in 1920.
• The proton carries a positive charge of +1.602 x 10^-19 Coulombs and has a mass approximately 1836 times that of an electron.
• Neutron:
• The neutron was discovered by James Chadwick in 1932.
• He bombarded Beryllium with alpha particles and observed a new type of radiation that consisted of uncharged particles with a mass slightly greater than that of a proton.
• The discovery of the neutron explained why the atomic mass was often more than double the atomic number and helped in the understanding of isotopes.
• Positron:
• The positron is the antiparticle of the electron, discovered by Carl D. Anderson in 1932 while studying cosmic rays.
• It has the same mass as an electron but carries a positive charge. It was the first evidence of antimatter.
• Rutherford’s Atomic Model:
• Developed in 1911 following the Alpha-particle Scattering Experiment (Gold Foil Experiment).
• This model superseded Thomson's model by proposing that the atom has a tiny, dense, positively charged nucleus at its center, with electrons revolving around it, much like planets around the sun.

Important Points

• Chronology of Discovery: The electron (1897) was discovered first, followed by the proton (identified in 1919), and finally the neutron (1932).
• Charge-to-Mass Ratio: J.J. Thomson is also famous for measuring the charge-to-mass ratio (e/m) of the electron, which he found to be constant regardless of the gas used in the discharge tube.
• Nobel Prize: J.J. Thomson was awarded the Nobel Prize in Physics in 1906 for his work on the conduction of electricity by gases and the discovery of the electron.

The correct answer is 384 kJ.

Key Points

• The heat produced in a resistor when an electric current flows through it is governed by Joule’s Law of Heating. The formula used to calculate this heat energy (H) is H = I²Rt, where I is the current, R is the resistance, and t is the time duration.
• In the given problem, the value of the current (I) is 4 A and the resistance (R) of the resistor is 50 Ω (ohms).
• The time (t) provided is 8 minutes. Since the standard SI unit for time in physics calculations is seconds, we must convert it: 8 minutes = 8 × 60 = 480 seconds.
• By substituting these values into the formula, we get H = (4)² × 50 × 480. This simplifies to 16 × 50 × 480, which equals 384,000 Joules (J).
• To express the final answer in kilojoules (kJ), we divide the value by 1,000. Therefore, 384,000 J / 1,000 = 384 kJ. Hence, the correct answer is 384 kJ.

Additional Information

• Joule's Law of Heating:
• This law was formulated by the English physicist James Prescott Joule in the 1840s. It states that the heat produced in a conductor is directly proportional to the square of the current, the resistance of the conductor, and the time for which the current flows.
• The mathematical expression is H = I²Rt. If the current is doubled, the heat produced increases by four times, assuming resistance and time remain constant.
• Resistance (R):
• Resistance is the property of a material to oppose the flow of electric current. Its SI unit is the Ohm (Ω).
• In a resistor, the electrical energy lost due to collisions between electrons and the atoms of the conductor is converted into thermal energy (heat).
• Electric Power and Heat:
• Electric Power (P) is the rate at which electrical energy is consumed or converted into other forms. It is given by P = I²R or P = VI.
• Heat energy can also be expressed as the product of Power and Time (H = P × t). In this case, the power dissipated is 800 Watts (W), and for 480 seconds, the total energy is 384,000 J.
• Practical Applications of the Heating Effect:
• Domestic Appliances: Devices such as electric irons, electric heaters, geysers, and toasters utilize high-resistance coils (often made of Nichrome) to generate significant heat.
• Electric Bulbs: The filament of an incandescent bulb (usually made of Tungsten) is heated to such a high temperature that it begins to emit light.
• Electric Fuse: A safety device consisting of a wire with a low melting point. When excessive current flows, the heat produced (I²Rt) melts the fuse wire, breaking the circuit and protecting appliances from damage.

Important Points

• Unit Conversions: Always ensure that time is in seconds and current is in Amperes to get the energy in Joules. 1 kiloJoule (kJ) is equivalent to 1,000 Joules (J).
• Factors Affecting Heat:
• If Resistance is increased, heat increases linearly.
• If Current is increased, heat increases exponentially (squared).
• If Time is increased, heat increases linearly.
• Nature of the Energy: The heat produced is a form of waste energy in many electronic circuits (like computers), requiring cooling systems like fans or heat sinks, but it is the primary desired output in heating appliances.

The correct answer is The geometric centre of the spherical surface of which the mirror is a part.

Key Points

• The reflecting surface of a spherical mirror forms a part of a sphere. This sphere has a centre, and this point is called the Centre of Curvature of the spherical mirror.
• It is represented by the letter C. It is important to note that the centre of curvature is not a part of the mirror; it lies outside its reflecting surface.
• For a concave mirror, the centre of curvature lies in front of it. This point is the geometric centre of the hollow sphere of glass of which the mirror is a part. Hence, option 3 is correct.
• The midpoint of the mirror's aperture or the geometric centre of the reflecting surface itself is called the Pole (P), not the centre of curvature. Hence, option 1 is incorrect.
• The point where parallel rays of light meet after reflection is the Principal Focus (F). Hence, option 2 is incorrect.
• The image of an object at infinity is formed at the Principal Focus of the mirror. Hence, option 4 is incorrect.

Additional Information

• Radius of Curvature (R): The radius of the sphere of which the reflecting surface of a spherical mirror forms a part is called the Radius of Curvature. It is the distance between the Pole (P) and the Centre of Curvature (C).
• Principal Axis: An imaginary straight line passing through the Pole and the Centre of Curvature of a spherical mirror. It is normal to the mirror at its pole.
• Principal Focus (F): For a concave mirror, it is the point on the principal axis where all the incident rays that are parallel to the axis actually converge after reflection.
• Focal Length (f): The distance between the Pole and the Principal Focus of a spherical mirror is called the focal length. For mirrors with small apertures, the radius of curvature is found to be equal to twice the focal length (R = 2f).
• Aperture: The diameter of the reflecting surface of a spherical mirror is called its aperture. In elementary physics, we usually consider mirrors whose aperture is much smaller than its radius of curvature.

Important Points

• Concave Mirror: A spherical mirror whose reflecting surface is curved inwards (faces towards the centre of the sphere) is called a concave mirror. It is also known as a converging mirror because it converges light rays.
• Convex Mirror: A spherical mirror whose reflecting surface is curved outwards is called a convex mirror. The centre of curvature for a convex mirror lies behind the mirror. It is also known as a diverging mirror.
• Light Path through C: Any ray of light passing through the centre of curvature of a concave mirror (or directed in the direction of the centre of curvature of a convex mirror) is reflected back along the same path because the incident ray falls on the mirror along the normal to the reflecting surface.

The correct answer is It gives a different smell in acidic and basic medium.

Key Points

• Olfactory indicators are substances whose smell or odour changes depending on whether they are placed in an acidic or basic medium.
• Clove oil is a classic example of an olfactory indicator used in chemistry laboratories to distinguish between acids and bases.
• When clove oil is added to an acidic solution, its characteristic pungent smell is retained and does not vanish.
• When clove oil is treated with a basic solution, such as Sodium Hydroxide (NaOH), the characteristic smell of the oil is lost or cannot be detected.
• This property of changing odour based on the pH of the medium makes it a reliable tool for chemical testing. Hence, the correct answer is option 4.

Additional Information

• Vanilla Essence: Similar to clove oil, vanilla essence acts as an olfactory indicator. It maintains its pleasant aroma in acidic conditions but loses the smell when mixed with a base.
• Onion: Finely chopped onions can also be used as olfactory indicators. The smell of onion is destroyed when it reacts with a basic solution, whereas it persists in an acidic solution.
• Utility for Visually Impaired: Olfactory indicators are extremely important for visually impaired students, as they allow the identification of acids and bases through the sense of smell rather than relying on colour changes used in traditional indicators like litmus or pH paper.
• Natural Indicators: Apart from olfactory types, there are natural indicators like litmus (from Lichens), Turmeric, and Red Cabbage which change colour to indicate the nature of the substance.
• Synthetic Indicators: These are man-made chemicals like Phenolphthalein and Methyl Orange. Phenolphthalein remains colourless in acid but turns pink in basic solutions.

Important Points

• The main chemical component in clove oil responsible for its odour is Eugenol.
• In a basic medium, the hydroxide ions react with the acidic functional groups of the indicator molecules (like Eugenol), changing their chemical structure and consequently neutralizing the scent.
• Acids have a pH less than 7, while bases have a pH greater than 7.
• The fuming or effervescence mentioned in other options usually relates to reactions involving carbonates (releasing CO2) or strong oxidizing agents, rather than the standard behavior of an olfactory indicator.
• Testing for acids and bases using smell is a qualitative analysis technique that provides immediate results without the need for complex electronic equipment.

The correct answer is -20 cm.

Key Points

• In optics, the magnification (m) produced by a lens is defined as the ratio of the image distance (v) to the object distance (u). The formula is expressed as m = v / u.
• According to the question, the magnification (m) is -2 and the image distance (v) is 40 cm. Substituting these values into the formula gives -2 = 40 / X, where X represents the object distance.
• By rearranging the equation to solve for X, we get X = 40 / -2, which results in X = -20 cm. Hence, the statement that the object distance is -20 cm is correct.
• The negative sign in the result -20 cm indicates that the object is placed in front of the lens, which is consistent with the standard Cartesian sign convention where the object distance is always taken as negative.
• Since the magnification is negative (-2), it signifies that the image formed by the convex lens is real and inverted. Hence, the correct answer is -20 cm.

Additional Information

• Convex Lens Characteristics:
• A convex lens is also known as a converging lens because it converges parallel rays of light passing through it.
• It can form both real and virtual images depending on the position of the object relative to the focal point.
• In this specific case, the magnification of -2 tells us the image is magnified (twice the size of the object) and inverted.
• Sign Convention for Lenses:
• Object distance (u): Always negative as the object is placed to the left of the lens.
• Image distance (v): Positive for real images (formed on the right side) and negative for virtual images (formed on the left side).
• Focal length (f): For a convex lens, the focal length is always positive, while for a concave lens, it is negative.
• Lens Formula:
• The relationship between focal length (f), object distance (u), and image distance (v) is given by 1/f = 1/v - 1/u.
• Using the values from the question (v = 40 cm, u = -20 cm), the focal length would be calculated as 1/f = 1/40 - 1/(-20) = 1/40 + 1/20 = 3/40, meaning f = 40/3 = 13.33 cm.

Important Points

• Nature of Magnification:
• If |m| > 1, the image is enlarged.
• If |m| < 1, the image is diminished.
• If m is negative, the image is real and inverted.
• If m is positive, the image is virtual and erect.
• Practical Applications:
• Convex lenses are used in microscopes, telescopes, magnifying glasses, and cameras.
• They are also used in spectacles to correct hypermetropia (long-sightedness).

The correct answer is 10 Ω.

Key Points

• In an electrical circuit, when resistors are connected in parallel, the reciprocal of the equivalent resistance (Req) is the sum of the reciprocals of the individual resistances.
• The mathematical formula for two resistors (R1 and R2) in parallel is: 1/Req = 1/R1 + 1/R2.
• According to the question, the equivalent resistance of the two resistors in parallel is 5 Ω.
• When one resistor is removed, the circuit resistance increases to 10 Ω. This indicates that the value of the remaining resistor (let's assume R1) is 10 Ω.
• By substituting the given values into the parallel formula: 1/5 = 1/10 + 1/R2.
• Rearranging the equation to solve for the unknown resistor: 1/R2 = 1/5 - 1/10.
• Taking the common denominator: 1/R2 = (2 - 1) / 10, which results in 1/R2 = 1/10.
• Thus, the value of the other resistor R2 is 10 Ω.
• Hence, the correct answer is 10 Ω.

Additional Information

• Parallel Combination Features:
• The potential difference (voltage) across each resistor in a parallel combination remains the same.
• The total current in the circuit is the sum of the currents flowing through each individual branch.
• The equivalent resistance is always smaller than the smallest individual resistance in the parallel network.
• Application: Household electrical wiring is done in parallel so that each appliance can operate independently with the same voltage.
• Series Combination Features:
• In a series circuit, resistors are connected end-to-end, and the total resistance (Rs) is the simple sum: Rs = R1 + R2 + ... + Rn.
• The current flowing through each resistor in a series circuit is the same.
• The total voltage is divided across the resistors based on their resistance values.
• Ohm's Law:
• Proposed by Georg Simon Ohm in 1827, it states that the current (I) passing through a conductor is directly proportional to the voltage (V) across it, provided physical conditions like temperature remain constant (V = I × R).

Important Points

• Identical Resistors: If n identical resistors, each with resistance R, are connected in parallel, the equivalent resistance is calculated as Req = R / n.
• In this specific problem, we have two 10 Ω resistors. Applying the shortcut: Req = 10 / 2 = 5 Ω, which perfectly matches the initial condition provided in the question.
• Resistance Factors: The resistance of a conductor depends on its length (directly proportional), cross-sectional area (inversely proportional), the nature of the material (resistivity), and temperature.
• Unit of Resistance: The SI unit of resistance is the Ohm, symbolized by the Greek letter Ω (Omega).

The correct answer is Reverberation helps in improving the clarity of sound..

Key Points

• Reverberation of sound is the phenomenon where a sound persists for some time even after the source of the sound has stopped. It occurs as a result of the multiple reflections of sound. Therefore, the fourth statement (4) is correct.
• When sound is produced in a large hall with hard walls, it reflects repeatedly from the walls, ceiling, and floor, causing the sound to be heard for a prolonged duration. Therefore, the third statement (3) is correct.
• Due to excessive reverberation, a new sound begins before the previous one finishes, causing the sounds to overlap with one another. This reduces the clarity of the sound and causes a disruption in conversation. Therefore, the second statement (2) is correct.
• Reverberation does not help in improving the clarity of sound; rather, it makes the sound blurred. If reverberation exceeds a certain limit, it becomes difficult to understand speech or music. Therefore, the first statement (1) is incorrect or not correct.
• Hence, the correct answer is 1.

Additional Information

• Reverberation Time: The time required for the intensity of sound to decrease by 60 decibels (60 dB) after the sound source has stopped is called the reverberation time. For an ideal hall, this time should neither be too long nor too short.
• Sound Absorbent Materials: To control reverberation in large auditoriums or cinema halls, compressed fibreboard, curtains, carpets, and acoustic tiles are used. These materials absorb the sound instead of reflecting it.
• What is meant by Echo: When a sound reflects off an obstacle and comes back to our ears, it is called an echo. To hear a distinct echo, the distance between the source of sound and the obstacle must be at least 17.2 meters.
• Practical Example: In a cinema hall, reverberation makes the sound feel deep and powerful, but to control it properly, the walls are covered with sound-absorbent fabrics or boards so that the clarity of speech is not lost.

Important Points

• Laws of Reflection of Sound: Just like light, in the reflection of sound, the angle of incidence is equal to the angle of reflection.
• Disadvantages of High Reverberation: If there is too much reverberation in school classrooms or meeting halls, the speaker's words become unclear, which is termed as acoustically poor.
• Architectural Design: Modern architects follow the principles of acoustics when designing auditoriums so that reverberation and echoes can be properly controlled.

The correct answer is Aerenchyma.

Key Points

• Aerenchyma is a specialized type of parenchyma tissue characterized by the presence of large air cavities or intercellular spaces.
• It is primarily found in aquatic plants (also known as hydrophytes), such as Lotus, Water Hyacinth (Eichhornia), and Hydrilla.
• The main function of these air cavities is to provide buoyancy to the plant, which helps them float on or near the water surface. Hence, option 2 is correct.
• Beyond buoyancy, aerenchyma also facilitates the internal diffusion of gases like oxygen and carbon dioxide between the aerial parts and the submerged tissues, which is vital for respiration and photosynthesis in an aquatic environment.
• The cells in this tissue are typically thin-walled and arranged in a way that creates a sponge-like network within the cortex of roots and stems.

Additional Information

• Chlorenchyma:
• This is a type of parenchyma that contains chlorophyll and is responsible for performing photosynthesis.
• It is commonly found in the mesophyll of leaves and the outer green layers of herbaceous stems. Hence, option 1 is incorrect.
• Tracheid:
• Tracheids are elongated, tube-like cells that form a part of the xylem (complex permanent tissue).
• These cells are dead at maturity, have lignified walls, and are responsible for the conduction of water and minerals in vascular plants. Hence, option 3 is incorrect.
• Companion Cell:
• These are specialized parenchymatous cells located in the phloem and are always associated with sieve tube elements.
• They possess a dense cytoplasm and a large nucleus to help regulate the pressure gradient and metabolic activities required for the translocation of food. Hence, option 4 is incorrect.

Important Points

• Parenchyma: It is the most common simple permanent tissue in plants, consisting of relatively unspecialized cells with thin cellulose walls. They are living cells and serve functions like storage, secretion, and support.
• Plant Tissue Classification:
• Meristematic Tissues: Responsible for growth through active cell division (Apical, Lateral, and Intercalary).
• Simple Permanent Tissues: Composed of a single type of cell (Parenchyma, Collenchyma, and Sclerenchyma).
• Complex Permanent Tissues: Composed of more than one type of cell working together as a unit (Xylem and Phloem).
• Collenchyma: Provides mechanical support and flexibility to parts like the petiole of a leaf, allowing them to bend without breaking.
• Sclerenchyma: Consists of dead cells with very thick lignin deposits, providing rigidity and strength to the plant body (found in husk of coconut).

The correct answer is Ribosome.

Key Points

• Ribosomes are the only organelles mentioned that are non-membrane bound. They are granular structures composed of ribonucleic acid (RNA) and proteins, serving as the primary site for protein synthesis in both prokaryotic and eukaryotic cells.
• Chromoplasts are a type of plastid responsible for pigment synthesis and storage. They contain fat-soluble carotenoid pigments like carotene and xanthophylls. These organelles are enclosed by a double membrane.
• Vacuoles are large, membrane-bound sacs found in the cytoplasm. In plant cells, they can occupy up to 90 percent of the volume and are bound by a single membrane called the tonoplast.
• Lysosomes are vesicular structures formed by the Golgi apparatus. They are membrane-bound and contain various hydrolytic enzymes such as lipases, proteases, and carbohydrases, which are active at an acidic pH.
• Because Chromoplasts, Vacuoles, and Lysosomes all possess at least one surrounding lipid layer, they are classified as membrane-bound, whereas the Ribosome lacks this structure.

Additional Information

• Eukaryotic Cells: These cells are characterized by the presence of a membrane-bound nucleus and other specialized structures called organelles. The cytoplasm of these cells is compartmentalized by these membranes.
• Double Membrane-bound Organelles:
• Mitochondria: Known as the powerhouse of the cell, involved in ATP production.
• Chloroplasts: Found in plants and algae, responsible for photosynthesis.
• Nucleus: Contains the genetic material (DNA) and is surrounded by the nuclear envelope.
• Single Membrane-bound Organelles:
• Endoplasmic Reticulum (ER): Involved in lipid and protein synthesis.
• Golgi Apparatus: Functions in the packaging and secretion of macromolecules.
• Peroxisomes: Small vesicles involved in oxidative reactions and detoxification.
• Non-membrane-bound Organelles:
• Apart from Ribosomes, other examples include Centrioles (found in animal cells and involved in cell division) and the Nucleolus (located inside the nucleus, involved in rRNA synthesis).

Important Points

• Discovery: Ribosomes were first observed as dense particles under an electron microscope by George Palade in the year 1953.
• Svedberg Unit (S): Ribosomes are measured by their sedimentation coefficient. Eukaryotic ribosomes are 80S (composed of 60S and 40S subunits), while prokaryotic ribosomes are 70S (composed of 50S and 30S subunits).
• Location: In eukaryotes, ribosomes are found freely in the cytoplasm, attached to the Rough Endoplasmic Reticulum (RER), or inside organelles like mitochondria and plastids.

The correct answer is 0.2 m/s2 and 9 km.

Key Points

• The boy starts from rest, which means the initial velocity (u) is 0 m/s. This is a fundamental concept in kinematics where "rest" denotes no motion at the start time (t = 0).
• The final velocity (v) reached by the boy is 60 m/s. The time taken to reach this velocity is 5 minutes. To use standard physical formulas, this time must be converted into seconds: 5 minutes multiplied by 60 seconds equals 300 seconds.
• Acceleration (a) is defined as the rate of change of velocity. Using the formula a = (v - u) / t, we substitute the values: a = (60 - 0) / 300, which simplifies to 60 / 300. This results in an acceleration of 0.2 m/s2.
• The distance (s) covered under uniform acceleration can be calculated using the second equation of motion: s = ut + (1/2)at². Since the initial velocity is 0, the formula becomes s = 0.5 × 0.2 × (300)².
• Calculating the distance gives s = 0.1 × 90,000, which equals 9,000 meters. To convert this value into kilometers, we divide by 1,000, resulting in exactly 9 km. Hence, the calculated values for acceleration and distance are 0.2 m/s2 and 9 km respectively.

Additional Information

• Equations of Motion: These equations describe the behavior of physical systems in terms of their motion as a function of time. They are valid only when acceleration is constant (uniform). The three primary equations are:
• v = u + at (Velocity-Time relation).
• s = ut + 1/2 at² (Position-Time relation).
• v² = u² + 2as (Position-Velocity relation).
• Acceleration: It is a vector quantity, meaning it has both magnitude and direction. The SI unit for acceleration is meters per second squared (m/s2). If the velocity increases, the acceleration is positive; if it decreases, it is known as retardation or negative acceleration.
• Uniform vs. Non-Uniform Motion: In uniform motion, an object covers equal distances in equal intervals of time (zero acceleration). In uniformly accelerated motion, the velocity changes by equal amounts in equal intervals of time, which is the case in this specific running race scenario.
• Unit Conversions: In physics problems, consistency of units is vital. Velocity is typically measured in m/s or km/h. To convert km/h to m/s, multiply by 5/18. To convert minutes to seconds, multiply by 60. Failure to convert units often leads to incorrect numerical results.

Important Points

• Velocity-Time Graphs: For an object starting from rest with uniform acceleration, the v-t graph is a straight line passing through the origin. The slope of this line represents the acceleration, while the area under the curve represents the total distance covered.
• Practical Application: In sports and automotive engineering, calculating acceleration is crucial for performance analysis. For instance, a sprinter's ability to reach top speed (final velocity) from the blocks (rest) in a specific time frame determines their acceleration rate.
• Scalar vs Vector: While distance is a scalar quantity (only magnitude), displacement is a vector. In a straight-line running race, the magnitude of distance and displacement are the same, allowing for the direct use of the equations of motion to find the path length of 9 km.

The correct answer is Plant cell.

Key Points

• Plant cells, specifically specialized cells like guard cells, change their shape by regulating the amount of water present within them through a process called osmosis.
• When water flows into the guard cells, they swell and become turgid, which causes the stomata (small pores on leaves) to open. Conversely, when water leaves the cells, they shrink and become flaccid, leading to the closure of the stomata. Hence, statement 1 is correct.
• The movement of water is driven by changes in solute concentration, often involving the active transport of potassium ions (K+) into the vacuole, which draws water inward.
• In contrast, animal cells such as cardiac muscle cells, striated muscle cells, and smooth muscle cells do not change shape by water volume; instead, they rely on specialized contractile proteins like actin and myosin that slide past each other. Hence, statements 2, 3, and 4 are incorrect.
• This mechanism in plants is vital for transpiration and gas exchange (CO2 and O2) required for photosynthesis.

Additional Information

• Turgor Pressure: This is the force within the plant cell that pushes the plasma membrane against the cell wall. It is maintained by the osmotic flow of water and provides structural integrity to non-woody plants.
• Central Vacuole: In a mature plant cell, the large central vacuole can occupy up to 90% of the cell volume. It acts as the primary storage for water and maintains turgidity.
• Plasmolysis: This occurs when a plant cell is placed in a hypertonic solution, causing water to leave the cell and the protoplast to shrink away from the cell wall.
• Muscle Contraction: In Cardiac, Striated (Skeletal), and Smooth muscle cells, shape change (contraction) is triggered by electrical impulses and the release of Calcium ions (Ca2+), utilizing ATP as the energy source.
• Cell Wall: Unlike animal cells, plant cells possess a rigid cell wall made of cellulose, which prevents the cell from bursting when it takes in large amounts of water.

Important Points

• Osmoregulation: This is the process by which plants maintain the water potential and osmotic pressure of their fluids, ensuring that metabolic activities continue even under varying environmental moisture levels.
• Stomatal Regulation: The guard cells are unique epidermal cells because they contain chloroplasts and can respond to light intensity and carbon dioxide levels to adjust the water content and cell shape.
• Hydrostatic Pressure: The reliance on hydrostatic pressure for movement is common in the plant kingdom (e.g., the closing of Mimosa pudica leaves) but is generally absent in vertebrate muscle mechanics.

The correct answer is Potassium.

Key Points

• The reactivity series, also known as the activity series, is an empirical, calculated, and analytically hierarchical arrangement of metals in descending order of their chemical reactivity.
• Potassium (chemical symbol K, atomic number 19) is placed at the top or the starting of this series because it is the most reactive metal among the common elements.
• Metals at the top of the series, such as Potassium and Sodium, lose their valence electrons very easily to form positive ions (cations), making them highly electropositive.
• Because of its extreme reactivity, Potassium reacts violently even with cold water to produce hydrogen gas and potassium hydroxide, releasing enough heat to ignite the hydrogen. Hence, statement 3 is the correct option.
• In contrast, Aluminium is placed lower than Potassium, Sodium, and Magnesium. Iron is situated in the middle of the series, and Gold is located at the very bottom as it is one of the least reactive metals.

Additional Information

• Displacement Reactions: A fundamental application of the reactivity series is predicting displacement reactions, where a more reactive metal (like Potassium or Zinc) will displace a less reactive metal (like Copper or Silver) from its salt solution.
• Reaction with Water and Acids:
• Metals like Potassium, Sodium, and Calcium react with cold water.
• Metals like Magnesium react with hot water.
• Metals like Aluminium, Zinc, and Iron react only with steam.
• Gold, Silver, and Platinum do not react with water or dilute acids at all.
• Extraction Methods: The position in the series determines the extraction process. Metals at the top (Potassium to Aluminium) are extracted via electrolysis of their molten ores. Metals in the middle (Zinc to Lead) are reduced using carbon (coke). Metals at the bottom (Gold and Silver) are often found in their native state in nature.
• Noble Metals: Gold (Au) and Platinum (Pt) are referred to as noble metals because they remain unaffected by air, moisture, and most acidic reagents, which is why they are used in jewelry and electronics.

Important Points

• Storage of Reactive Metals: Due to their high affinity for oxygen and moisture, Potassium and Sodium are stored under kerosene oil to prevent accidental fires or oxidation in the open air.
• Hydrogen Reference: Although it is a non-metal, Hydrogen is included in the reactivity series to serve as a reference point. Metals placed above hydrogen can displace it from dilute acids to form hydrogen gas, while those below it cannot.
• Order of the Series (Descending): Potassium > Sodium > Calcium > Magnesium > Aluminium > Zinc > Iron > Tin > Lead > [Hydrogen] > Copper > Mercury > Silver > Gold.
• Chemical Symbols and Atomic Numbers:
• Potassium (K): 19
• Aluminium (Al): 13
• Iron (Fe): 26
• Gold (Au): 79

The correct answer is Vinegar.

Key Points

• A heterogeneous mixture is a mixture in which the composition is not uniform throughout and the different components can be observed as separate phases.
• Vinegar: It is a homogeneous mixture (solution) of acetic acid and water. Because the acetic acid dissolves completely in water, the mixture has a uniform appearance and composition throughout. Hence, this is not a heterogeneous mixture and is the correct answer.
• Water in oil: Water and oil are immiscible liquids. When mixed, they form two distinct layers or droplets, making the composition non-uniform. Hence, statement 1 is a heterogeneous mixture.
• Sand and sugar: This is a mixture of two different solid substances. The grains of sand and sugar do not blend into a single phase and can be easily identified or separated. Hence, statement 3 is a heterogeneous mixture.
• Sand and salt: Similar to sand and sugar, sand and salt crystals remain distinct from each other when mixed together in a solid state. Hence, statement 4 is a heterogeneous mixture.

Additional Information

• Homogeneous Mixtures:
• These mixtures have a uniform composition and a single phase.
• Common examples include Air (a mixture of gases like Nitrogen and Oxygen), Alloys (like Steel or Brass), and Saltwater.
• In these mixtures, the particles are distributed at the molecular or atomic level.
• Heterogeneous Mixtures:
• These mixtures consist of physically distinct parts and often have visible boundaries between components.
• Examples include Suspensions (like Muddy water), Colloids (like Milk or Fog), and mechanical mixtures like Trail mix.
• The components can typically be separated using physical methods such as filtration, sorting, or decantation.
• Vinegar Properties:
• Vinegar usually contains about 5 percent to 8 percent acetic acid by volume.
• It is widely used in food preservation (pickling), cooking, and as a mild cleaning agent due to its acidic nature.
• The pH value of commercial vinegar is typically around 2.4 to 3.4.
• Miscibility and Solubility:
• The ability of a substance to form a homogeneous mixture depends on its solubility and chemical polarity.
• Water is a polar solvent, while Oil is non-polar, which is why they do not mix and instead form a heterogeneous mixture.

The correct answer is Two.

Key Points

• Gregor Johann Mendel, during his pioneering inheritance studies in the mid-19th century, proposed that every trait in a sexually reproducing organism is controlled by two copies of a discrete unit called a factor.
• These factors, which we now identify as genes, are passed from parents to offspring. Mendel suggested that an individual receives one factor from each parent, resulting in a pair that determines the expressed characteristic.
• This concept was fundamental to his Law of Segregation, which states that during the formation of gametes (eggs and sperm), the two copies of each factor separate so that each gamete carries only one factor for each trait.
• Upon fertilization, the two factors are restored in the zygote, ensuring that the offspring has a complete set of two factors, maintaining the diploid nature of the organism.
• The presence of two copies allows for variations in traits, as the factors can be identical (homozygous) or different (heterozygous).

Additional Information

• Pisum sativum (Garden Pea): Mendel chose the garden pea for his experiments conducted between 1856 and 1863. This plant was ideal because it had seven distinct pairs of contrasting traits, such as tall vs. dwarf stems and round vs. wrinkled seeds.
• Alleles: In modern genetics, the different forms of a factor are called alleles. For example, the gene for height has two alleles: T (tall) and t (dwarf).
• Dominant and Recessive: Mendel observed that in a pair of different factors, one could mask the expression of the other. The expressed factor is dominant, while the masked one is recessive.
• Genotype vs. Phenotype: The genotype refers to the actual genetic makeup (the two factors/alleles present, e.g., Tt), whereas the phenotype is the observable physical characteristic (e.g., a tall plant).
• Monohybrid Cross: By crossing plants with two different versions of a single trait, Mendel consistently observed a 3:1 ratio in the F2 generation, which statistically proved the existence of paired factors.

Important Points

• Publication and Rediscovery: Mendel published his findings in 1866 in the paper "Experiments on Plant Hybridization." However, his work remained largely ignored until its rediscovery in 1900 by Hugo de Vries, Carl Correns, and Erich von Tschermak.
• Diploidy: The proposal of two factors aligns with the biological reality of homologous chromosomes. Humans and many other organisms are diploid, meaning they have two sets of chromosomes, one from each parent.
• Mendel’s Laws: His studies led to the formulation of three major laws: the Law of Dominance, the Law of Segregation, and the Law of Independent Assortment.
• Significance: The realization that two copies of a factor control inheritance provided the mathematical basis for the science of genetics, allowing scientists to predict the probability of traits appearing in future generations.

The correct answer is Uterus.

Key Points

• Contraceptive devices like the loop or the copper-T are categorized as Intrauterine Devices (IUDs). These are specifically designed to be inserted and placed within the uterus of the female reproductive system. Hence, option 1 is correct.
• The uterus, also known as the womb, is the ideal location for these devices because it is where implantation of a fertilized egg occurs. By occupying this space, the IUD prevents the embryo from attaching to the uterine wall.
• These devices must be inserted by doctors or expert nurses through the vagina and the cervical canal into the uterine cavity. They are not intended for placement in the heart, kidney, or ovaries, as these organs are not involved in the direct process of conception or gestation.
• The primary mechanism of devices like the Copper-T is to release copper ions, which increase the phagocytosis of sperms within the uterus. This makes the environment hostile to spermatozoa, effectively preventing fertilization.
• IUDs are currently one of the most widely accepted methods of contraception in India, particularly for women who want to delay pregnancy or space out births between children.

Additional Information

• Types of Intrauterine Devices (IUDs):
• Non-medicated IUDs: An example is the Lippes Loop. These work by causing a minor inflammatory response in the uterus that prevents implantation.
• Copper-releasing IUDs: Common examples include CuT (Copper-T), Cu7, and Multiload 375. The copper ions released suppress sperm motility and the fertilizing capacity of the sperm.
• Hormone-releasing IUDs: Examples include Progestasert and LNG-20. These release progestogens that make the uterus unsuitable for implantation and make the cervix hostile to sperm by thickening the cervical mucus.
• Effectiveness and Duration:
• IUDs are highly effective, with a failure rate of less than 1%.
• Depending on the specific model, a Copper-T can remain effective for 3, 5, or even 10 years, making it a long-term reversible contraceptive method.
• Biological Roles of Other Options:
• Heart: The heart is a muscular organ responsible for pumping blood throughout the circulatory system. It has no role in reproductive contraception.
• Kidney: The kidneys are vital organs of the excretory system, responsible for filtering waste products from the blood to form urine.
• Ovary: The ovaries are the primary female sex organs that produce ova (eggs) and secrete hormones like estrogen and progesterone. While they are part of the reproductive system, IUDs are not placed inside them.

Important Points

• Side Effects: Common side effects of IUD insertion can include irregular bleeding, cramping, or pelvic inflammatory disease in rare cases.
• Reversibility: One of the biggest advantages of the Copper-T or loop is that fertility returns to normal almost immediately after the device is removed by a healthcare professional.
• Protection: It is important to note that while IUDs are excellent for preventing pregnancy, they do not provide protection against Sexually Transmitted Diseases (STDs) or HIV/AIDS; only barrier methods like condoms provide such protection.

The correct answer is ultraviolet.

Key Points

• The Ozone layer is a thin part of the Earth's atmosphere that absorbs almost all of the sun's harmful ultraviolet (UV) radiation.
• It is primarily found in the lower portion of the stratosphere, approximately 15 to 35 kilometers above Earth's surface.
• The ozone molecule is made up of three oxygen atoms (O3), which are formed when UV light hits ordinary oxygen molecules (O2) and splits them into individual atoms that then combine with other O2 molecules.
• By absorbing UV-B and UV-C radiation, the ozone layer prevents these high-energy rays from reaching the ground, where they could cause skin cancer, cataracts, and damage to DNA in humans and animals.
• It also plays a crucial role in protecting phytoplankton and terrestrial plants, which are the base of the food chain and sensitive to intense radiation.

Additional Information

• The Montreal Protocol:
• This is an international treaty signed in 1987 to protect the ozone layer by phasing out the production of numerous substances responsible for ozone depletion.
• The main chemicals targeted are Chlorofluorocarbons (CFCs) and Halons, which were commonly used in refrigeration and aerosol sprays.
• Measurement of Ozone:
• The concentration of ozone in the atmosphere is measured in Dobson Units (DU).
• A "hole" in the ozone layer is technically defined as an area where the concentration drops below 220 Dobson Units.
• Types of Radiation:
• Infrared (IR) Radiation: These are long-wavelength rays that are mostly experienced as heat; they are absorbed by greenhouse gases like carbon dioxide.
• Gamma Rays: These are extremely high-frequency electromagnetic bursts that are typically blocked by the Earth's upper atmosphere (thermosphere).
• Cosmic Rays: These are high-energy particles originating from outside the solar system, largely diverted by the Earth's magnetic field (magnetosphere).
• World Ozone Day:
• The United Nations General Assembly designated September 16 as the International Day for the Preservation of the Ozone Layer.

Important Points

• Formation: Ozone is naturally produced through the photochemical reaction known as the Chapman Cycle.
• Location: While stratospheric ozone is beneficial (protecting life), ground-level ozone (tropospheric ozone) is a pollutant and a primary component of smog.
• Recent Trends: Due to the success of the Montreal Protocol, the ozone layer is currently healing and is expected to return to 1980 levels by the mid-21st century.

The correct answer is Ammeter.

Key Points

• Archimedes’ principle states that when a body is partially or completely immersed in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the body.
• Ammeter: This is an instrument used to measure electric current in a circuit. It operates based on the principle of electromagnetism, where a current-carrying coil placed in a magnetic field experiences a torque (Lorentz Force). It has no relation to fluid displacement or buoyancy. Hence, statement 2 is the correct answer as it is not based on Archimedes' principle.
• Submarine: These vessels use ballast tanks to manage their buoyancy. By filling these tanks with water, the submarine's weight increases to exceed the buoyant force, allowing it to sink. To surface, compressed air pushes the water out, making the submarine lighter than the displaced water. Hence, statement 1 is based on the principle.
• Hydrometer: This instrument is used for measuring the relative density or specific gravity of liquids. It is designed to float in a liquid, and the depth to which it sinks depends on the density of the fluid, directly applying the law of buoyancy. Hence, statement 3 is based on the principle.
• Lactometer: This is a specialized type of hydrometer used specifically to test the purity of milk. It works by measuring the density of milk compared to water; if the milk is adulterated, its specific gravity changes, which is reflected in the lactometer's reading. Hence, statement 4 is based on the principle.

Additional Information

• Buoyancy: It is the upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid.
• Ammeter Connection: An ammeter is always connected in series within a circuit to measure the total flow of electrons. It is designed with very low resistance so that it does not significantly alter the current it is intended to measure.
• Ship Design: Large ships made of steel float because they are hollowed out, ensuring the total volume of water they displace weighs more than the ship itself, illustrating a primary application of Archimedes' principle.
• Relative Density: It is a dimensionless quantity defined as the ratio of the density of a substance to the density of a reference material (usually water at 4 degrees Celsius).

Important Points

• Discovery: The principle was discovered by the Greek mathematician Archimedes of Syracuse in the 3rd century BC. Legend says he discovered it while stepping into a bath, leading him to shout "Eureka!"
• Mathematical Expression: The buoyant force (Fb) can be calculated as Fb = ρ × V × g, where ρ (rho) is the density of the fluid, V is the volume of the displaced fluid, and g is the acceleration due to gravity (approximately 9.8 m/s²).
• Floating and Sinking: An object will float if its density is less than the fluid's density and sink if its density is greater. If the densities are equal, the object will remain suspended at its current depth (neutral buoyancy).

The correct answer is a = (v - u)/t.

Key Points

• Acceleration is defined as the rate of change of velocity of an object with respect to time. It describes how quickly the speed or direction of a moving body changes.
• In the given context, u represents the initial velocity (the velocity at the start of the time interval), v represents the final velocity (the velocity at the end of the time interval), and t represents the time taken for this change to occur.
• The formula for acceleration (a) is derived from the change in velocity divided by the time interval: a = (Final Velocity - Initial Velocity) / Time. Hence, the statement a = (v - u) / t is correct.
• If an object moves with constant velocity, the acceleration is zero because the difference between the final and initial velocity (v - u) is null.
• Acceleration is a vector quantity, meaning it possesses both magnitude and direction. It points in the same direction as the change in velocity.

Additional Information

• Types of Acceleration:
• Positive Acceleration: This occurs when the final velocity (v) is greater than the initial velocity (u). It implies that the object is speeding up in the direction of motion.
• Negative Acceleration (Retardation/Deceleration): This occurs when the final velocity (v) is less than the initial velocity (u). In this case, the acceleration value is negative, indicating the object is slowing down.
• Uniform Acceleration: When the velocity of an object changes by equal amounts in equal intervals of time, the motion is said to be uniformly accelerated. A classic example is a ball falling freely under gravity.
• SI Unit and Dimensions:
• The Standard International (SI) unit for acceleration is meter per second squared (m/s²).
• In terms of dimensional analysis, it is represented as [M⁰ L¹ T⁻²].
• Equations of Motion:
• The formula a = (v - u) / t is often rearranged to form the first equation of motion: v = u + at.
• These equations were famously described by Sir Isaac Newton and are fundamental to Classical Mechanics.

Important Points

• Graphical Representation: On a Velocity-Time (v-t) graph, the slope of the line represents the acceleration. A steeper slope indicates a higher rate of acceleration, while a horizontal line indicates zero acceleration (constant velocity).
• Newton's Second Law: The concept of acceleration is central to Newton's Second Law of Motion, which states that Force (F) is equal to mass (m) multiplied by acceleration (a), expressed as F = ma.
• Instantaneous vs. Average: Average acceleration is calculated over a finite time interval, whereas instantaneous acceleration is the acceleration at a specific moment in time, calculated using calculus as the derivative of velocity with respect to time (dv/dt).
• Directional Changes: Even if the speed of an object remains constant, it is still accelerating if its direction changes. A prime example is Uniform Circular Motion, where the object undergoes centripetal acceleration directed toward the center of the circle.

The correct answer is Larger surface area increases evaporation.

Key Points

• Evaporation is defined as a surface phenomenon where molecules in a liquid state spontaneously transition into a gaseous state at any temperature below the boiling point.
• The rate of evaporation is directly proportional to the surface area exposed to the atmosphere. When the surface area is increased, a larger number of molecules are positioned at the surface, providing more opportunities for them to escape into the air. Hence, statement 4 is correct.
• In a smaller surface area, fewer molecules are exposed to the environment at any given time, which leads to a significant reduction in the rate of vapor formation. Hence, statement 1 is incorrect.
• Evaporation occurs most effectively in open containers where vapor can diffuse away. In a closed container, the process eventually reaches dynamic equilibrium, where the rate of evaporation equals the rate of condensation, resulting in no net loss of liquid. Hence, statement 2 is incorrect.
• Scientific principles of thermodynamics confirm that the physical dimensions and exposure of the liquid surface are critical determinants of how fast a substance evaporates. Hence, statement 3 is incorrect.

Additional Information

• Factors Affecting Evaporation:
• Temperature: As the temperature of a liquid increases, the kinetic energy of its molecules also increases. More molecules acquire the energy needed to overcome intermolecular forces, thereby increasing the rate of evaporation.
• Humidity: Humidity is the measure of water vapor present in the air. If the atmospheric humidity is high, the air’s capacity to hold more moisture is limited, which decreases the rate at which a liquid evaporates.
• Wind Speed: An increase in wind speed helps in moving the vapor particles away from the surface of the liquid quickly. This prevents the air above the liquid from becoming saturated, thus speeding up evaporation.
• Nature of the Liquid: Different liquids evaporate at different rates based on their chemical properties. Volatile liquids like acetone or alcohol have weaker bonds between molecules compared to water, causing them to evaporate much faster.
• Practical Examples:
• Drying Clothes: We spread out wet clothes on a line to increase the surface area, which ensures they dry faster than if they were bunched up.
• Cooling in Saucers: Hot tea or coffee is often poured into a saucer to cool it down quickly. The larger surface area of the saucer promotes rapid evaporation, which is an endothermic process that carries away heat.

Important Points

• Molecular Mechanism: Unlike boiling, which is a bulk phenomenon occurring throughout the liquid, evaporation only involves molecules at the topmost layer. These surface molecules absorb energy from the surroundings to break free from the liquid phase.
• Cooling Effect: Evaporation always causes cooling. This happens because the high-energy molecules leave the liquid, reducing the average kinetic energy of the remaining molecules, which results in a drop in temperature.
• Latent Heat of Vaporization: The amount of heat energy required to change 1 kg of a liquid into gas at atmospheric pressure without a change in temperature is known as the latent heat of vaporization.

The correct answer is K > Mg > Al > Zn.

Key Points

• The Reactivity Series of metals, also known as the Activity Series, is an arrangement of metals in the descending order of their chemical reactivity. It helps in predicting the outcome of displacement reactions and the extraction methods for metals.
• Potassium (K) is an Alkali metal belonging to Group 1 of the periodic table. It is highly electropositive and is one of the most reactive metals, reacting vigorously with water and air. It occupies a position at the very top of the reactivity series.
• Magnesium (Mg) is an Alkaline earth metal in Group 2. While it is highly reactive, its ability to lose electrons is lower than that of Group 1 metals like Potassium. It reacts slowly with cold water but rapidly with boiling water or steam.
• Aluminium (Al) belongs to Group 13. It is less reactive than Magnesium. Although it is quite reactive, it often forms a thin, protective layer of Aluminium Oxide on its surface, which prevents further reaction. In the series, it follows Magnesium.
• Zinc (Zn) is a Transition metal. It is less reactive than Aluminium and Magnesium. It does not react with cold water but reacts with steam to form zinc oxide and hydrogen gas.
• Based on the tendency of these metals to lose electrons and form positive ions, the correct decreasing order of reactivity is Potassium (K) > Magnesium (Mg) > Aluminium (Al) > Zinc (Zn). Hence, the correct answer is K > Mg > Al > Zn.

Additional Information

• Displacement Reactions:
• A more reactive metal has the power to displace a less reactive metal from its salt solution. For example, if Magnesium is added to a Zinc Sulphate solution, it will displace Zinc because Mg is higher in the reactivity series.
• Conversely, Zinc cannot displace Magnesium from its salt solution because it is lower in the series.
• Reaction with Acids:
• Metals placed above Hydrogen in the reactivity series (like K, Mg, Al, and Zn) can displace Hydrogen gas from dilute hydrochloric acid or sulphuric acid.
• The rate of evolution of hydrogen bubbles decreases as we move from Potassium to Zinc in the series.
• Periodic Table Trends:
• In the periodic table, metallic character (reactivity) generally increases as we go down a group due to increasing atomic size and decreasing ionization energy.
• Reactivity generally decreases across a period from left to right as the effective nuclear charge increases, making it harder for atoms to lose electrons.

Important Points

• Storage of Potassium: Due to its extreme reactivity with oxygen and moisture in the air, Potassium (K) and Sodium (Na) are always stored under Kerosene to prevent accidental fires.
• Galvanization: Zinc (Zn) is used in the process of Galvanization. Since Zinc is more reactive than Iron (Fe), it acts as a sacrificial layer, corroding first and protecting the underlying iron from rusting.
• Reduction Methods:
• High-reactivity metals (K, Na, Ca, Mg, Al) are extracted via Electrolytic Reduction of their molten chlorides or oxides.
• Medium-reactivity metals (Zn, Fe, Pb) are usually extracted by reducing their oxides with Carbon (Coke).
• Amphoteric Nature: Both Aluminium and Zinc are special because they react with both acids and bases to produce hydrogen gas, showing amphoteric properties.

Shortcut Trick

Let the efficiency of the emptying hole be −3 units/hour (empties the tank of 3 units in 1 hour).

To maintain the level, 3 filling pipes are required ⇒ Efficiency of 3 filling pipes = +3 units/hour ⇒ Efficiency of 1 filling pipe = +1 unit/hour.

To fill the tank of 3 units in 3 hours ⇒ Required net efficiency = +1 unit/hour.

Net efficiency = (Number of filling pipes × 1) − 3 = 1 ⇒ Total filling pipes = 4.

Number of extra pipes required = 4 − 3 = 1 pipe.

∴ The correct answer is 1.

Alternate Method

Given:

Time taken by the emptying hole to empty the full tank = 1 hour

Number of filling pipes required to maintain the water level = 3

Formula Used:

Work = Efficiency × Time

Calculations:

Let the capacity of the tank be 3 units.

⇒ Efficiency of the emptying hole = −3 units/hour

To maintain the water level (net efficiency = 0):

⇒ Efficiency of 3 filling pipes − Efficiency of emptying hole = 0

⇒ Efficiency of 3 filling pipes = 3 units/hour

⇒ Efficiency of 1 filling pipe = 1 unit/hour

Let the total number of filling pipes required to fill the empty tank in 3 hours be n.

⇒ Net efficiency × Time = Total Capacity

⇒ (n × 1 − 3) × 3 = 3

⇒ n − 3 = 1

⇒ n = 4

⇒ More pipes required = 4 − 3 = 1

∴ The correct answer is 1.

Additional Information

Maintaining the Water Level

When the water level remains constant, the rate of inflow is exactly equal to the rate of outflow, resulting in zero net work done.

Inlet and Outlet Pipes

Inlet pipes perform positive work (fill the tank), while outlet pipes or leakage holes perform negative work (empty the tank).

Shortcut Trick Ratio of Cost Price (CP) to Selling Price (SP) = 200 : 280 = 5 : 7. Profit units = 7 − 5 = 2 units. Profit percentage = (Profit units ÷ CP units) × 100. Profit % = (2 ÷ 5) × 100 = 40%. ∴ The correct answer is 40%.

Alternate Method Given: Cost Price (CP) = ₹200, Selling Price (SP) = ₹280. Formulas: Profit = SP − CP | Profit % = (Profit ÷ CP) × 100

⇒ Profit = 280 − 200 = ₹80. ⇒ Profit % = (80 ÷ 200) × 100. ⇒ Profit % = 0.4 × 100 = 40%. ∴ The correct answer is 40%.

Additional Information Profit and Loss Basics Profit occurs when the Selling Price is greater than the Cost Price. It is always calculated on the Cost Price unless specified otherwise. Loss Percentage Loss occurs when CP > SP. Formula: Loss % = [(CP − SP) ÷ CP] × 100.

Shortcut Trick

Use identity: 64x² − 49 = (8x − 7)(8x + 7)

Cancel (8x − 7) ⇒ expression becomes 8x + 7 − 8x

Remaining term = 7

No need to expand fully

∴ Answer = 7

Alternate Method

Given:

(64x² − 49) ÷ (8x − 7) − 8x

Concept Used:

Algebraic identity and simplification

Calculation:

⇒ (64x² − 49) ÷ (8x − 7) − 8x

⇒ [(8x − 7)(8x + 7)] ÷ (8x − 7) − 8x

⇒ (8x + 7) − 8x

⇒ 7

∴ The correct answer is 7.

Additional Information

Difference of Squares

a² − b² = (a − b)(a + b)

Factorization

Break expressions into factors to simplify fractions

Cancellation Rule

Common factors in numerator and denominator can be cancelled

Quick Trick

Always check for identities before expanding expressions

Shortcut Trick

To find an equivalent ratio, multiply or divide both terms of the base ratio by the same non-zero number.

Given ratio = 5 : 3

Multiply both terms by 10 → (5 × 10) : (3 × 10) = 50 : 30.

∴ The correct answer is 50 : 30.

Alternate Method

Given: Base ratio = 5 : 3

Formula Used: Ratio a : b is equivalent to (a × k) : (b × k)

⇒ Option 1: 50 : 30 → Divide by 10 → 5 : 3 (Matches)

⇒ Option 2: 48 : 33 → Divide by 3 → 16 : 11 (Does not match)

⇒ Option 3: 53 : 31 → Simplest form is 53 : 31 (Does not match)

⇒ Option 4: 125 : 85 → Divide by 5 → 25 : 17 (Does not match)

∴ The correct answer is 50 : 30.

Additional Information

Equivalent Ratios

Ratios that express the same relationship between numbers. They can be obtained by multiplying or dividing both terms by the same constant k.

Simplest Form of Ratio

A ratio is in its simplest form when the HCF of both terms is 1. Example: 5 : 3 is the simplest form of 50 : 30.

Ratio Comparison

To compare two ratios a:b and c:d, convert them to fractions a/b and c/d and compare their values.

Shortcut Trick

Given cosec A = 5/4, so sin A = 4/5 and cos A = 3/5 (using 3-4-5 triplet).

sin 2A = 2 × sin A × cos A = 2 × (4/5) × (3/5) = 24/25.

cos 2A = 1 − 2sin²A = 1 − 2 × (16/25) = (25 − 32)/25 = −7/25.

Value = (24/25) + 2 × (−7/25) = (24 − 14)/25 = 10/25 = 2/5.

∴ The correct answer is 2/5.

Alternate Method

Given: cosec A = 5/4

Formula Used: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A

⇒ cosec A = 5/4 ⇒ sin A = 4/5

⇒ cos²A = 1 − sin²A = 1 − (4/5)² = 1 − 16/25 = 9/25

⇒ cos A = 3/5

⇒ sin 2A = 2 × (4/5) × (3/5) = 24/25

⇒ cos 2A = (3/5)² − (4/5)² = 9/25 − 16/25 = −7/25

⇒ sin 2A + 2cos 2A = 24/25 + 2 × (−7/25)

⇒ 24/25 − 14/25 = 10/25 = 2/5

∴ The correct answer is 2/5.

Additional Information

Double Angle Formulas

sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Pythagorean Triplets

For any right-angled triangle, Base² + Perpendicular² = Hypotenuse². Common triplets include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Tangent Double Angle

tan 2A = (2 tan A) / (1 − tan²A). Also, sin 2A = (2 tan A) / (1 + tan²A) and cos 2A = (1 − tan²A) / (1 + tan²A).

Shortcut Trick

Apply the British Flag Theorem for any point P related to rectangle ABCD:

PA² + PC² = PB² + PD²

87² + 96² = 12² + PD²

7569 + 9216 = 144 + PD²

16785 − 144 = PD² ⇒ PD = √16641 = 129 cm

∴ The correct answer is 129 cm.

Alternate Method

Given: PA = 87 cm, PB = 12 cm, PC = 96 cm.

Formula Used: For any point P inside a rectangle ABCD, the sum of the squares of the distances from P to the opposite vertices is equal: PA² + PC² = PB² + PD².

Calculations:

⇒ PA² + PC² = PB² + PD²

⇒ 87² + 96² = 12² + PD²

⇒ 7569 + 9216 = 144 + PD²

⇒ 16785 = 144 + PD²

⇒ PD² = 16785 − 144

⇒ PD² = 16641

⇒ PD = √16641

⇒ PD = 129 cm

∴ The correct answer is 129 cm.

Additional Information

British Flag Theorem

This theorem states that for any point P (inside, outside, or on the boundary) of a rectangle ABCD, the relation PA² + PC² = PB² + PD² always holds true.

Rectangle Properties

A rectangle is a quadrilateral with four right angles. Its diagonals are equal in length and bisect each other, which is the fundamental reason behind the British Flag Theorem derivation using Pythagoras Theorem.

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c².

Shortcut Trick

Recognize identity: (a³ − b³) ÷ (a² + ab + b²) = a − b

Here 0.1452 = 0.66 × 0.22 ⇒ denominator = a² + ab + b²

So expression = a − b

= 0.66 − 0.22 = 0.44

∴ Answer = 0.44

Alternate Method

Given:

((0.66)³ − (0.22)³) ÷ ((0.66)² + 0.1452 + (0.22)²)

Concept Used:

Identity: a³ − b³ = (a − b)(a² + ab + b²)

Calculation:

⇒ Let a = 0.66, b = 0.22

⇒ Check middle term: ab = 0.66 × 0.22 = 0.1452

⇒ So denominator = a² + ab + b²

⇒ Using identity:

⇒ a³ − b³ = (a − b)(a² + ab + b²)

⇒ Substitute:

⇒ Expression = [(a − b)(a² + ab + b²)] ÷ (a² + ab + b²)

⇒ Cancel common term

⇒ Expression = a − b

⇒ = 0.66 − 0.22

⇒ = 0.44

∴ The correct answer is 0.44.

Additional Information

Algebraic Identity

a³ − b³ = (a − b)(a² + ab + b²)

Pattern Recognition

Check if middle term equals product ab

Simplification Trick

Cancel common expressions to reduce computation

Speed Tip

Avoid expanding cubes when identity applies

Shortcut Trick

Net efficiency = Filling rate − Emptying rate

Total capacity = 16 units (LCM of 8 and 16)

Efficiency of Filling pipe = 2 and Emptying pipe = 1

Time to fill half tank (8 units) = 8 ÷ (2 − 1) = 8 minutes

∴ The correct answer is 8 minutes.

Alternate Method

Given: Filling time = 8 minutes, Emptying time = 16 minutes.

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

Calculations:

⇒ Total capacity of the tank = 16 units

⇒ Capacity to be filled (one-half) = 16 ÷ 2 = 8 units

⇒ Net efficiency when both pipes are open = 2 − 1 = 1 unit/min

⇒ Required time = 8 units ÷ 1 unit/min = 8 minutes

∴ The correct answer is 8 minutes.

Additional Information

Net Filling Rate

If pipe A fills in x hours and pipe B empties in y hours (where y > x), the net rate is (1/x − 1/y) per hour.

Pipes and Cisterns Logic

Filling pipes are considered positive efficiency, while emptying pipes (leaks) are considered negative efficiency.

Shortcut Trick

Area in 1 roll = 2 × (22/7) × 1 × (21/22) = 6 m².

Total effort in 600 rolls = 600 × 6 = 3600 m².

Given 10% overlap (rolled twice), Total Effort = 110% of Ground Area.

Total Ground Area = 3600 ÷ 1.1 = 3272.73 m².

∴ The correct answer is 3272.73 sq. m.

Alternate Method

Given:

Radius (r) of the road roller = 1 m

Length (h) of the road roller = 21/22 m

Number of rolls = 600

Condition: 10% of the ground is rolled twice.

Formula Used:

Curved Surface Area (CSA) of a cylinder = 2 × π × r × h

Calculations:

⇒ Area covered in 1 rotation (CSA) = 2 × (22/7) × 1 × (21/22)

⇒ CSA = 2 × (1/7) × 21 = 2 × 3 = 6 m²

⇒ Total coverage effort by 600 rolls = 600 × 6 = 3600 m²

⇒ Let the actual total ground area be A.

⇒ Total effort = (90% of A × 1 roll) + (10% of A × 2 rolls)

⇒ 3600 = 0.9A + 0.2A = 1.1A

⇒ A = 3600 ÷ 1.1 = 3272.7272...

⇒ A ≈ 3272.73 m²

∴ The correct answer is 3272.73 sq. m.

Additional Information

Curved Surface Area (CSA) of Cylinder

It is the area of the curved part of the cylinder, calculated as 2πrh.

Total Surface Area (TSA) of Cylinder

It includes the CSA and the area of both circular bases: 2πr(h + r).

Volume of Cylinder

The total space occupied by the cylinder: πr²h.

Shortcut Trick

Discount of 75% means the Selling Price is 25% of the Cost Price.

SP = 25% of 3,300 = 1/4 × 3,300.

SP = 825.

∴ The correct answer is ₹825.

Alternate Method

Given: Cost Price (CP) = ₹3,300, Discount Percentage = 75%.

Formula: Selling Price (SP) = CP − (Discount% × CP).

⇒ Discount Amount = 75/100 × 3,300

⇒ Discount Amount = 0.75 × 3,300 = ₹2,475

⇒ Selling Price (SP) = 3,300 − 2,475

⇒ Selling Price (SP) = ₹825

∴ The correct answer is ₹825.

Additional Information

Selling Price calculation from Discount

When discount is offered on the Cost Price (as per problem statement), SP = CP × (100 − Discount%)/100.

Relationship between CP and SP

If SP < CP, there is a loss. Loss = CP − SP.

Shortcut Trick

Combine ratios M : N = 6 : 8 and N : T = 9 : 6 by making N common.

M : N : T = (6 × 9) : (8 × 9) : (8 × 6) = 54 : 72 : 48 = 9 : 12 : 8.

Total ratio units = 9 + 12 + 8 = 29 units.

Share of N = (12 ÷ 29) × 3,596 = 12 × 124 = ₹1,488.

∴ The correct answer is ₹1,488.

Alternate Method

Given: Total amount = ₹3,596, M : N = 6 : 8, N : T = 9 : 6.

Formula Used: Combined Ratio a : b : c and Share = (Individual Ratio ÷ Sum of Ratios) × Total Amount.

⇒ M : N = 6 : 8 = 3 : 4

⇒ N : T = 9 : 6 = 3 : 2

⇒ To equalize N, multiply first ratio by 3 and second by 4:

⇒ M : N = 9 : 12 and N : T = 12 : 8

⇒ M : N : T = 9 : 12 : 8

⇒ Total Parts = 9 + 12 + 8 = 29

⇒ Share of N = (12 ÷ 29) × 3,596

⇒ Share of N = 12 × 124 = 1,488

∴ The correct answer is ₹1,488.

Additional Information

Ratio Balancing

To combine A : B and B : C, multiply the terms so that the value of B becomes the LCM of its values in both ratios.

Compound Ratio

The compound ratio of a : b and c : d is (a × c) : (b × d).

Division of Proportions

If an amount X is divided in ratio a : b : c, the share of the second person is [b ÷ (a + b + c)] × X.

Shortcut Trick

Use the multiplying factor method for successive changes.

Final Price = 100 × 0.65 × 0.40 × 0.90 × 1.65

Final Price = 100 × 0.3861 = 38.61

Net change = 38.61 − 100 = −61.39%

∴ The correct answer is Decreases by 61.39%.

Alternate Method

Assume original price is 100 units.

⇒ Price after 1st month = 100 − 35% of 100 = 65

⇒ Price after 2nd month = 65 − 60% of 65 = 65 × 0.40 = 26

⇒ Price after 3rd month = 26 − 10% of 26 = 26 × 0.90 = 23.4

⇒ Price after 4th month = 23.4 + 65% of 23.4 = 23.4 × 1.65 = 38.61

⇒ Net change = Final Price − Original Price = 38.61 − 100 = −61.39

⇒ Percentage change = (−61.39 ÷ 100) × 100 = −61.39%

∴ The correct answer is Decreases by 61.39%.

Additional Information

Successive Percentage Formula

For two changes a% and b%, the net change is (a + b + ab/100)%. For more than two, calculate for the first two and then combine the result with the next.

Multiplying Factor (MF)

Increase by x% means MF = (1 + x/100). Decrease by x% means MF = (1 − x/100). Final value = Initial value × MF₁ × MF₂ × ... × MF_n.

Percentage Decrease vs Increase

If the final value is less than the initial value, it is a decrease. Percentage Change = [(Final − Initial) ÷ Initial] × 100.

Shortcut Trick

Assume a base mean of 40 and calculate deviations from it.

Deviations: (35−40), (26−40), (56−40), (41−40), (56−40), (46−40), (34−40), (16−40), (68−40)

⇒ −5, −14, +16, +1, +16, +6, −6, −24, +28

Net Deviation = −5 − 14 + 16 + 1 + 16 + 6 − 6 − 24 + 28 = +18

Actual Mean = Assumed Mean + (Net Deviation ÷ Total Count) = 40 + (18 ÷ 9) = 42.

∴ The correct answer is 42.

Alternate Method

Given: Observations are 35, 26, 56, 41, 56, 46, 34, 16, and 68.

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

⇒ Total number of observations (n) = 9

⇒ Sum of observations = 35 + 26 + 56 + 41 + 56 + 46 + 34 + 16 + 68

⇒ Sum = 378

⇒ Arithmetic Mean = 378 ÷ 9

⇒ Arithmetic Mean = 42

∴ The correct answer is 42.

Additional Information

Properties of Arithmetic Mean

The sum of deviations of a set of observations from their arithmetic mean is always zero (∑(x_i − x̄) = 0).

Combined Mean

If two groups have sizes n₁, n₂ and means x̄₁, x̄₂, the combined mean is (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).

Effect of Operations

If each observation is increased, decreased, multiplied, or divided by a constant k, the mean also changes by the same constant k.

Shortcut Trick

Relative Speed (Same direction) = 93 − 33 = 60 kmph

Convert to m/s: 60 × (5/18) = 50/3 m/s

Total Distance to cover = Sum of lengths = 163 + 117 = 280 m

Time Taken = Distance ÷ Relative Speed = 280 ÷ (50/3) = 28 × 0.6 = 16.8 sec

∴ The correct answer is 16.8 sec.

Alternate Method

Given:

Length of Train 1 (L₁) = 163 m

Length of Train 2 (L₂) = 117 m

Speed of Faster Train (S₁) = 93 km/hr

Speed of Slower Train (S₂) = 33 km/hr

Formula Used:

Time = (L₁ + L₂) ÷ (Relative Speed)

Calculations:

⇒ Relative Speed = S₁ − S₂ (Same direction)

⇒ Relative Speed = 93 − 33 = 60 km/hr

⇒ Relative Speed in m/s = 60 × (5/18) = 50/3 m/s

⇒ Total Distance = 163 + 117 = 280 m

⇒ Time taken = 280 ÷ (50/3)

⇒ Time taken = (280 × 3) ÷ 50 = 840 ÷ 50

⇒ Time taken = 16.8 sec

∴ The correct answer is 16.8 sec.

Additional Information

Relative Speed Concepts

When two objects move in the same direction, Relative Speed = S₁ − S₂. When they move in opposite directions, Relative Speed = S₁ + S₂.

Speed 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 Distance

Whether moving in the same or opposite direction, the total distance covered to cross each other is always the sum of their lengths (L₁ + L₂).

Shortcut Trick

Net Profit/Loss% = a + b + (a × b) ÷ 100

Here, a = +50 (Mark-up) and b = −52 (Discount).

Net Change = 50 − 52 + (50 × (−52)) ÷ 100

Net Change = −2 − (2600 ÷ 100) = −2 − 26 = −28%.

Negative sign indicates a loss of 28%.

∴ The correct answer is 28%.

Alternate Method

Given → Mark-up = 50% above CP, Discount = 52% on MP. Formulas: MP = CP × (100 + Mark-up%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = (CP − SP) ÷ CP × 100

⇒ Let Cost Price (CP) = 100. ⇒ Marked Price (MP) = 100 + 50% of 100 = 150. ⇒ Selling Price (SP) = 150 − 52% of 150 = 150 − (52 × 1.5) = 150 − 78 = 72. ⇒ Loss = CP − SP = 100 − 72 = 28. ⇒ Loss percentage = (28 ÷ 100) × 100 = 28%.

∴ The correct answer is 28%.

Additional Information

Relation between CP and MP

The ratio of CP to MP can be expressed as: CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%).

Effective Change Formula

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

Shortcut Trick

Divide each dimension of the cuboid by the side of the cube to find the number of cubes along each edge.

Number of cubes = (60 ÷ 5) × (45 ÷ 5) × (30 ÷ 5)

Number of cubes = 12 × 9 × 6 = 648

∴ The correct answer is 648 cubes.

Alternate Method

Given:

Length (L) of cuboid = 60 cm

Breadth (B) of cuboid = 45 cm

Height (H) of cuboid = 30 cm

Side (a) of smaller cube = 5 cm

Formula Used:

Number of cubes = (Volume of cuboid) ÷ (Volume of one cube)

Volume of cuboid = L × B × H

Volume of cube = a³

Calculations:

⇒ Volume of cuboid = 60 × 45 × 30 = 81,000 cm³

⇒ Volume of one cube = 5 × 5 × 5 = 125 cm³

⇒ Number of cubes = 81,000 ÷ 125

⇒ Number of cubes = 648

∴ The correct answer is 648 cubes.

Additional Information

Total Surface Area of a Cuboid

The total surface area is given by the formula 2(LB + BH + HL).

Diagonal of a Cuboid

The longest rod that can be placed inside a cuboid is its diagonal, calculated as √(L² + B² + H²).

Volume Conservation

When a solid is melted or cut into smaller pieces, the total volume remains constant, provided there is no wastage.

Shortcut Trick

Summing equations: 5 pens + 5 pencils = 75 ⇒ 1 pen + 1 pencil = 15

Subtracting equations: 1 pen − 1 pencil = 5

Solving for pencil: (15 − 5) ÷ 2 = 5

Cost of 3 pencils = 3 × 5 = 15

∴ The correct answer is ₹15.

Alternate Method

Given:

Cost of 2 pens and 3 pencils = ₹35

Cost of 3 pens and 2 pencils = ₹40

Method: Linear Equation Elimination

Let the cost of one pen be x and one pencil be y.

⇒ 2x + 3y = 35 --- (Equation 1)

⇒ 3x + 2y = 40 --- (Equation 2)

Multiply Equation 1 by 3 and Equation 2 by 2 to eliminate x:

⇒ 6x + 9y = 105 --- (Equation 3)

⇒ 6x + 4y = 80 --- (Equation 4)

Subtract Equation 4 from Equation 3:

⇒ (6x − 6x) + (9y − 4y) = 105 − 80

⇒ 5y = 25

⇒ y = 25 ÷ 5 = 5

⇒ Cost of 3 pencils = 3y = 3 × 5 = 15

∴ The correct answer is ₹15.

Additional Information

Elimination Method

A process of solving simultaneous equations by adding or subtracting them to remove one variable, usually after multiplying by a constant.

Substitution Method

Solving one equation for one variable (e.g., x = 15 − y) and substituting it into the other equation to find the value of the second variable.

Consistent Systems

A system of linear equations that has at least one solution (intersecting lines) is called a consistent system.

Shortcut Trick

The Empirical Relationship between the measures of central tendency is given by:

Mode = 3 × Median − 2 × Mean

⇒ Mode = 3(70) − 2(40) = 210 − 80 = 130

∴ The correct answer is 130.

Alternate Method

Given:

⇒ Mean = 40

⇒ Median = 70

Formula Used:

⇒ Mode = 3 × Median − 2 × Mean

Calculations:

⇒ Substituting the given values in the formula:

⇒ Mode = 3 × (70) − 2 × (40)

⇒ Mode = 210 − 80

⇒ Mode = 130

∴ The correct answer is 130.

Additional Information

Symmetrical Distribution

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

Positively Skewed Distribution

When the tail of the distribution is longer on the right side, it is positively skewed. In this case, Mean > Median > Mode.

Negatively Skewed Distribution

When the tail is longer on the left side, it is negatively skewed. In this case, Mean < Median < Mode (as seen in this problem).

Memory Trick

Always remember the coefficients by the alphabetical order: Mode = 3 × Median − 2 × Mean (Coefficients 3 and 2 follow the word length of Median and Mean).

Shortcut Trick

Let number = x

20% of x + 96 = x ⇒ 0.2x + 96 = x

⇒ 0.8x = 96 ⇒ x = 120

80% of x = 0.8 × 120

∴ Answer = 96

Alternate Method

Given:

20% of a number added to 96 equals the number

Formula used:

Percentage = (Part ÷ Whole) × 100

Calculation:

⇒ Let number = x

⇒ 20% of x = (20/100) × x = 0.2x

⇒ According to question:

⇒ 0.2x + 96 = x

⇒ x − 0.2x = 96

⇒ 0.8x = 96

⇒ x = 96 ÷ 0.8 = 120

⇒ 80% of x = 0.8 × 120

⇒ = 96

∴ The correct answer is 96.

Additional Information

Percentage Conversion

x% = x/100

Equation Formation

Translate word problems into algebraic equations

Shortcut Insight

If 20% + 96 = 100%, then 80% = 96

Direct Logic

Remaining percentage gives direct value

Shortcut Trick

Total percentage spent = 20% + 35% + 25% = 80%

Savings percentage = 100% − 80% = 20%

Total Savings = 20% of ₹1,20,000

⇒ 0.20 × 1,20,000 = ₹24,000

∴ The correct answer is ₹24,000.

Alternate Method

Given:

Total Income = ₹1,20,000

Expenditure on Accommodation = 20%

Expenditure on Household = 35%

Expenditure on Other Activities = 25%

Calculations:

⇒ Total Expenditure = 20% + 35% + 25%

⇒ Total Expenditure = 80%

⇒ Total Savings % = 100% − 80% = 20%

⇒ Value of Savings = (20 / 100) × 1,20,000

⇒ Value of Savings = 0.2 × 1,20,000

⇒ Value of Savings = ₹24,000

∴ The correct answer is ₹24,000.

Additional Information

Percentage Change

Percentage change is calculated as (New Value − Old Value) / Old Value × 100.

Successive Percentage

If expenses were 'of the remaining amount', we use the formula: Final = Initial × (1 − r₁/100) × (1 − r₂/100).

Base Change Rule

If A is x% more than B, then B is [x / (100 + x)] × 100% less than A.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 80 + 30 = 110 km

Total Time = (80 ÷ 40) + (30 ÷ 10) = 2 + 3 = 5 hours

Average Speed = 110 ÷ 5 = 22 km/hr

∴ The correct answer is 22 km/hr.

Alternate Method

Given:

Distance 1 (d₁) = 80 km, Speed 1 (s₁) = 40 km/hr

Distance 2 (d₂) = 30 km, Speed 2 (s₂) = 10 km/hr

Formula Used:

Average Speed = (d₁ + d₂) ÷ (t₁ + t₂)

⇒ Time taken for first part (t₁) = Distance ÷ Speed = 80 ÷ 40 = 2 hours

⇒ Time taken for second part (t₂) = Distance ÷ Speed = 30 ÷ 10 = 3 hours

⇒ Total Distance = 80 + 30 = 110 km

⇒ Total Time = 2 + 3 = 5 hours

⇒ Average Speed = 110 ÷ 5 = 22 km/hr

∴ The correct answer is 22 km/hr.

Additional Information

Average Speed (Equal Distances)

When a journey is divided into two equal distances with speeds u and v, the average speed is 2uv ÷ (u + v).

Relative Speed

For objects moving in the same direction, relative speed is the difference (u − v). In the opposite direction, it is the sum (u + v).

Conversion Factor

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

Shortcut Trick

Rate difference between Bank B and Bank A = 7.5% − 3.5% = 4% per annum.

Total difference in interest over 2 years = 4% × 2 = 8%.

Interest difference = 8% of Principal (₹2,40,000).

Calculation: 2,40,000 × (8 ÷ 100) = 2,400 × 8 = ₹19,200.

∴ The correct answer is ₹19,200.

Alternate Method

Given: Principal (P) = ₹2,40,000, Rate A (R₁) = 3.5%, Rate B (R₂) = 7.5%, Time (T) = 2 years.

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

⇒ SI paid to Bank A = (2,40,000 × 3.5 × 2) ÷ 100

⇒ SI paid to Bank A = 2,400 × 7 = ₹16,800

⇒ SI paid to Bank B = (2,40,000 × 7.5 × 2) ÷ 100

⇒ SI paid to Bank B = 2,400 × 15 = ₹36,000

⇒ Difference = 36,000 − 16,800

⇒ Difference = ₹19,200

∴ The correct answer is ₹19,200.

Additional Information

Simple Interest Formula

The basic formula for Simple Interest is SI = (P × R × T) ÷ 100, where P is Principal, R is rate per annum, and T is time in years.

Interest Difference Shortcut

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

Simple vs Compound Interest

In SI, interest is calculated only on the initial principal, whereas in CI, interest is calculated on the principal plus the accumulated interest of previous periods.

Shortcut Trick

Solve brackets first: (18 + 6) = 24 and (8 − 3) = 5

Expression becomes: [24 ÷ 3 × 2 − 4] × [5 + 9] − 2 × 5

= [8 × 2 − 4] × 14 − 10

= [16 − 4] × 14 − 10 = 12 × 14 − 10

∴ Answer = 158

Alternate Method

Given:

[(18 + 6) ÷ 3 × 2 − 4] × [5 + 3²] − 2 × (8 − 3)

Concept Used:

BODMAS rule

Calculation:

⇒ [(18 + 6) ÷ 3 × 2 − 4] × [5 + 3²] − 2 × (8 − 3)

⇒ [24 ÷ 3 × 2 − 4] × [5 + 3²] − 2 × (8 − 3)

⇒ [24 ÷ 3 × 2 − 4] × [5 + 9] − 2 × (8 − 3)

⇒ [8 × 2 − 4] × 14 − 2 × (8 − 3)

⇒ [16 − 4] × 14 − 2 × 5

⇒ 12 × 14 − 2 × 5

⇒ 168 − 2 × 5

⇒ 168 − 10

⇒ 158

∴ The correct answer is 158.

Additional Information

BODMAS Full Form

Brackets, Orders, Division, Multiplication, Addition, Subtraction

Order of Operations

Solve brackets first, then powers, then ÷ and ×, then + and −

Left to Right Rule

For × and ÷, solve from left to right

Simplification Tip

Always simplify step-by-step without skipping operations

Shortcut Trick

Ratio of current ages of Elder son : Younger son = 3 : 1

Let current ages be 3x and x. After 5 years: (3x + 5) = 2 × (x + 5)

3x + 5 = 2x + 10 ⇒ x = 5

Elder son's age = 3 × 5 = 15 years

Mother's age = Elder son + 20 = 15 + 20 = 35 years

∴ The correct answer is 35 years.

Alternate Method

Given: Current age ratio = 3 : 1 | Mother = Elder + 20 | After 5 yrs: Elder = 2 × Younger

Let current age of Younger son = x

⇒ Current age of Elder son = 3x

⇒ After 5 years, Elder son = 3x + 5 and Younger son = x + 5

⇒ 3x + 5 = 2(x + 5)

⇒ 3x + 5 = 2x + 10

⇒ x = 5

⇒ Current age of Elder son = 3 × 5 = 15 years

⇒ Current age of Mother = 15 + 20 = 35 years

∴ The correct answer is 35 years.

Additional Information

Age Difference Property

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

Ratio Balancing

To solve age problems involving two different ratios at two different times, ensure the unit difference between the individuals remains the same.

Linear Equations in Ages

Always define the present age as the base variable 'x' to avoid confusion when moving forward (x + n) or backward (x − n) in time.

Shortcut Trick

Divide 1000 by 45 to find the remainder.

1000 ÷ 45 ⇒ Quotient = 22, Remainder = 10.

Smallest number to be added = Divisor − Remainder.

Required number = 45 − 10 = 35.

∴ The correct answer is 35.

Alternate Method

Given: Number = 1000, Divisor = 45.

Formula Used: Number to be added = (Divisor − Remainder)

Performing division of 1000 by 45:

⇒ 1000 = 45 × 22 + 10

⇒ Here, Remainder (R) = 10

⇒ Divisor (D) = 45

⇒ Smallest number to be added = D − R

⇒ 45 − 10 = 35

∴ The correct answer is 35.

Additional Information

Division Algorithm

Dividend = (Divisor × Quotient) + Remainder. The remainder is always less than the divisor.

Smallest number to subtract

To make a number exactly divisible by another, the smallest number to be subtracted is simply the Remainder.

Divisibility Rule of 45

A number is divisible by 45 if it is simultaneously divisible by both 5 (ends in 0 or 5) and 9 (sum of digits is divisible by 9).

The position and positional values or English Alphabets are:

Given: WPGB is related to AKKW in a certain way based on the English alphabetical order. In the same way, MVWH is related to QQAC. 4

The logic followed here is:

Similarly,

Thus, YGIS is related to CBMN, following the same logic.

Hence, the correct answer is "Option 1".

BODMAS Rule:

Decoding the given information:

Given equation: 441 D 444 C 21 A 21 B 441 = ?

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

⇒ 441 − 444 + 21 ÷ 21 × 441 = ?

⇒ 441 − 444 + 1 × 441 = ?

⇒ 441 − 444 + 441 = ?

⇒ 882 − 444 = ?

⇒ 438 = ?

Hence, the correct answer is "Option 4".

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

Now,

Odd digits, each of which is immediately preceded by an even digit and also immediately followed by an odd digit.

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

Thus, there are 3 odd digits, each of which is immediately preceded by an even digit and also immediately followed by an odd digit.

Hence, the correct answer is "Option 3".

Given: Seven boxes M, N, O, P, Q, R and S are kept one over the other but not necessarily in the same order.

1. Only two boxes are kept between M and O.
2. Only N is kept above S.
3. No box is kept below O.
4. R is kept at some place below P but at some place above Q. 

Thus, Box P is kept third above Q.

Hence, the correct answer is "Option 2".

Given Table:

According to the given conditions:

Given: % 5 * 7 $

Here, Rule (iii) applies because the first element is a symbol (%) and the fourth element is an odd number (7).

% → coded as ©

5 → coded as J

• → coded as F

7 → coded as ©

$ → coded as E

So, % 5 * 7 $ → Coded as ©JF©E

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: QCW, SDU, UES, WFQ, ?

The logic followed here is:

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

Hence, the correct answer is "Option 2".

Given: Seven people B, C, D, I, J, K, and L are sitting around a circular table facing the centre of the table.

1. L sits third to the left of I.
2. Only one person sits between K and B, when counted from the left of B.
3. C sits third to the right of D.
4. C sits to the immediate right of L.
5. J is not an immediate neighbour of L.

Thus, three people sit between J and B, when counted from the right of J.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: VK 12 is related to ZP 16 in a certain way. In the same way, RG 15 is related to VL 19. 

The logic followed here is:

Similarly,

Thus, MB 24 is related to QG 28, following the same logic.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: CER121, IKX98, OQD75, UWJ52, ?

The logic followed here is:

Thus, ACP29 will replace the question mark (?) in the given series to make it logically complete.

Hence, the correct answer is "Option 1".