RRB Group D Official Paper (Held On: 19 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is 49.

Key Points

• The resistance of a conductor is directly proportional to its length (L) and inversely proportional to its area of cross-section (A). The formula is given by R = ρ(L/A), where ρ is the resistivity of the material.
• When a wire of total resistance R is cut into 7 equal parts, the length of each part becomes 1/7th of the original length. Since the material and area of cross-section remain the same, the resistance of each small part (let it be r) is r = R / 7.
• In a parallel combination, the reciprocal of the equivalent resistance (R') is the sum of the reciprocals of the individual resistances. Since all 7 parts are identical, the formula becomes 1/R' = 1/r + 1/r + 1/r + 1/r + 1/r + 1/r + 1/r.
• Substituting the value of r, we get 1/R' = 7 / (R/7), which simplifies to 1/R' = 49 / R.
• To find the ratio R / R', we rearrange the equation: R / R' = 49. Hence, the calculated ratio is 49.

Additional Information

• Factors affecting Resistance: The resistance of a wire depends on the nature of the material, the temperature, the length, and the cross-sectional area. If the temperature is constant, resistance is solely determined by dimensions and resistivity.
• Series Connection: If these 7 parts were connected in series instead of parallel, the equivalent resistance would be the sum of individual resistances (R/7 + R/7 + ... 7 times), which would result back in the original resistance R.
• Parallel Connection Features: In a parallel circuit, the voltage across each resistor remains the same, while the total current is the sum of the currents through each branch. This connection effectively increases the total area of cross-section, thereby reducing the equivalent resistance.
• Resistivity (ρ): It is an intrinsic property of the material. Cutting the wire into pieces does not change its resistivity, as it only depends on the material type (e.g., copper, aluminum) and temperature.
• Ohm's Law: This law states that the current through a conductor between two points is directly proportional to the voltage across the two points, provided physical conditions remain constant (V = IR).

Important Points

• Unit of Resistance: The SI unit of resistance is the Ohm, represented by the Greek letter Ω.
• Mathematical Generalization: If a wire of resistance R is cut into n equal parts and connected in parallel, the new resistance R' will always be R / (n^2). In this specific question, n = 7, so R' = R / 49.
• Ratio Analysis: The ratio R / R' represents how many times the resistance has decreased. A ratio of 49 indicates a significant drop in resistance, which is typical when moving from a single long wire to multiple parallel paths.
• Applications: This principle is used in power distribution and household wiring, where appliances are connected in parallel to ensure low equivalent resistance and independent operation.

The correct answer is SO.

Key Points

• Washing Soda is a chemical compound with the formula Na₂CO₃.10H₂O, which is chemically known as Sodium Carbonate Decahydrate.
• The elements that constitute washing soda are Sodium, Carbon, Oxygen, and Hydrogen.
• The symbol for Sodium is Na. This is derived from its Latin name, Natrium. Hence, the symbol given in option 4 is correct.
• The symbol for Carbon is C. It is a non-metallic element belonging to Group 14 of the Periodic Table. Hence, the symbol given in option 2 is correct.
• The symbol for Oxygen is O. It is a highly reactive non-metal and oxidizing agent. Hence, the symbol given in option 1 is correct.
• The symbol SO is not an elemental symbol. S represents Sulfur and O represents Oxygen. Together they represent Sulfur Monoxide, which is not an element present in the standard chemical description of individual symbols for washing soda's components. Hence, option 3 is the incorrect symbol.

Additional Information

• Chemical Composition: Washing Soda contains two atoms of Sodium, one atom of Carbon, and three atoms of Oxygen in its anhydrous form (Soda Ash), along with ten molecules of water of crystallization.
• Production: It is predominantly manufactured using the Solvay Process (also known as the ammonia-soda process), which uses Sodium Chloride, Ammonia, and Limestone as raw materials.
• Physical Properties: It is a white, crystalline solid that is soluble in water. It undergoes efflorescence, where it loses water molecules when exposed to dry air to become Sodium Carbonate Monohydrate.
• Applications:
• It is extensively used in glass, soap, and paper industries.
• It is used for removing the permanent hardness of water.
• It serves as a cleansing agent for domestic purposes and as a laboratory reagent.
• Atomic Details:
• Sodium (Na): Atomic number 11, located in Period 3.
• Carbon (C): Atomic number 6, located in Period 2.
• Oxygen (O): Atomic number 8, located in Period 2.

Important Points

• Differentiation: It is important to distinguish Washing Soda from Baking Soda. While Washing Soda is Sodium Carbonate (Na₂CO₃), Baking Soda is Sodium Bicarbonate (NaHCO₃).
• Nature: Sodium Carbonate is a basic salt because it is formed by the reaction of a strong base (Sodium Hydroxide) and a weak acid (Carbonic Acid).
• Historical Context: Historically, it was extracted from the ashes of plants growing in sodium-rich soils, which is why the anhydrous form is still commonly referred to as Soda Ash.

The correct answer is Mass is the amount of matter in an object.

Key Points

• Mass is defined as the fundamental measure of the amount of matter contained within an object. It is an intrinsic property that does not change regardless of the object's position in the universe. Hence, statement 2 is correct.
• Weight is the force exerted on an object due to gravity. It is calculated using the formula W = m × g, where 'm' is mass and 'g' is the acceleration due to gravity. Therefore, weight is directly dependent on the gravitational pull of the location. Hence, statement 1 is incorrect.
• The SI unit of mass is the kilogram (kg), while the SI unit of weight is the Newton (N) because weight is a measure of force. Since they represent different physical quantities, their units are not the same. Hence, statement 3 is incorrect.
• The mass of an object remains constant throughout the universe unless matter is physically added to or removed from the object. It does not fluctuate based on altitude, latitude, or different celestial bodies. Hence, statement 4 is incorrect.
• While mass is a scalar quantity (having only magnitude), weight is a vector quantity because it has both magnitude and a direction pointing towards the center of gravity.

Additional Information

• Gravitational Variations:
• The acceleration due to gravity (g) on Earth is approximately 9.8 m/s². However, on the Moon, gravity is about 1/6th of that on Earth.
• Consequently, an object with a mass of 60 kg will have a weight of 588 N on Earth but only about 98 N on the Moon, even though its mass remains exactly 60 kg.
• Measurement Tools:
• Mass is typically measured using a physical balance or a beam balance, which compares an unknown mass with a known standard mass.
• Weight is measured using a spring balance, which measures the extension of a spring caused by the gravitational pull on the object.
• Inertia:
• Mass is also a measure of inertia, which is the resistance of an object to any change in its state of motion. The greater the mass, the higher the inertia, making it harder to accelerate or stop the object.

Important Points

• Center of Gravity: This is the point through which the entire weight of a body acts. For a regular-shaped object with uniform density, this point usually coincides with its geometric center.
• Zero Weight: An object can experience weightlessness (zero weight) in free fall or in deep space far from any celestial body, but its mass can never be zero as long as it exists as matter.
• Relativistic Mass: According to Albert Einstein's theory of relativity, the mass of an object can appear to increase when it travels at speeds approaching the speed of light, though for all general science and classical mechanics applications, mass is considered constant.

The correct answer is the same type of atoms.

Key Points

• Molecules of an element are constituted by the same type of atoms. This means that all atoms within the molecule have the same atomic number and the same number of protons in their nuclei.
• A molecule is the smallest unit of a substance (element or compound) that can exist independently and retain the chemical properties of that substance.
• In the case of elements, atoms of the same element chemically bond together to form a molecule. For example, two atoms of Hydrogen combine to form a Hydrogen molecule (H2). Hence, the statement regarding the same type of atoms is correct.
• Molecules of compounds, on the other hand, are formed by the different types of atoms (atoms of different elements) joining together in fixed proportions, such as Water (H2O) or Carbon Dioxide (CO2).
• The number of atoms constituting a molecule is known as its atomicity. Depending on the element, the molecule may contain one, two, or more identical atoms.

Additional Information

• Classification of Molecules based on Atomicity:
• Monoatomic Molecules: These consist of a single atom. Most Noble Gases like Helium (He), Neon (Ne), and Argon (Ar) exist in this form because they are chemically unreactive.
• Diatomic Molecules: These are composed of two atoms of the same element. Common examples include Oxygen (O2), Nitrogen (N2), Hydrogen (H2), and Chlorine (Cl2).
• Triatomic Molecules: These consist of three atoms. A prominent example is Ozone (O3), which is an allotrope of oxygen.
• Polyatomic Molecules: These contain more than three atoms. For instance, Phosphorus exists as P4 (tetraatomic) and Sulphur typically exists as S8 (octatomic).
• Chemical Bonding: The atoms in a molecule are held together by strong chemical bonds, primarily covalent bonds in the case of non-metal elemental molecules, where electrons are shared between the atoms.
• Allotropes: Some elements can exist in two or more different physical forms in the same state, known as allotropes. For example, Carbon exists as Diamond, Graphite, and Fullerenes, all of which are made of the same type of atoms but arranged differently.
• Historical Context: John Dalton proposed the Atomic Theory in 1808, stating that elements consist of indivisible particles called atoms. Later, Amedeo Avogadro clarified the distinction between atoms and molecules in 1811.

Important Points

• It is important to distinguish between an atom (the smallest particle of an element) and a molecule (the smallest particle of a substance capable of independent existence).
• The molecular mass of an element is calculated by multiplying the atomic mass of the atom by its atomicity. For example, the molecular mass of O2 is 16 x 2 = 32 u.
• While molecules of elements consist of the same type of atoms, the physical and chemical properties of the molecule can differ significantly from the individual atoms (e.g., individual oxygen atoms are highly reactive radicals, while O2 is a stable gas).
• Metals and some other elements do not have a simple molecular structure but consist of a giant lattice of atoms held together; however, in chemical calculations, they are often treated as monoatomic units.

The correct answer is Aperture.

Key Points

• The Aperture of a spherical mirror refers to the diameter of the reflecting curved surface. It represents the effective width of the mirror that is available for reflecting light rays.
• Physically, the aperture determines the amount of light that falls on the mirror and is subsequently reflected. A mirror with a larger aperture collects more light, which results in the formation of a brighter image compared to a mirror with a smaller aperture.
• In most cases, the reflecting surface of a spherical mirror has a circular boundary, and the aperture is simply the diameter of this circular outline.
• It is important to distinguish aperture from the radius of curvature. While the radius of curvature relates to the depth or steepness of the curve, the aperture relates to the linear size of the opening of the mirror.
• In optics, a "small aperture" mirror is one where the diameter of the reflecting surface is much smaller than its radius of curvature. This condition is crucial for ensuring that paraxial rays form a sharp focus.

Additional Information

• Pole (P): The geometric center of the reflecting surface of a spherical mirror is called its Pole. It is a point that lies directly on the surface of the mirror.
• Center of Curvature (C): A spherical mirror is considered a part of a hollow sphere. The center of this original sphere is called the Center of Curvature. Note that this point lies outside the reflecting surface of the mirror.
• Radius of Curvature (R): The distance between the Pole (P) and the Center of Curvature (C) is the Radius of Curvature. It is the radius of the sphere of which the mirror forms a part.
• Principal Axis: This is an imaginary straight line passing through the Pole and the Center of Curvature of the spherical mirror. It serves as the reference line for measuring distances in the Cartesian sign convention.
• Principal Focus (F): For a concave mirror, it is the point on the principal axis where rays parallel to the axis converge after reflection. For a convex mirror, it is the point from which such rays appear to diverge.

Important Points

• Focal Length (f): The distance between the Pole (P) and the Principal Focus (F) is called the Focal Length. For mirrors with a small aperture, there is a mathematical relationship: R = 2f, meaning the radius of curvature is twice the focal length.
• Concave Mirror: Often called a converging mirror, its reflecting surface is curved inwards. It can form both real and virtual images depending on the object's position. It is used in headlights, shaving mirrors, and by dentists.
• Convex Mirror: Known as a diverging mirror, its reflecting surface is curved outwards. It always forms a virtual, erect, and diminished image, providing a wider field of view. It is primarily used as a rear-view mirror in vehicles.
• Mirror Formula: The relationship between the object distance (u), image distance (v), and focal length (f) is given by the equation: 1/v + 1/u = 1/f.

The correct answer is To form a protective oxide layer.

Key Points

• Anodising is a sophisticated electrolytic process used to increase the thickness of the natural oxide layer on the surface of metal parts, most commonly aluminium.
• During this process, a clean aluminium article is made the anode in an electrolytic bath, usually containing an acid such as dilute sulphuric acid.
• When an electric current is passed through the electrolyte, oxygen gas is evolved at the anode, which reacts with the aluminium to form a thick, uniform, and protective layer of aluminium oxide (Al2O3).
• The primary objective of this process is to provide corrosion resistance and enhance the durability of the metal by creating a barrier against environmental factors. Hence, the correct answer is to form a protective oxide layer.
• Unlike paint or plating, this oxide layer is fully integrated with the underlying aluminium substrate, so it cannot chip or peel off.

Additional Information

• Aesthetic and Decorative Uses:
• The anodic oxide layer has a porous structure immediately after the process. These pores can be used to absorb dyes or pigments.
• This allows aluminium products to be finished in a wide variety of colors while retaining the metallic texture, which is common in consumer electronics and cookware.
• Electrical Properties:
• While aluminium is an excellent conductor of electricity, aluminium oxide is an insulator.
• Therefore, anodising actually decreases the surface electrical conductivity, which can be useful in specific engineering applications where insulation is required.
• Surface Hardness:
• The anodised layer is significantly harder than the base metal. This provides excellent abrasion resistance and wear resistance, extending the lifespan of components used in machinery and transportation.
• Adhesion:
• The process provides a much better adhesion surface for paints, primers, and glues compared to bare metal, making it a critical pre-treatment step in many manufacturing workflows.

Important Points

• Common Applications: Anodised aluminium is extensively used in aerospace for aircraft skins, architecture for window frames and roofs, and in automotive parts for its lightweight and rust-proof nature.
• Maintenance: Because the oxide layer is chemically stable, anodised surfaces require very low maintenance and are resistant to UV radiation, meaning the color does not fade easily over years of exposure.
• Environmental Safety: Anodising is considered an environmentally safe metal finishing process as it does not involve heavy metals like lead or cadmium, and the chemical by-products are often neutralized before disposal.
• Comparison: While aluminium forms a natural oxide film when exposed to air, this natural layer is very thin (approx 2-3 nanometers). Industrial anodising can increase this thickness to 5 to 25 micrometers or more, depending on the application.

The correct answer is Advanced sunrise and delayed sunset is due to atmospheric refraction.

Key Points

• Atmospheric Refraction is the phenomenon responsible for both advanced sunrise and delayed sunset. This occurs because the Earth's atmosphere is not uniform; it consists of layers with varying optical densities and refractive indices.
• Advanced Sunrise: The sun is visible to us about 2 minutes before the actual sunrise. When the sun is slightly below the horizon, the light rays coming from it travel from rarer to denser layers of the atmosphere and bend toward the normal. To an observer on Earth, these rays appear to come from a position above the horizon. Hence, the statement regarding advanced sunrise being due to refraction is correct.
• Delayed Sunset: Similarly, for the same reason of atmospheric refraction, the sun remains visible for about 2 minutes after it has already crossed below the horizon. This creates the illusion that the sun is still present in the sky. Hence, the statement regarding delayed sunset being due to refraction is correct.
• Daylight Extension: Because of these two effects, the total duration of daylight is increased by approximately 4 minutes (2 minutes in the morning and 2 minutes in the evening).
• Evaluation of Options:
• Statement 1 suggests scattering causes advanced sunrise, which is scientifically inaccurate. Hence, statement 1 is incorrect.
• Statement 2 correctly identifies atmospheric refraction as the cause for both events. Hence, statement 2 is correct.
• Statement 3 attributes the timing difference to the scattering of light, which actually causes the color changes, not the position shift. Hence, statement 3 is incorrect.
• Statement 4 claims it is due to Earth’s rotation only; while rotation causes the day-night cycle, the timing discrepancy is an optical phenomenon. Hence, statement 4 is incorrect.

Additional Information

• Mechanism of Refraction: As light travels from the vacuum of space into the Earth's atmosphere, it enters a denser medium. According to Snell's Law, light bends toward the normal as it moves into a medium with a higher refractive index. The air layers closer to the Earth's surface are denser than the upper layers, causing continuous bending of sunlight.
• Apparent vs. Actual Position: Due to refraction, the apparent position of the sun is slightly higher than its actual position. This effect is most pronounced when the sun is near the horizon (sunrise and sunset).
• Scattering of Light: It is important to distinguish refraction from scattering. Scattering (specifically Rayleigh scattering) is the process where gas molecules in the atmosphere redirect shorter wavelengths of light (blue/violet). This is why the sky appears blue and why the sun appears reddish during sunrise and sunset, as red light has a longer wavelength and is scattered the least.
• Twinkling of Stars: This is another classic example of atmospheric refraction. Starlight undergoes multiple refractions as it passes through different layers of the atmosphere with varying temperatures and densities, causing the apparent position of the star to fluctuate slightly, which we perceive as twinkling.
• Flattening of the Sun: Atmospheric refraction also causes the sun to appear flattened or oval-shaped during sunrise and sunset because the bottom edge of the sun is refracted more than the top edge.

Important Points

• Refractive Index Variation: The refractive index of air depends on its temperature and pressure. Colder air is denser and has a higher refractive index than warmer air.
• Horizon Definition: The actual sunrise is defined as the moment when the sun physically crosses the horizon. Atmospheric refraction allows us to see it before this physical event occurs.
• Optical Illusions in Nature: Other phenomena related to refraction include mirages in deserts or on hot roads, where light bends due to temperature gradients in the air near the ground.

The correct answer is Sensory neurons.

Key Points

• Sensory neurons, also known as afferent neurons, are specialized nerve cells that convert external stimuli from the environment into internal electrical impulses.
• In a reflex arc, these neurons are the primary components responsible for receiving signals from receptors (such as those in the skin, eyes, or ears) and transmitting them toward the Central Nervous System (CNS).
• They function as the "input" mechanism of the nervous system, detecting changes such as temperature, pressure, pain, or light. Hence, the statement that they receive and transmit signals from external stimuli is correct.
• Structurally, most sensory neurons are pseudounipolar, meaning they have an axon that fans out into two branches: one to the peripheral nervous system and another to the spinal cord.
• The transmission happens rapidly to ensure that the body can react to potentially harmful stimuli without the need for immediate conscious thought from the brain.

Additional Information

• Motor Neurons (Efferent Neurons):
• These neurons carry nerve impulses away from the Central Nervous System toward effectors, such as muscles or glands.
• Their primary role is to initiate a physical response, such as muscle contraction or hormone secretion, based on the signals received from the CNS.
• Relay Neurons (Interneurons):
• Located entirely within the spinal cord and brain, relay neurons serve as the connecting link between sensory neurons and motor neurons.
• They process the incoming sensory information and "relay" the signal to the appropriate motor pathway to complete the reflex arc.
• Effector Muscle:
• The effector is the organ, tissue, or cell (often a skeletal muscle) that responds to the impulse from the motor neuron.
• In a typical withdrawal reflex, the effector muscle contracts to move a limb away from a stimulus like a hot flame or a sharp needle.
• The Reflex Arc Pathway:
• The standard sequence involves: Receptor → Sensory Neuron → Integration Center (Relay Neuron in Spinal Cord) → Motor Neuron → Effector.
• This pathway bypasses the cerebral cortex of the brain for the initial reaction, which significantly reduces response time.

Important Points

• Speed of Transmission: Reflexes are involuntary and nearly instantaneous. For example, the patellar reflex (knee-jerk) occurs in approximately 15 to 30 milliseconds.
• Synapse: The gap between two neurons where chemical neurotransmitters are released to pass the signal is called a synapse. In a monosynaptic reflex, there is only one synapse (between sensory and motor), while polysynaptic reflexes involve one or more relay neurons.
• Location: The cell bodies of sensory neurons are typically clustered in the dorsal root ganglia located just outside the spinal cord.
• Role in Survival: The primary biological function of the reflex arc is protection and homeostasis, allowing for immediate avoidance of injury.

The correct answer is all tall (Tt).

Key Points

• According to Mendel's Laws of Inheritance, when a homozygous tall pea plant (TT) is crossed with a homozygous short pea plant (tt), it represents a monohybrid cross, which focuses on a single trait (stem height).
• During the process of meiosis, the parent plants produce gametes. The tall plant (TT) produces gametes that all carry the T allele, while the short plant (tt) produces gametes that all carry the t allele.
• In the F1 (First Filial) generation, the fusion of these gametes results in offspring that all have the heterozygous genotype (Tt).
• Because the Tall (T) allele is completely dominant over the Short (t) allele, the recessive trait (shortness) is masked in this generation.
• As a result, 100% of the F1 offspring will exhibit the tall phenotype. Hence, option 1 is correct.

Additional Information

• Gregor Johann Mendel: Known as the Father of Genetics, he was an Austrian monk who conducted hybridization experiments on garden peas (Pisum sativum) for seven years between 1856 and 1863.
• Law of Dominance: This principle states that in a heterozygous individual (like Tt), the dominant allele will express itself, while the recessive allele remains hidden. The recessive trait only appears when two copies of the recessive allele are present (tt).
• Genotype vs. Phenotype: Genotype refers to the genetic makeup of an organism (e.g., TT, Tt, tt), whereas Phenotype refers to the observable physical characteristics (e.g., Tall or Short).
• Homozygous and Heterozygous: An organism is homozygous when it has two identical alleles for a gene (TT or tt). It is heterozygous when it has two different alleles (Tt).

Important Points

• F2 Generation: When the F1 plants (Tt) are allowed to self-pollinate, the F2 (Second Filial) generation is produced. In this generation, the recessive trait reappears.
• Phenotypic Ratio: The F2 generation typically shows a 3:1 ratio, where 75% of the plants are tall and 25% are short.
• Genotypic Ratio: The genetic distribution in the F2 generation is 1:2:1, consisting of 1 homozygous tall (TT), 2 heterozygous tall (Tt), and 1 homozygous short (tt).
• Seven Contrasting Traits: Mendel studied seven pairs of contrasting traits in pea plants: Stem height (Tall/Short), Seed shape (Round/Wrinkled), Seed color (Yellow/Green), Flower color (Violet/White), Pod shape (Full/Constricted), Pod color (Green/Yellow), and Flower position (Axial/Terminal).

The correct answer is Na.

Key Points

• The chemical symbol for Sodium is Na. This symbol is derived from its Latin name, Natrium. Hence, option 1 is correct.
• Sodium is a chemical element with the atomic number 11, which means it contains 11 protons in its atomic nucleus.
• It is located in Group 1 of the Periodic Table, placing it among the Alkali Metals.
• In terms of its position, it is found in Period 3 and the s-block of the modern periodic table.
• The atomic mass of Sodium is approximately 22.989 u.
• Under standard conditions of temperature and pressure, sodium is a soft, silvery-white, and highly reactive metal.

Additional Information

• Physical Properties:
• Sodium is so soft that it can be easily cut with a knife at room temperature.
• It has a density of 0.97 g/cm³, which makes it less dense than water, allowing it to float.
• Its melting point is relatively low for a metal, at 97.79°C (208.02°F), and its boiling point is 883°C.
• Chemical Reactivity:
• Sodium is extremely reactive, particularly with water. When it comes into contact with water, it reacts exothermically to produce sodium hydroxide (NaOH) and flammable hydrogen gas.
• Due to this high reactivity, sodium is never found as a pure element in nature; it must be stored under mineral oil or kerosene to prevent reaction with moisture or oxygen in the air.
• Industrial and Domestic Uses:
• The most common compound is Sodium Chloride (NaCl), which is used globally as table salt and for de-icing roads.
• Sodium Carbonate (Na2CO3), also known as washing soda, is vital in the manufacture of glass, paper, and detergents.
• Sodium Bicarbonate (NaHCO3), or baking soda, is widely used in the food industry and as a fire extinguisher component.
• Biological Importance:
• Sodium is an essential electrolyte for the human body. It is critical for maintaining osmotic pressure, fluid balance, and the transmission of nerve impulses.
• The sodium-potassium pump is a vital mechanism in cell biology that moves ions across cell membranes.

Important Points

• Origin of the Name: The element was first isolated by Humphry Davy in the year 1807 through the electrolysis of caustic soda (NaOH).
• Comparison with Other Options:
• S is the chemical symbol for Sulfur (atomic number 16), which is a non-metal. Hence, option 3 is incorrect.
• Sd and So are not valid chemical symbols for any known element in the IUPAC periodic table. Hence, options 2 and 4 are incorrect.
• Valency: Sodium has a valency of 1 because it has one electron in its outermost shell (electron configuration: 2, 8, 1). It tends to lose this electron to achieve a stable octet configuration, forming a cation (Na+).
• Abundance: Sodium is the sixth most abundant element in the Earth's crust, making up about 2.6% of its mass.

The correct answer is -210 J.

Key Points

• The Work-Energy Theorem states that the net work done by the forces acting on a body is equal to the change in its kinetic energy.
• In this problem, the given values are: Mass of the object (m) = 20 kg, Initial velocity (u) = 5 m/s, and Final velocity (v) = 2 m/s.
• The Initial Kinetic Energy (Ki) is calculated using the formula 1/2 * m * u^2. Substituting the values: 1/2 * 20 * (5 * 5) = 10 * 25 = 250 Joules.
• The Final Kinetic Energy (Kf) is calculated using the formula 1/2 * m * v^2. Substituting the values: 1/2 * 20 * (2 * 2) = 10 * 4 = 40 Joules.
• The Work Done (W) is the difference between the final and initial kinetic energy (W = Kf - Ki). Therefore, W = 40 - 250 = -210 J.
• The negative sign indicates that the force applied was a retarding force, acting in the direction opposite to the motion of the mass.

Additional Information

• Kinetic Energy:
• It is the energy possessed by an object due to its motion. The formula for kinetic energy is K.E = 1/2 mv^2, where m is mass and v is velocity.
• Since velocity is squared in the formula, kinetic energy is always a non-negative value, but the change in kinetic energy can be negative.
• Work Done:
• Work is defined as the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
• The SI unit of work is the Joule (J). One Joule is the work done by a force of 1 Newton causing a displacement of 1 meter.
• Positive, Negative, and Zero Work:
• Positive Work: When the force and displacement are in the same direction (angle is 0 degrees).
• Negative Work: When the force and displacement are in opposite directions (angle is 180 degrees), such as friction.
• Zero Work: When the force is perpendicular to the displacement (angle is 90 degrees), such as a coolie carrying a load on his head while walking horizontally.

Important Points

• Mass (m): It is a scalar quantity representing the amount of matter in an object, measured in kilograms (kg). In this case, it is 20 kg.
• Velocity (v): It is a vector quantity representing the rate of change of displacement, measured in meters per second (m/s).
• Energy Conservation: While the Work-Energy theorem focuses on kinetic energy, the total mechanical energy (sum of potential and kinetic energy) remains constant in a conservative field if no external work is done.
• Application: The concept of negative work is frequently used in calculating braking distances for vehicles and understanding air resistance or drag forces in physics.

The correct answer is A satellite orbiting the Earth.

Key Points

• Uniform circular motion is defined as the motion of an object traveling at a constant speed along a circular path. Although the speed is constant, the velocity of the object is constantly changing because its direction of motion changes at every point on the path.
• A satellite orbiting the Earth at a constant altitude is a primary example of uniform circular motion. The gravitational force exerted by the Earth acts as the centripetal force, which keeps the satellite in its circular orbit while its speed remains relatively constant. Hence, statement 3 is correct.
• A person walking in a straight line is an example of rectilinear motion. Since the path is not circular, it does not qualify as circular motion. Hence, statement 1 is incorrect.
• A car moving on a straight road also represents rectilinear motion. Even if the car maintains a constant speed, the lack of a curved or circular path means it cannot be uniform circular motion. Hence, statement 4 is incorrect.
• A ball thrown vertically upwards is an example of non-uniform linear motion. The ball is subject to a constant acceleration due to gravity (approximately 9.8 m/s2), causing its speed to decrease as it rises and increase as it falls, all while moving in a straight vertical line. Hence, statement 2 is incorrect.

Additional Information

• Velocity and Acceleration: In uniform circular motion, even though the magnitude of velocity (speed) is constant, there is a continuous acceleration known as centripetal acceleration. This acceleration is always directed toward the center of the circular path.
• Centripetal Force: This is the "center-seeking" force required to maintain any circular motion. The formula for this force is F = mv2/r, where m is mass, v is velocity, and r is the radius of the path. In the case of a satellite, this force is provided by gravity.
• Angular Velocity: This refers to the rate at which an object rotates around a center. It is usually denoted by the Greek letter omega (ω). The relationship between linear speed (v) and angular speed is v = rω.
• Work Done: In perfect uniform circular motion, the work done by the centripetal force is zero. This is because the force acting on the object is always perpendicular (90 degrees) to the direction of its displacement (which is tangential to the circle).
• Real-world Examples: Other common examples of this motion include the motion of the tips of the hands of a clock, a point on a ceiling fan rotating at a constant speed, and the Moon revolving around the Earth.

Important Points

• Direction of Velocity: In circular motion, the direction of velocity at any point is along the tangent to the circle at that specific point.
• Time Period: The time taken by an object to complete one full revolution of the circular path is called its Time Period (T). It can be calculated using the formula T = 2πr / v.
• Frequency: The number of revolutions completed by the object in one second is called its frequency (f), which is the reciprocal of the time period (f = 1/T).
• Nature of Acceleration: It is important to remember that uniform circular motion is an accelerated motion despite having a constant speed, because the velocity vector changes direction constantly.

The correct answer is Both statements A and B are correct.

Key Points

• Statement A describes the chemical reaction between Ethanoic acid and sodium hydroxide. Ethanoic acid (also known as acetic acid) has the chemical formula CH3COOH, and when it reacts with the strong base sodium hydroxide (NaOH), it undergoes a double displacement reaction to form sodium acetate (CH3COONa) and water (H2O). Hence, statement A is correct.
• The chemical equation for this reaction is: CH3COOH + NaOH → CH3COONa + H2O.
• Statement B identifies the type of reaction. A neutralisation reaction is a specific type of chemical reaction where an acid and a base react with each other to produce a salt and water. Since ethanoic acid is an acid and sodium hydroxide is a base, their interaction fits this definition perfectly. Hence, statement B is correct.
• In this neutralisation process, the hydrogen ion (H+) from the acid combines with the hydroxide ion (OH-) from the base to form water, while the remaining ions form the ionic salt.
• Hence, the correct answer is Both statements A and B are correct.

Additional Information

• Ethanoic Acid (Acetic Acid):
• It is a weak organic acid that is a primary component of vinegar (typically 5% to 8% by volume).
• In its pure form, it is called glacial acetic acid because it freezes into ice-like crystals at a temperature of 16.6 degrees Celsius.
• It is widely used in the food industry as a preservative and in the textile industry for the production of synthetic fibers like rayon.
• Sodium Hydroxide (Caustic Soda):
• It is a strong inorganic base and is highly alkaline.
• It is deliquescent, meaning it absorbs moisture and carbon dioxide from the air.
• Major applications include the manufacture of soaps, detergents, paper, and as a chemical reagent in laboratories.
• Sodium Acetate:
• It is the sodium salt of acetic acid and is often used as a buffering agent to maintain a constant pH in chemical reactions.
• It is used in heating pads and hand warmers because of its ability to release heat upon crystallization from a supersaturated solution.
• In the food industry, it is used as a seasoning (sodium diacetate) to give a salt-and-vinegar flavor to snacks.

Important Points

• pH Levels in Neutralisation: Although the reaction is a neutralisation, the resulting pH of the solution depends on the strength of the reactants. When a weak acid (like ethanoic acid) reacts with a strong base (like sodium hydroxide), the resulting salt solution is slightly alkaline, with a pH greater than 7.
• Indicators: To observe this neutralisation in a laboratory, indicators like phenolphthalein are used. The solution turns from pink (in base) to colorless as the acid is added to reach the equivalence point.
• Exothermic Nature: Most neutralisation reactions are exothermic, meaning they release heat energy to the surroundings during the process.

The correct answer is works perfectly with hard water.

Key Points

• Soap is chemically defined as the sodium or potassium salt of long-chain fatty acids (such as stearic, palmitic, or oleic acid).
• A soap molecule has a dual nature: a hydrophilic head (the ionic -COO⁻Na⁺ or -COO⁻K⁺ part) that interacts with water and a hydrophobic tail (the long hydrocarbon chain) that avoids water but attaches to grease and oil. Hence, statements 3 and 4 are associated with soap.
• When soap is dissolved in water, the molecules orient themselves into spherical clusters known as micelles. In a micelle, the hydrophobic tails are trapped in the interior away from water, while the hydrophilic heads remain on the surface in contact with the aqueous environment. Hence, statement 1 is associated with soap.
• Soap fails to work effectively in hard water. Hard water contains high concentrations of calcium (Ca²⁺) and magnesium (Mg²⁺) ions. These ions react with soap to form an insoluble, curdy precipitate known as scum. Hence, statement 2 is NOT associated with soap and is the correct answer.
• The formation of scum wastes soap and makes it difficult for the soap to create lather, which is essential for the cleansing process.

Additional Information

• Cleansing Action of Soap:
• The hydrophobic end of the soap attaches to the dirt or oil on clothes, while the hydrophilic end stays in the water.
• This creates an emulsion where the oil is lifted from the surface and suspended in water, allowing it to be washed away during rinsing.
• Saponification:
• This is the chemical reaction used to produce soap. It involves the hydrolysis of fats (triglycerides) with a strong alkali like Sodium Hydroxide (NaOH) for hard soaps or Potassium Hydroxide (KOH) for soft soaps.
• The process typically yields glycerol as a byproduct, which is often used in the cosmetic industry.
• Synthetic Detergents:
• Unlike soaps, detergents are sodium salts of long-chain benzene sulphonic acids or alkyl hydrogen sulphates.
• The primary advantage of detergents is that they work perfectly with hard water because their calcium and magnesium salts are soluble in water and do not form scum.
• Detergents are widely used in shampoos and products for cleaning clothes because of their superior performance in varied water conditions.

Important Points

• Environmental Impact: While soaps are biodegradable and broken down by bacteria in sewage systems, some older types of synthetic detergents were non-biodegradable, leading to water pollution and foam in rivers. Modern detergents are designed to be more environmentally friendly.
• pH Level: Soap solutions are generally basic (alkaline) in nature, which can sometimes irritate sensitive skin. Potassium soaps are usually milder than sodium soaps.
• Hardness of Water: It is measured in parts per million (ppm) or degrees. Temporary hardness is caused by bicarbonates, while permanent hardness is caused by chlorides and sulphates of calcium and magnesium.

The correct answer is They prevent maturation of primary oocyte..

Key Points

• Oral contraceptive pills are hormonal preparations used by females to prevent pregnancy. They contain either progestogens alone or a combination of progestogen and estrogen.
• The primary mechanism of these pills is the inhibition of ovulation. By providing a steady supply of synthetic hormones, they suppress the secretion of Follicle Stimulating Hormone (FSH) and Luteinizing Hormone (LH) from the pituitary gland.
• Since FSH is responsible for stimulating the growth of follicles, its suppression ensures that the primary oocyte does not undergo maturation into a secondary oocyte. Without maturation, no egg is released from the ovary. Hence, statement 2 is correct.
• Physical barriers like condoms, diaphragms, or surgical methods like tubectomy (which blocks the fallopian tube) are responsible for creating physical obstructions. Oral pills do not create any such physical barrier. Hence, statement 1 is incorrect.
• Chemicals that are toxic to sperms are known as spermicides, which are usually available as creams, jellies, or foams. Oral pills do not act as toxins to sperms. Hence, statement 3 is incorrect.
• Mechanical barriers used in the vagina or cervix, such as vaults or cervical caps, block the passage of sperms. Oral pills function biochemically through the bloodstream and do not physically block the vaginal passage. Hence, statement 4 is incorrect.

Additional Information

• Method of Administration: Pills are typically taken daily for a period of 21 days, starting within the first five days of the menstrual cycle. This is followed by a 7-day break during which menstruation occurs, after which the pattern is repeated.
• Saheli: This is a famous oral contraceptive for females developed by the Central Drug Research Institute (CDRI) in Lucknow, India. It is unique because it is a non-steroidal preparation and is taken once a week, offering high contraceptive value with very few side effects.
• Alteration of Cervical Mucus: Apart from inhibiting ovulation, oral pills also increase the thickness and quality of cervical mucus. This thick mucus acts as a barrier that prevents or retards the entry of sperms into the uterus.
• Implantation Inhibition: These hormones also alter the nature of the endometrium (the inner lining of the uterus), making it unsuitable for the implantation of a fertilized egg, providing an additional layer of protection against pregnancy.

Important Points

• Hormonal Feedback: The pills work on the principle of negative feedback. High levels of exogenous progesterone and estrogen inhibit the hypothalamus from releasing GnRH, which effectively shuts down the hormonal trigger for the mid-cycle LH surge required for ovulation.
• Effectiveness: When taken correctly, oral contraceptives are 99% effective. However, their efficacy can be reduced if doses are missed or if certain antibiotics are taken simultaneously.
• Medical Uses: Beyond contraception, these pills are often prescribed to regulate menstrual cycles, reduce menstrual cramps, and manage symptoms of Polycystic Ovary Syndrome (PCOS).
• Emergency Contraceptive Pills (ECPs): These are different from regular daily pills; they contain higher doses of levonorgestrel and must be taken within 72 hours of unprotected intercourse to prevent conception.

The correct answer is Electron.

Key Points

• The Electron is a subatomic particle that carries a negative elementary electric charge. It was the first subatomic particle to be discovered, identified by J.J. Thomson in 1897 during his experiments with cathode rays. Hence, the correct answer is Electron.
• In terms of mass, an electron is extremely light, weighing approximately 9.109 x 10^-31 kilograms. This is roughly 1/1837th of the mass of a proton, which is why it is considered to have negligible mass in most atomic calculations.
• Protons are subatomic particles found in the nucleus of an atom with a positive charge of +1 and a mass of approximately 1 atomic mass unit (amu). Since they have a positive charge, this option is incorrect. Hence, statement 3 is incorrect.
• Neutrons are also located in the nucleus and have no electric charge (they are neutral). Their mass is slightly greater than that of a proton, approximately 1 amu. Because they lack a negative charge, this option is incorrect. Hence, statement 4 is incorrect.
• Nucleons is a collective term for the particles that reside within the atomic nucleus, specifically protons and neutrons. Since this term refers to a group containing positively charged and neutral particles with significant mass, it does not fit the description. Hence, statement 2 is incorrect.

Additional Information

• Atomic Structure: Atoms are composed of a dense, positively charged nucleus surrounded by a cloud of electrons. The electrons are held in place by the electromagnetic force, specifically the attraction between the negative electrons and the positive protons.
• Discovery of the Nucleus: In 1911, Ernest Rutherford performed the Gold Foil Experiment, which led to the conclusion that most of the atom's mass and its positive charge are concentrated in a tiny central region called the nucleus.
• Electron Configuration: Electrons reside in specific energy levels called shells (K, L, M, N...). The arrangement of electrons in these shells determines the chemical properties of the element. The outermost electrons are known as valence electrons.
• Subatomic Charges: The magnitude of the charge of an electron is exactly equal to the magnitude of the charge of a proton (1.602 x 10^-19 Coulombs), but with an opposite sign. This balance ensures that an atom with an equal number of protons and electrons remains electrically neutral.
• Quantum Nature: According to quantum mechanics, electrons exhibit wave-particle duality. This means they possess characteristics of both particles and waves, a concept pioneered by Louis de Broglie.

Important Points

• Bohr Model: Proposed by Niels Bohr in 1913, this model suggested that electrons orbit the nucleus in fixed circular paths or orbits, which helped explain the spectral lines of hydrogen.
• Mass Number vs. Atomic Number: The Atomic Number (Z) is the number of protons, while the Mass Number (A) is the sum of protons and neutrons. Electrons are excluded from the mass number calculation due to their negligible weight.
• Isotopes: These are atoms of the same element that have the same number of protons but different numbers of neutrons. This changes the mass of the atom but usually does not significantly change its chemical behavior, which is governed by electrons.
• Ionization: When an atom gains or loses electrons, it becomes an ion. Losing electrons creates a cation (positive charge), while gaining electrons creates an anion (negative charge).

The correct answer is Unbalanced.

Key Points

• For a rocket to move upward from its launchpad, the forces acting on it must be unbalanced. According to Newton's First Law of Motion, an object at rest will stay at rest unless acted upon by an external unbalanced force.
• When a rocket's engines ignite, they produce an upward force known as thrust. For the rocket to lift off, this thrust must be greater than the downward force of gravity (the rocket's weight).
• The net force is the vector sum of all forces. In this case, Net Force = Thrust - (Weight + Air Resistance). Since the rocket accelerates upward, the net force is positive and non-zero, making the forces unbalanced. Hence, the correct option is Unbalanced.
• Newton's Second Law of Motion states that Acceleration (a) = Net Force (F) / Mass (m). Without an unbalanced force, the acceleration would be zero, and the rocket would either remain stationary or move at a constant velocity.
• If the forces were balanced, the upward thrust would exactly equal the downward gravity and drag, resulting in no change in the rocket's state of motion. Hence, the option 'Balanced' is incorrect.

Additional Information

• Newton's Third Law of Motion: This law is the fundamental principle behind rocket propulsion. It states that for every action, there is an equal and opposite reaction. The rocket pushes exhaust gases downward at high velocity (action), and these gases push the rocket upward (reaction).
• Thrust: This is the mechanical force generated by the rocket's propulsion system. It depends on the mass flow rate of the exhaust and the exit velocity of the gases. To achieve orbit, engines must maintain high thrust levels for an extended duration.
• Gravity: Throughout the flight, gravity acts as a constant downward force. Its magnitude decreases slightly as the rocket moves further from Earth's center, but it remains a primary factor the rocket must overcome using fuel and combustion.
• Air Resistance (Drag): As the rocket travels through the atmosphere, air molecules collide with it, creating friction or drag. This force acts in the opposite direction of the rocket's motion and increases with velocity and atmospheric density.
• Mass Variation: Unlike many other vehicles, a rocket's mass changes significantly during flight because it consumes a massive amount of propellant (fuel and oxidizer). As mass decreases and thrust remains constant, the acceleration increases dramatically.

Important Points

• Vector Quantities: Force and acceleration are vector quantities, meaning they have both magnitude and direction. The direction of the unbalanced force determines the direction of the acceleration.
• Inertia: The tendency of the rocket to resist changes in its state of motion is called inertia, which is directly proportional to its mass. High inertia requires a larger unbalanced force to achieve the same acceleration.
• Equilibrium: An object is in mechanical equilibrium when the sum of forces is zero (balanced). A rocket sitting on a launchpad before ignition is in static equilibrium, where the normal force from the ground balances gravity.
• Historical Context: Sir Isaac Newton published these laws in his work Philosophiae Naturalis Principia Mathematica in 1687, which laid the foundation for modern aerospace engineering and physics.

The correct answer is Insects.

Key Points

• Abiotic resistance refers to a plant's ability to survive and grow under non-living environmental stresses. These factors are physical or chemical rather than biological.
• Salinity is an abiotic factor characterized by an excessive concentration of salts (like Sodium Chloride) in the soil. It affects the osmotic potential of the plant, making it difficult to absorb water. Hence, this is an abiotic resistance factor.
• Heat is a critical abiotic stress caused by high temperatures that exceed the optimal range for plant enzymes. It can lead to protein denaturation and reduced photosynthetic efficiency. Hence, this is an abiotic resistance factor.
• Drought is an abiotic factor resulting from a prolonged shortage of water. Plants develop resistance through mechanisms like deep root systems, stomatal closure, and osmotic adjustment. Hence, this is an abiotic resistance factor.
• Insects are living organisms (animals) that cause biological damage to crops through feeding, egg-laying, or by acting as disease vectors. Stresses caused by living entities are classified as biotic stresses, not abiotic. Hence, Insects is the correct answer as it is NOT an abiotic resistance factor.

Additional Information

• Biotic Stress in Crops:
• This involves damage caused by living organisms such as fungi, bacteria, viruses, nematodes, weeds, and insects.
• Examples include the Desert Locust (pest), Late Blight in potatoes (fungal), and Rice Tungro Virus.
• Resistance is often achieved through Biotechnology, such as the development of Bt Cotton which produces a toxin to kill certain pests.
• Abiotic Stress Factors:
• These are environmental conditions that limit crop productivity. Besides salinity, heat, and drought, they include frost/cold, flooding, radiation, and heavy metal toxicity (like Lead or Arsenic).
• According to agricultural data, abiotic stresses are responsible for more than 50% of the global yield loss in major crops like wheat, rice, and maize.
• Plant Adaptations:
• Plants respond to abiotic stress by producing Heat Shock Proteins (HSPs) or secondary metabolites like proline to maintain cellular stability.
• Xerophytes are specifically adapted to drought, while Halophytes are plants naturally resistant to salinity.

Important Points

• Food Security: Developing multi-stress resistant varieties is essential for maintaining global food security in the face of climate change.
• Measurement Units: Soil salinity is often measured by Electrical Conductivity (EC), while drought intensity is monitored using the Palmer Drought Severity Index (PDSI).
• Breeding Techniques: Modern Genome Editing tools like CRISPR-Cas9 are being used to enhance abiotic resistance in crops by targeting specific stress-responsive genes.
• Classification: It is important to distinguish between Resistance (ability to prevent the stress) and Tolerance (ability to endure the stress without significant yield loss).

The correct answer is Red.

Key Points

• In a domestic electric circuit, the Live wire (also known as the Phase wire) is traditionally colored Red. This wire is responsible for carrying the electric current from the power source to the various appliances in the house.
• The Live wire is maintained at a high potential, which is typically 220 Volts (in India) relative to the earth or neutral wire.
• According to the conventional color coding used in domestic wiring, the three wires are identified as: Red for Live, Black for Neutral, and Green for Earth. Hence, the statement regarding Red being the live wire is correct.
• Safety devices such as the Fuse and the Switch are always connected to the Live wire. This ensures that when the fuse blows or the switch is turned off, the appliance is completely disconnected from the high-voltage source.
• It is crucial to identify the Live wire correctly during electrical repairs to avoid Electric Shocks, as touching the live wire while grounded can lead to a fatal current passing through the body.

Additional Information

• Neutral Wire:
• The Neutral wire is generally Black in the old convention and Blue in the new international convention.
• Its primary function is to complete the circuit by providing a Return Path for the current. It is ideally maintained at Zero Potential (0V).
• Earth Wire:
• The Earth wire is usually Green or Green-Yellow. It is a safety wire connected to a large Copper Plate buried deep in the earth near the house.
• It provides a Low-resistance Path for any leakage current, ensuring that if the live wire touches the metallic body of an appliance, the current flows into the ground rather than through the user.
• New Color Convention:
• Modern electrical installations often follow the International Electrotechnical Commission (IEC) standards.
• Under these new rules, the Live wire is Brown, the Neutral wire is Blue, and the Earth wire is Green with a Yellow stripe.
• Short Circuit and Overloading:
• A Short Circuit occurs when the Live wire and the Neutral wire come into direct contact due to damaged insulation, causing a sudden surge in current.
• Overloading happens when too many high-power appliances are connected to a single circuit, drawing more current than the wires can safely handle.

Important Points

• Standard Supply in India: The domestic electricity supply in India is Alternating Current (AC) with a frequency of 50 Hz and a standard voltage of 220V.
• Three-Pin Plug: In a standard Three-pin Plug, the top pin is the Earth pin. It is designed to be Thicker and Longer than the other two pins (Live and Neutral) so that the grounding connection is established first before the power is supplied.
• Color Variations: While Red is the most common color for live wires in single-phase domestic circuits, in three-phase industrial circuits, the live wires are often colored Red, Yellow, and Blue.

The correct answer is B, C.

Key Points

• The number of neutrons in an atom is calculated by subtracting the atomic number (Z) from the mass number (A). The formula is: Number of Neutrons = A - Z.
• For Boron (B), the most common isotope has an atomic number of 5 and a mass number of approximately 11. Therefore, the number of neutrons is 11 - 5 = 6.
• For Carbon (C), the most common isotope (Carbon-12) has an atomic number of 6 and a mass number of 12. Therefore, the number of neutrons is 12 - 6 = 6.
• Comparing the two, both Boron (B) and Carbon (C) possess 6 neutrons in their most stable forms. Hence, statement 2 is correct.
• In contrast, for Lithium (Li), the atomic number is 3 and the mass number is 7, giving 4 neutrons. For Beryllium (Be), the atomic number is 4 and the mass number is 9, giving 5 neutrons. Hence, statement 1 is incorrect.

Additional Information

• Atomic Number (Z): It represents the number of protons in the nucleus of an atom. It is unique to every element and determines its position in the Modern Periodic Table. For example, Helium has 2 protons, while Lithium has 3.
• Mass Number (A): It is the total sum of protons and neutrons (collectively called nucleons) in the nucleus. It is usually denoted as a whole number in calculations. For instance, the mass number of Beryllium is 9.
• Isotones: Elements or atoms that have the same number of neutrons but different atomic numbers and mass numbers are called isotones. In this question, Boron-11 and Carbon-12 are examples of isotones.
• Isotopes: These are atoms of the same element having the same atomic number but different mass numbers. For example, Carbon-12 and Carbon-14 are isotopes of Carbon.
• Isobars: These are atoms of different chemical elements that have the same mass number but different atomic numbers. An example is Argon-40 and Calcium-40.

Important Points

• Discovery of Neutrons: The neutron was discovered by the British physicist James Chadwick in the year 1932. It is a subatomic particle with no net electric charge and a mass slightly greater than that of a proton.
• Elements Comparison:
• Helium (He): Atomic Number 2, Mass Number 4, Neutrons = 2.
• Lithium (Li): Atomic Number 3, Mass Number 7, Neutrons = 4.
• Beryllium (Be): Atomic Number 4, Mass Number 9, Neutrons = 5.
• Boron (B): Atomic Number 5, Mass Number 11, Neutrons = 6.
• Carbon (C): Atomic Number 6, Mass Number 12, Neutrons = 6.
• The distribution of electrons, protons, and neutrons determines the chemical and physical properties of the element. Neutrons play a critical role in the stability of the nucleus.

The correct answer is Lysosomes contain digestive enzymes.

Key Points

• Lysosomes are membrane-bound vesicular structures formed by the process of packaging in the Golgi apparatus. They are often referred to as the "suicidal bags" of the cell.
• These organelles contain a wide variety of hydrolytic enzymes (hydrolases), such as lipases, proteases, and carbohydrases, which are optimally active at an acidic pH. These enzymes are responsible for digesting carbohydrates, proteins, lipids, and nucleic acids. Hence, statement 2 is correct.
• The synthesis of fats (lipids) and steroids primarily takes place in the Smooth Endoplasmic Reticulum (SER), not in the lysosomes. Hence, statement 1 is incorrect.
• Protein synthesis is the primary function of ribosomes, which may be found floating freely in the cytoplasm or attached to the Rough Endoplasmic Reticulum (RER). Lysosomes do not synthesize proteins; they degrade them. Hence, statement 3 is incorrect.
• Growth hormones are signaling molecules produced by specialized endocrine glands, such as the pituitary gland. They are not contained or produced within lysosomes. Hence, statement 4 is incorrect.

Additional Information

• Structure and Discovery:
• Lysosomes were discovered by the Belgian cytologist Christian de Duve in 1955.
• They are typically spherical in shape and are enclosed by a single unit membrane to prevent the powerful enzymes from leaking and destroying the rest of the cell.
• Mechanism of Action:
• Lysosomes maintain an internal acidic environment (around pH 4.5 to 5.0) through proton pumps in their membranes that transport H+ ions from the cytosol.
• They are involved in autophagy (digesting the cell's own damaged organelles) and heterophagy (digesting external materials like bacteria or food particles taken in via endocytosis).
• Types of Lysosomes:
• Primary Lysosomes: Newly formed vesicles from the Golgi body containing inactive enzymes.
• Secondary Lysosomes: Formed by the fusion of a primary lysosome with a phagosome or food vacuole; this is where active digestion occurs.
• Residual Bodies: Lysosomes containing undigested material that are eventually eliminated via exocytosis.
• Clinical Significance:
• Malfunctioning of lysosomal enzymes can lead to Lysosomal Storage Diseases (LSDs), such as Tay-Sachs disease or Gaucher's disease, where undigested substances accumulate and damage tissues.
• They are highly abundant in animal cells, particularly in White Blood Cells (leukocytes) like macrophages, which utilize lysosomal enzymes to destroy pathogens.

The correct answer is Energy storage.

Key Points

• The analogy focuses on the biological function of specific substances in plants. Just as Chlorophyll is defined by its role as a green pigment, Starch is defined by its role as energy storage.
• Starch is a polysaccharide carbohydrate consisting of a large number of glucose units joined by glycosidic bonds. It is the primary way autotrophic plants store excess energy produced during metabolism.
• During the process of photosynthesis, plants synthesize glucose. Since glucose is soluble in water and can affect the osmotic pressure of cells, plants convert it into insoluble starch for long-term storage in organs like roots, tubers, and seeds.
• In the analogy, Chlorophyll is the substance that captures light energy from the sun, while Starch is the chemical form in which that energy is kept for later use, such as during the night or dormant periods.
• Starch is the most common carbohydrate in the human diet and is contained in large amounts in staple foods such as potatoes, maize (corn), rice, and wheat.

Additional Information

• Structure of Starch:
• It comprises two types of molecules: Amylose and Amylopectin.
• Amylose is a linear chain of glucose and makes up about 20% to 30% of the structure, while Amylopectin is highly branched and makes up the remaining 70% to 80%.
• Comparison with Glycogen:
• While plants use starch for energy storage, animals and humans use glycogen.
• Glycogen is stored primarily in the liver and skeletal muscles and is broken down into glucose when the body requires a quick energy boost.
• Chlorophyll Characteristics:
• It is located within the thylakoid membranes of the chloroplasts.
• The central atom in a chlorophyll molecule is Magnesium (Mg). It absorbs wavelengths of light in the blue and red regions but reflects green light, which gives plants their characteristic color.
• Biochemical Testing:
• The presence of starch can be easily detected using Iodine solution. A chemical reaction occurs that turns the solution to a blue-black color if starch is present.
• This test is a fundamental experiment in botany and biochemistry to demonstrate the occurrence of photosynthesis in leaves.

Important Points

• Carbohydrate Classification: Starch belongs to the complex carbohydrate group because it is a polymer of monosaccharides. Unlike simple sugars, it provides sustained energy release.
• Industrial Uses: Beyond nutrition, starch is used in the paper industry for sizing, in the textile industry for stiffening fabrics, and as a thickening agent in the food industry.
• Evolutionary Significance: The ability of plants to store energy as starch allowed them to survive in seasonal climates where sunlight or water might be scarce for several months of the year.

The correct answer is Using sound-absorbent materials.

Key Points

• Reverberation is defined as the persistence of sound in an enclosed space after the original source of sound has stopped. It occurs due to multiple reflections of sound waves from surfaces like walls, ceilings, and floors.
• To reduce reverberation, the surfaces of the room or auditorium must be covered with sound-absorbent materials. These materials reduce the amount of reflected energy by converting sound energy into heat energy within the material. Hence, statement 4 is correct.
• Using hard concrete floors would actually increase reverberation because concrete is a rigid and non-porous material that reflects nearly all incident sound waves. Hence, statement 1 is incorrect.
• Removing furniture increases the volume of open space and exposes more reflective surfaces. Items like sofas, cushioned chairs, and wooden desks act as obstacles and partial absorbers; removing them makes the room more "live" and increases the reverberation time. Hence, statement 2 is incorrect.
• Marble walls are highly polished, dense, and hard surfaces. They have a very low absorption coefficient, meaning they reflect sound waves very efficiently, which would lead to prolonged reverberation and potential echoes. Hence, statement 3 is incorrect.
• Commonly used absorbent materials include compressed fiberboard, glass wool, heavy curtains, and carpets. These are specifically designed with porous structures to trap sound waves.

Additional Information

• Mechanism of Absorption: When sound waves strike porous materials, the air molecules vibrate inside the tiny pores. The friction between the air molecules and the walls of the pores converts the vibrational energy into thermal energy, thereby reducing the reflected sound.
• Acoustic Panelling: In cinemas and concert halls, walls are often covered with perforated acoustic panels. These panels have small holes that allow sound to enter and get trapped in the insulating material behind them.
• Impact of Soft Furnishings: Materials such as draperies, upholstered seats, and wall-to-wall carpeting are effective in residential and commercial settings to maintain acoustic comfort and prevent speech unintelligibility.
• Absorption Coefficient: This is a numerical value ranging from 0 to 1. A value of 0 represents a perfect reflector (no absorption), while a value of 1 represents a perfect absorber. Materials like open windows have a coefficient of 1.0 as no sound is reflected back.

Important Points

• Reverberation Time (RT): This is the time duration required for the sound pressure level to decrease by 60 decibels (dB) after the source has stopped. This standard is often referred to as RT60.
• Sabine's Formula: The reverberation time is calculated using Sabine's Equation, which states that the time is directly proportional to the volume of the room and inversely proportional to the total absorption area.
• Optimal Times: Different environments require different reverberation times. For example, a lecture hall requires a low RT (around 0.5 to 1.0 seconds) for speech clarity, whereas a cathedral or concert hall for pipe organ music may have an RT of over 2.0 seconds to create a "full" sound.
• Echo vs Reverberation: An echo is a distinct reflection that reaches the ear at least 0.1 seconds after the original sound. Reverberation is the blending of many reflections so close together that they cannot be distinguished as separate sounds.

The correct answer is Synergids.

Key Points

• Synergids are two specialized cells located at the micropylar end of the embryo sac in angiosperms. They are part of the egg apparatus, which also includes the egg cell (female gamete).
• The primary function of synergids is to guide the pollen tube toward the female gamete. This guidance is achieved through chemotropic attraction, where the synergids secrete specific chemical substances that direct the pollen tube's growth.
• Inside the synergids, there is a specialized cellular structure called the filiform apparatus. These are thick, finger-like projections of the cell wall that play a vital role in directing the pollen tube into the embryo sac and facilitating the release of sperm nuclei.
• During the process of fertilization, one of the synergids usually starts to degenerate before or upon the entry of the pollen tube, creating a passage for the male gametes to reach the egg cell and the central cell. Hence, the statement that synergids help in guiding the pollen tube is correct.

Additional Information

• Antipodal Cells:
• These are a group of three cells located at the chalazal end (opposite to the micropyle) of the embryo sac.
• In most plants, their function is to provide nutrition to the developing embryo sac, although they typically degenerate before or shortly after fertilization.
• Endosperm:
• The endosperm is the nutritive tissue formed as a result of triple fusion (one male gamete fusing with two polar nuclei) during double fertilization.
• It is usually triploid (3n) and serves as a food source for the embryo during seed development and germination.
• Style:
• The style is the long, slender tube-like part of the carpel (or pistil) that connects the stigma to the ovary.
• While the pollen tube must grow through the tissues of the style to reach the ovule, the style itself does not provide the final chemical guidance into the embryo sac; that role belongs to the synergids.
• Double Fertilization:
• This is a characteristic feature of flowering plants (angiosperms) where two male gametes are released into the embryo sac.
• One male gamete fuses with the egg cell to form a diploid zygote (2n), and the other fuses with the central cell containing two polar nuclei to form the primary endosperm nucleus (PEN).

Important Points

• The embryo sac in a typical angiosperm is an 8-nucleate, 7-celled structure.
• The micropyle is the small opening in the integuments of the ovule through which the pollen tube usually enters.
• The chemotropic signals involved in guidance often include calcium ions and specific proteins secreted by the filiform apparatus.

The correct answer is mixing food with saliva.

Key Points

• The tongue is a vital muscular organ located in the oral cavity (mouth) that functions as a primary accessory organ in the digestive system.
• One of its most important roles is mixing food with saliva. This process ensures that food is sufficiently moistened and lubricated for easy passage through the esophagus. Hence, the correct answer is mixing food with saliva.
• The tongue facilitates the formation of a bolus, which is a small, rounded mass of chewed food. By manipulating the food against the palate and teeth, it prepares the food for deglutition (swallowing).
• It acts as a sensory organ equipped with taste buds located on various types of papillae (such as fungiform and circumvallate). These receptors detect sweet, sour, salty, bitter, and umami flavors, which triggers the brain to signal the stomach to begin producing gastric juices.
• During the process of mastication (chewing), the tongue constantly repositions food between the teeth to ensure it is ground efficiently.

Additional Information

• Chemical Digestion in the Mouth:
• While the tongue mixes the food, the salivary glands (parotid, submandibular, and sublingual) secrete saliva containing the enzyme salivary amylase (also known as ptyalin).
• This enzyme initiates the chemical breakdown of complex carbohydrates (starch) into simpler sugars like maltose.
• Anatomy of the Tongue:
• The tongue is composed of skeletal muscle, allowing for voluntary control and highly flexible movements.
• It is anchored to the floor of the mouth by a vertical fold of mucous membrane called the lingual frenulum.
• Misconceptions in Digestion:
• The tongue does not secrete digestive enzymes; these are produced by the salivary glands, stomach, and pancreas.
• The teeth, not the tongue, are responsible for the mechanical action of crushing food into smaller pieces.
• The absorption of nutrients occurs predominantly in the small intestine, specifically through the villi and microvilli in the jejunum and ileum.
• Non-Digestive Functions:
• Beyond digestion, the tongue is essential for phonation (speech) and helps in cleaning the teeth by dislodging food particles after a meal.

Shortcut Trick

Relative Speed = Total distance ÷ Time taken

Total distance = 282 + 248 = 530 m

Relative Speed (in m/s) = 530 ÷ 12 = 265/6 m/s

Relative Speed (in km/hr) = (265/6) × (18/5) = 53 × 3 = 159 km/hr

Speed of second train = 159 − 68 = 91 km/hr

∴ The correct answer is 91 km/hr.

Alternate Method

Given:

Length of first train (L₁) = 282 m

Length of second train (L₂) = 248 m

Time taken (t) = 12 seconds

Speed of first train (S₁) = 68 km/hr

Formula Used:

Relative Speed = (L₁ + L₂) / Time

Conversion: km/hr to m/s = Multiply by 5/18

Calculations:

⇒ Let the speed of the second train be S₂ km/hr.

⇒ Relative speed = (68 + S₂) km/hr

⇒ (68 + S₂) × (5/18) = (282 + 248) / 12

⇒ (68 + S₂) × (5/18) = 530 / 12

⇒ 68 + S₂ = (530 / 12) × (18 / 5)

⇒ 68 + S₂ = (106) × (1.5)

⇒ 68 + S₂ = 159

⇒ S₂ = 159 − 68

⇒ S₂ = 91 km/hr

∴ The correct answer is 91 km/hr.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: S_rel = S₁ + S₂.

Same Direction Relative Speed

When two objects move in the same direction, their relative speed is the difference between their speeds: S_rel = |S₁ − S₂|.

Train Crossing Objects

To cross a stationary pole or man, a train covers its own length. To cross another train or platform, it covers the sum of both lengths.

Shortcut Trick

Total days from 5 February 2023 to 19 April 2023 = 23 (Feb) + 31 (Mar) + 19 (Apr) = 73 days.

Time (T) in years = 73 ÷ 365 = 1/5 year.

Simple Interest = (Principal × Rate × Time) ÷ 100

SI = (2000 × 7 × 1/5) ÷ 100 = 4 × 7 = 28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate of Interest (R) = 7% per annum

Time (T) = 5 February 2023 to 19 April 2023

Formula Used:

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

Calculations:

⇒ Calculation of days: Feb (28 − 5) = 23, March = 31, April = 19

⇒ Total days = 23 + 31 + 19 = 73 days

⇒ T (in years) = 73 ÷ 365 = 1/5 year

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (2000 × 7) ÷ 500

⇒ SI = 4 × 7 = ₹28

∴ The correct answer is ₹28.

Additional Information

Simple Interest Formula

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

Time Conversion for Interest

When time is given in days, divide by 365 (ordinary year) or 366 (leap year). 2023 is not a leap year, so we use 365.

Counting Days Rule

Standard banking practice excludes the day of deposit but includes the day of withdrawal (or vice versa).

Amount Calculation

The final Amount (A) is the sum of the Principal and the Interest: A = P + SI.

Shortcut Trick

If (m − n) is even, then both m and n must be either both even or both odd.

Case 1: Let m = 3 and n = 1 (both odd) ⇒ m² − n² = 9 − 1 = 8 (divisible by 4).

Case 2: Let m = 4 and n = 2 (both even) ⇒ m² − n² = 16 − 4 = 12 (divisible by 4).

In both cases, the result is divisible by 4.

∴ The correct answer is 4.

Alternate Method

Given: m and n are positive integers and (m − n) is even.

Formula Used: m² − n² = (m − n)(m + n)

⇒ If (m − n) is even, then the difference between m and n is a multiple of 2.

⇒ This implies that m and n have the same parity (both even or both odd).

⇒ If both are even or both are odd, their sum (m + n) must also be even.

⇒ m² − n² = (m − n) × (m + n)

⇒ Since both factors are even, let (m − n) = 2a and (m + n) = 2b.

⇒ m² − n² = 2a × 2b = 4ab.

⇒ Therefore, the expression is always divisible by 4.

∴ The correct answer is 4.

Additional Information

Parity Rules

Even ± Even = Even; Odd ± Odd = Even; Even ± Odd = Odd.

Difference of Squares

For any integers, m² − n² is divisible by (m − n) and (m + n).

Divisibility of Consecutive Squares

The difference between squares of two consecutive odd integers is always divisible by 8.

Shortcut Trick

In a right-angled triangle, if tan A = 3/4, the sides represent a Pythagorean triplet (3, 4, 5).

Here, Perpendicular (P) = 3 and Base (B) = 4.

The Hypotenuse (H) is always 5 for this triplet.

Since cosec A = Hypotenuse / Perpendicular, we get 5/3.

∴ The correct answer is 5/3.

Alternate Method

Given: tan A = 3/4 in ΔABC right-angled at B.

Formula Used: tan A = Perpendicular / Base; cosec A = Hypotenuse / Perpendicular; H² = P² + B²

⇒ Let Perpendicular (BC) = 3k and Base (AB) = 4k.

⇒ AC² = AB² + BC²

⇒ AC² = (4k)² + (3k)² = 16k² + 9k² = 25k²

⇒ AC = √(25k²) = 5k

⇒ cosec A = AC / BC = 5k / 3k

⇒ cosec A = 5/3

∴ The correct answer is 5/3.

Additional Information

Trigonometric Ratios

Standard ratios: sin θ = P/H, cos θ = B/H, tan θ = P/B, cosec θ = H/P, sec θ = H/B, cot θ = B/P.

Pythagorean Triplets

Common sets of integers (a, b, c) such that a² + b² = c². Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

Reciprocal Identities

cosec A = 1 / sin A, sec A = 1 / cos A, cot A = 1 / tan A.

Shortcut Trick

The mode is the value that appears most frequently in a data set.

By scanning the given list: 1 appears 2 times, 3 appears 3 times, 5 appears 2 times, 9 appears 2 times, and 6 appears 5 times.

Since 6 has the highest frequency, it is the mode.

∴ The correct answer is 6.

Alternate Method

Given: Observations = {9, 6, 8, 1, 9, 6, 4, 5, 7, 1, 6, 5, 3, 6, 6, 3, 3}

Formula Used: Mode = Observation with the maximum frequency.

Frequency Distribution of the data:

⇒ 1 occurs 2 times

⇒ 3 occurs 3 times

⇒ 4 occurs 1 time

⇒ 5 occurs 2 times

⇒ 6 occurs 5 times

⇒ 7 occurs 1 time

⇒ 8 occurs 1 time

⇒ 9 occurs 2 times

⇒ Maximum Frequency = 5

⇒ The observation corresponding to the maximum frequency is 6.

∴ The correct answer is 6.

Additional Information

Types of Mode

A data set can be Unimodal (one mode), Bimodal (two modes), or Multimodal (more than two modes).

Empirical Formula

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

Median for Discrete Data

To find the median, arrange the data in ascending order. If N is odd, Median = ((N+1)/2)^th term. If N is even, Median is the average of (N/2)^th and (N/2 + 1)^th terms.

Shortcut Trick

Total amount to be deducted = ₹(62 + 17 + 79) = ₹158.

Remaining amount = ₹10,415 − ₹158 = ₹10,257.

Sum of ratio parts = 15 + 10 + 14 = 39 units.

1 unit = ₹10,257 ÷ 39 = ₹263.

Difference between Q and R (in ratio) = 14 − 10 = 4 units.

Difference in original shares = (4 × 263) + (79 − 17) = 1052 + 62 = ₹1,114.

∴ The correct answer is ₹1,114.

Alternate Method

Given: Total Sum = ₹10,415; Ratio after deductions = 15 : 10 : 14.

Formula: Original Share = (Ratio part × common factor) + deducted amount.

⇒ Total reduction = 62 + 17 + 79 = 158

⇒ New total sum = 10415 − 158 = 10257

⇒ 15x + 10x + 14x = 10257

⇒ 39x = 10257

⇒ x = 10257 ÷ 39 = 263

⇒ Original share of Q = (10 × 263) + 17 = 2630 + 17 = 2647

⇒ Original share of R = (14 × 263) + 79 = 3682 + 79 = 3761

⇒ Difference = 3761 − 2647 = 1114

∴ The correct answer is ₹1,114.

Additional Information

Ratio & Proportion Basics

If a total sum S is divided into a:b:c, then share of A = (a / (a+b+c)) × S.

Inverse Variation

If x varies inversely as y, then x × y = k (constant). In ratios, if a:b is the ratio of quantities, the inverse ratio is 1/a : 1/b.

Compound Ratio

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

Shortcut Trick

The product of the number of sides n and (n − 3) is always twice the number of diagonals.

n × (n − 3) = 2 × 665 = 1330

Now, find a value of n such that n × (n − 3) = 1330.

By checking the options, if n = 38, then 38 × (38 − 3) = 38 × 35 = 1330.

This satisfies the condition perfectly.

∴ The correct answer is 38.

Alternate Method

Given:

Number of diagonals of the regular polygon = 665

Formula Used:

Number of diagonals of an n-sided polygon = [n(n − 3)] / 2

⇒ 665 = [n(n − 3)] / 2

⇒ 1330 = n² − 3n

⇒ n² − 3n − 1330 = 0

⇒ n² − 38n + 35n − 1330 = 0

⇒ n(n − 38) + 35(n − 38) = 0

⇒ (n − 38)(n + 35) = 0

⇒ n = 38 or n = −35

Since the number of sides n must be a positive integer, we take n = 38.

∴ The correct answer is 38.

Additional Information

Sum of Interior Angles

The sum of all interior angles of an n-sided polygon is (n − 2) × 180°.

Exterior Angles Property

The sum of all exterior angles of any convex polygon is always 360°.

Regular Polygon Angles

For a regular polygon, each interior angle is [(n − 2) × 180°] / n and each exterior angle is 360° / n.

Shortcut Trick

Calculate the semi-perimeter (s) of the triangle: s = (22 + 18 + 24) ÷ 2 = 32 cm.

Triangle Area = √(32 × 10 × 14 × 8) = 32√35 cm².

Since Triangle Area = 1/3 × Parallelogram Area, then Parallelogram Area = 3 × 32√35 = 96√35 cm².

Height (h) = Area ÷ Base = (96 × √35) ÷ 24 = 4 × √35 ≈ 4 × 5.916 = 23.66 cm.

∴ The correct answer is 23.6 cm.

Alternate Method

Given: Sides of triangle (a, b, c) = 22 cm, 18 cm, 24 cm. Base of parallelogram (b_p) = 24 cm.

Formula Used:
Heron's Formula for Triangle Area = √(s(s−a)(s−b)(s−c))
Area of Parallelogram = Base × Height

⇒ Semi-perimeter of triangle (s) = (22 + 18 + 24) ÷ 2 = 32 cm

⇒ Area of triangle = √(32 × (32 − 22) × (32 − 18) × (32 − 24))

⇒ Area of triangle = √(32 × 10 × 14 × 8) = √35840 = 32√35 cm²

⇒ Area of parallelogram = 3 × Area of triangle = 3 × 32√35 = 96√35 cm²

⇒ Base × Height = 96√35

⇒ 24 × h = 96√35

⇒ h = (96 ÷ 24) × √35 = 4√35

⇒ h ≈ 4 × 5.916 = 23.664 cm

∴ The correct answer is 23.6 cm.

Additional Information

Heron's Formula

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

Parallelogram Properties

Area is calculated as Base × Perpendicular Height. The diagonal of a parallelogram divides it into two triangles of equal area.

Relationship between Triangle and Parallelogram

If a triangle and a parallelogram have the same base and height, the area of the triangle is exactly half the area of the parallelogram.

Shortcut Trick

Combined efficiency of P + Q = 1/12 units/hr.

Efficiency of P = 1/24 units/hr.

Efficiency of Q = 1/12 − 1/24 = 1/24 units/hr.

Time for Q to fill 1/4 cistern = (1/4) ÷ (1/24) = 6 hours.

∴ The correct answer is 6 hours.

Alternate Method

⇒ Total capacity of the cistern = 24 units (LCM of 12 and 24).

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

⇒ Efficiency of Q = 2 − 1 = 1 unit/hr.

⇒ Work to be done = 1/4 of total cistern = 24 × (1/4) = 6 units.

⇒ Time taken by Q = 6 ÷ 1 = 6 hours.

∴ The correct answer is 6 hours.

Additional Information

Time and Work Relation

Total Work = Efficiency × Time. Efficiency is inversely proportional to the time taken to complete a task.

Pipes and Cisterns

Filling pipes have positive efficiency, while outlet/leakage pipes have negative efficiency. Net work is the sum of their individual efficiencies.

LCM Method

Using the LCM of the given time durations as the total capacity of the tank simplifies calculations by avoiding fractions.

Shortcut Trick

Ratio 5 years ago (Meera : Kavya) = 3 : 1 (Difference = 2)

Ratio 7 years later (Meera : Kavya) = 2 : 1 (Difference = 1)

Equalize difference: 5 yrs ago = 3 : 1 | 7 yrs later = 4 : 2 (Multiplied by 2)

Ratio change 1 unit → 5 + 7 = 12 years

Meera 5 years ago = 3 × 12 = 36 years

Current age = 36 + 5 = 41 years

∴ The correct answer is 41 years.

Alternate Method

Let Meera and Kavya current ages be M and K respectively.

Given:

5 years ago: (M − 5) = 3(K − 5)

7 years later: (M + 7) = 2(K + 7)

⇒ M − 5 = 3K − 15 ⇒ M − 3K = −10 --- (i)

⇒ M + 7 = 2K + 14 ⇒ M − 2K = 7 --- (ii)

⇒ Subtracting (i) from (ii): (M − 2K) − (M − 3K) = 7 − (−10)

⇒ K = 17 years

⇒ Substitute K in (ii): M − 2(17) = 7

⇒ M − 34 = 7

⇒ M = 41 years

∴ The correct answer is 41 years.

Additional Information

Age Difference Constant

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

Ratio Method for Ages

When solving age problems using ratios, always ensure the numerical difference between the ratio parts of the two individuals is made equal across different time periods.

Linear Equation Method

Standard algebraic approach involves setting up equations for "n years ago" (x − n) and "n years later" (x + n).

Shortcut Trick

Discount = Marked Price − Selling Price

Discount = 657 − 526 = 131

Discount% = (Discount ÷ MP) × 100

Discount% = (131 ÷ 657) × 100 = 19.9391...

Rounding off to two decimal places, we get 19.94%.

∴ The correct answer is 19.94%.

Alternate Method

Given → Marked Price (MP) = ₹657, Selling Price (SP) = ₹526. Formulas: Discount = MP − SP | Discount% = (Discount ÷ MP) × 100

→ Discount = 657 − 526 = 131 → Discount% = (131 ÷ 657) × 100 → Discount% = 13100 ÷ 657 → Discount% ≈ 19.939% → Rounded Discount% = 19.94%

∴ The correct answer is 19.94%.

Additional Information

Marked Price and Discount

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

Selling Price Calculation

Selling Price can also be calculated if discount percentage is known: SP = MP × (100 − Discount%)/100.

Shortcut Trick

If each number in a series is multiplied by a constant (k), the new average is simply the old average multiplied by that same constant.

Old Average = 7

Multiplier (k) = 3

New Average = 7 × 3 = 21

∴ The correct answer is 21.

Alternate Method

Given:

Total numbers (n) = 13

Original average = 7

⇒ Sum of 13 numbers = Average × Count

⇒ Original Sum = 7 × 13 = 91

If each number is multiplied by 3, the total sum also increases 3 times:

⇒ New Sum = 91 × 3 = 273

⇒ New Average = New Sum ÷ Count

⇒ New Average = 273 ÷ 13

⇒ New Average = 21

∴ The correct answer is 21.

Additional Information

Effect of Basic Operations on Average

If every term is increased or decreased by x, the average increases or decreases by x respectively. If divided by x, the average is divided by x.

Weighted Average

When combining two groups, the weighted average = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂), where n is the count and a is the average.

Average of Consecutive Numbers

For any Arithmetic Progression (AP), the average is always the middle term or (First Term + Last Term) ÷ 2.

Shortcut Trick

Ratio of SP₁ to SP₂ is 1000 : 1400 = 5 : 7.

Since SP₁ represents a 5% loss, 5 units = (100 − 5)% = 95%.

Then 1 unit = 95 ÷ 5 = 19%.

SP₂ represents 7 units = 7 × 19% = 133%.

The profit percentage x = 133% − 100% = 33%.

∴ The correct answer is 33.

Alternate Method

Given: Selling Price 1 (SP₁) = ₹1,000 | Loss% = 5% | Selling Price 2 (SP₂) = ₹1,400. Formulas: CP = SP × 100 ÷ (100 − Loss%) | Profit% = (SP − CP) ÷ CP × 100.

⇒ Calculate CP: CP = 1000 × 100 ÷ (100 − 5) = 1000 × 100 ÷ 95. ⇒ CP = 100000 ÷ 95 = ₹1052.63 (approx) or 20000 ÷ 19. ⇒ Profit = SP₂ − CP = 1400 − (20000 ÷ 19). ⇒ Profit = (26600 − 20000) ÷ 19 = 6600 ÷ 19. ⇒ Profit% (x) = (6600 ÷ 19) ÷ (20000 ÷ 19) × 100. ⇒ x = (6600 ÷ 20000) × 100 = 33.

∴ The correct answer is 33.

Additional Information

Cost Price (CP)

The price at which an article is purchased. It is the base for calculating Profit or Loss percentages.

Selling Price (SP)

The price at which an article is sold. If SP > CP, there is a profit. If SP < CP, there is a loss.

Shortcut Trick

Income ratio = 5 : 8 | Expenditure ratio = 1 : 4

Difference in income parts = 8 − 5 = 3 | Difference in expenditure parts = 4 − 1 = 3

Since differences are same, (5 − 1) units = (8 − 4) units = 4 units = savings

⇒ 4 units = ₹75,000 → 1 unit = ₹18,750

Mohan income = 5 units = 5 × ₹18,750 = ₹93,750

∴ The correct answer is ₹93,750.

Alternate Method

Given: Ratio of Incomes = 5 : 8, Ratio of Expenditures = 1 : 4, Savings = ₹75,000 each.

Formula: Income − Expenditure = Savings

⇒ Let Mohan income be 5x and David income be 8x.

⇒ Expenditure of Mohan = 5x − 75,000

⇒ Expenditure of David = 8x − 75,000

⇒ (5x − 75,000) / (8x − 75,000) = 1 / 4

⇒ 4 × (5x − 75,000) = 1 × (8x − 75,000)

⇒ 20x − 3,00,000 = 8x − 75,000

⇒ 12x = 2,25,000

⇒ x = 18,750

⇒ Mohan income = 5x = 5 × 18,750 = 93,750

∴ The correct answer is ₹93,750.

Additional Information

Income, Expenditure and Savings Relationship

The fundamental relation is Income = Expenditure + Savings. If savings are constant, the change in income directly affects the expenditure.

Ratio Balancing Method

When a constant value is added or subtracted from both terms of a ratio, we balance the difference between ratio parts to solve quickly without variables.

Cross Multiplication Trick

For two ratios (a:b and c:d) and a constant difference (k), the variable can be found by: x = [k × (d − c)] / [ad − bc].

Shortcut Trick

Ratio of dimensions = 6 : 5 : 4 ⇒ Volume units = 6 × 5 × 4 = 120 units³

120 units³ = 810000 √2 ⇒ 1 unit³ = 6750 √2 = 3375 × (2 √2) = 15³ × (√2)³

Side ratio constant (1 unit) = 15 √2 cm

Total Surface Area units = 2 × (30 + 20 + 24) = 148 units²

Surface Area = 148 × (15 √2)² = 148 × (225 × 2) = 148 × 450 = 66600 cm²

∴ The correct answer is 66600 cm².

Alternate Method

Given:

Ratio of Length, Breadth and Height = 6 : 5 : 4

Volume of the Cuboid = 810000 √2 cm³

Formula Used:

Volume of Cuboid = Length × Breadth × Height

Total Surface Area = 2 × (LB + BH + HL)

Calculations:

⇒ Let Length = 6x, Breadth = 5x, and Height = 4x

⇒ Volume = 6x × 5x × 4x = 120x³

⇒ 120x³ = 810000 √2

⇒ x³ = 6750 √2 = 3375 × 2 × √2 = 3375 × (√2)³

⇒ x³ = 15³ × (√2)³ ⇒ x = 15 √2

⇒ Total Surface Area = 2 × (6x × 5x + 5x × 4x + 4x × 6x)

⇒ Surface Area = 2 × (30x² + 20x² + 24x²) = 2 × 74x² = 148x²

⇒ Surface Area = 148 × (15 √2)² = 148 × (225 × 2) = 148 × 450

⇒ Surface Area = 66600 cm²

∴ The correct answer is 66600 cm².

Additional Information

Diagonal of a Cuboid

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

Lateral Surface Area (LSA)

The area of the four vertical walls of a cuboid is given by LSA = 2 × Height × (Length + Breadth).

Volume and Surface Area of a Cube

For a cube with side 'a', the Volume is a³ and the Total Surface Area is 6a².

Shortcut Trick

Ratio of articles P : Q : R = 3 : 4 : 5

Total Value ratio = (280 × 3) : (180 × 4) : (100 × 5) = 840 : 720 : 500 = 2060 units

2060 units = ₹6,180 ⇒ 1 unit = 3

Articles of type Q = 4 × 3 = 12

∴ The correct answer is 12 articles.

Alternate Method

Given: Prices of P, Q, R are ₹280, ₹180, ₹100. Ratio of quantity = 3 : 4 : 5. Total expenditure = ₹6,180.

Calculations:

⇒ Let the number of articles be 3x, 4x, and 5x respectively.

⇒ (280 × 3x) + (180 × 4x) + (100 × 5x) = 6180

⇒ 840x + 720x + 500x = 6180

⇒ 2060x = 6180

⇒ x = 6180 ÷ 2060 = 3

⇒ Number of articles of type Q = 4x = 4 × 3 = 12

∴ The correct answer is 12 articles.

Additional Information

Ratio and Value Relationship

When quantities are given in a ratio (a:b:c) and their individual values are (v₁, v₂, v₃), the total value is calculated as (a × v₁ + b × v₂ + c × v₃) × k.

Unitary Method in Ratios

Finding the value of one 'part' or 'unit' by dividing the total actual value by the sum of ratio-product parts allows for quick calculation of any individual component.

Shortcut Trick

Net discount for 10% and 15% = 10 + 15 − (10 × 15) ÷ 100 = 23.5%.

Net discount for 23.5% and 7% = 23.5 + 7 − (23.5 × 7) ÷ 100.

= 30.5 − 1.645 = 28.855%.

The approximate single discount is 28.85%.

∴ The correct answer is 28.85%.

Alternate Method

Given → Discounts are d₁ = 7%, d₂ = 10%, and d₃ = 15%. Formulas: Single Equivalent Discount = 100 − [100 × (100 − d₁)/100 × (100 − d₂)/100 × (100 − d₃)/100]

⇒ Assume Marked Price (MP) = 100. ⇒ Price after 7% discount = 100 × 0.93 = 93. ⇒ Price after 10% discount = 93 × 0.90 = 83.7. ⇒ Price after 15% discount = 83.7 × 0.85 = 71.145. ⇒ Total Discount = 100 − 71.145 = 28.855%. ⇒ Approximate value = 28.85%.

∴ The correct answer is 28.85%.

Additional Information

Successive Discount Formula

For two discounts a% and b%, the single equivalent discount is (a + b − ab/100)%.

Marked Price & Discount

Discount is always calculated on the Marked Price (MP). Selling Price (SP) = MP − Discount.

Shortcut Trick

Total Work (LCM of 15, 18, 27) = 270 units.

Efficiencies: A = 18, B = 15, C = −10.

First 5 hr (A+B): (18 + 15) × 5 = 165 units.

Next 12 hr (B+C): (15 − 10) × 12 = 60 units.

Remaining: 270 − (165 + 60) = 45 units.

Time for B alone: 45 ÷ 15 = 3 hours.

∴ The correct answer is 3 hr.

Alternate Method

⇒ Work done by Pipe A and Pipe B in 5 hours = (18 + 15) × 5 = 33 × 5 = 165 units.

⇒ Work done by Pipe B and Pipe C in 12 hours = (15 − 10) × 12 = 5 × 12 = 60 units.

⇒ Total work done so far = 165 + 60 = 225 units.

⇒ Remaining work = Total capacity − Work done = 270 − 225 = 45 units.

⇒ Time taken by Pipe B alone = Remaining work ÷ Efficiency of B = 45 ÷ 15 = 3 hours.

∴ The correct answer is 3 hr.

Additional Information

Inlet and Outlet Pipes

Inlet pipes add water (positive efficiency) while outlet pipes remove water (negative efficiency). Net efficiency is the algebraic sum.

Work and Time Relationship

Total Work = Time taken × Efficiency. For cisterns, capacity is the LCM of individual filling/emptying times.

Shortcut Trick

To find the greatest possible scale to measure the dimensions exactly, find the HCF (Highest Common Factor) of the given dimensions.

Given dimensions: 16 m, 20 m, 24 m.

HCF(16, 20, 24) = 4 m.

∴ The correct answer is 4 m.

Alternate Method

Given:

Dimensions of the room = 16 m, 20 m, 24 m

Formula:

Length of the greatest possible scale = HCF (Highest Common Factor) of the dimensions

Calculations:

⇒ Prime factorization of 16 = 2 × 2 × 2 × 2 = 2⁴

⇒ Prime factorization of 20 = 2 × 2 × 5 = 2² × 5

⇒ Prime factorization of 24 = 2 × 2 × 2 × 3 = 2³ × 3

⇒ HCF(16, 20, 24) = 2² = 4

⇒ Greatest scale length = 4 m

∴ The correct answer is 4 m.

Additional Information

Highest Common Factor (HCF)

The HCF of two or more numbers is the greatest number that divides each of them exactly without leaving a remainder.

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all the given numbers.

HCF and LCM Relation

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

Shortcut Trick

Recognize the algebraic identity: (a + b)² − 4ab = (a − b)².

Here, a = 3x and b = 5y, so 4ab = 4 × 3x × 5y = 60xy.

The numerator becomes (3x − 5y)², which is identical to the denominator.

Expression = (3x − 5y)² ÷ (3x − 5y)² = 1.

∴ The correct answer is 1.

Alternate Method

Given: [(3x + 5y)² − 60xy] ÷ (3x − 5y)²

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

⇒ Expand the numerator: (3x)² + 2(3x)(5y) + (5y)² − 60xy

⇒ Numerator = 9x² + 30xy + 25y² − 60xy = 9x² − 30xy + 25y²

⇒ Expand the denominator: (3x − 5y)² = (3x)² − 2(3x)(5y) + (5y)²

⇒ Denominator = 9x² − 30xy + 25y²

⇒ Division = (9x² − 30xy + 25y²) ÷ (9x² − 30xy + 25y²)

⇒ 1

∴ The correct answer is 1.

Additional Information

Standard Algebraic Identities

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b² are basic expansions.

Relationship Identity

(a + b)² − (a − b)² = 4ab is often used to simplify complex polynomial divisions.

Alternative Relationship

(a + b)² + (a − b)² = 2(a² + b²) is another useful variant in simplification.

Shortcut Trick

Test the given options to satisfy both Malati and Rani conditions.

For Option (4): Rose = ₹120, Jasmine = ₹90.

Condition 1: (5 × 120) + (9 × 90) = 600 + 810 = ₹1410 (Satisfied).

Condition 2: (6 × 120) + (5 × 90) = 720 + 450 = ₹1170 (Satisfied).

∴ The correct answer is 120; 90.

Alternate Method

Let the price of 1 Rose plant be ₹ R and 1 Jasmine plant be ₹ J.

Given:

Case 1 (Malati): 5R + 9J = 1410 → (Eq. 1)

Case 2 (Rani): 6R + 5J = 1170 → (Eq. 2)

Multiply (Eq. 1) by 5 and (Eq. 2) by 9 to eliminate J:

⇒ 25R + 45J = 7050

⇒ 54R + 45J = 10530

Subtract the first resulting equation from the second:

⇒ (54R − 25R) = 10530 − 7050

⇒ 29R = 3480

⇒ R = 3480 ÷ 29 = 120

Substitute R = 120 in (Eq. 2):

⇒ 6(120) + 5J = 1170

⇒ 720 + 5J = 1170

⇒ 5J = 1170 − 720

⇒ 5J = 450

⇒ J = 90

∴ The correct answer is 120; 90.

Additional Information

Linear Equations in Two Variables

A pair of equations with two variables (like x and y) can be solved using Elimination or Substitution methods to find unique values.

Elimination Method

Multiply equations by suitable constants so that the coefficients of one variable become equal, allowing its removal through subtraction.

Verification

Always verify the calculated values by substituting them back into the original equations to ensure they satisfy the given conditions.

Shortcut Trick

Ratio of cost of TV to Washing Machine (WM) = 100 : 75 = 4 : 3.

Total initial cost for 4 WM and 5 TV = (4 × 3) + (5 × 4) = 12 + 20 = 32 units.

Change in WM cost = +87% of 12 = 10.44; Change in TV cost = −45% of 20 = −9.

Net change = 10.44 − 9 = 1.44 units.

Percentage change = (1.44 ÷ 32) × 100 = 4.5%.

∴ The correct answer is Increase by 4.5%.

Alternate Method

Formulas: Total Cost = (Price × Quantity) | Percentage Change = (New − Old) ÷ Old × 100

⇒ Let the cost of TV be 100. Then cost of Washing Machine = 75.

⇒ Total Initial Cost = (4 × 75) + (5 × 100) = 300 + 500 = 800.

⇒ New WM cost = 75 + (87% of 75) = 75 + 65.25 = 140.25.

⇒ New TV cost = 100 − (45% of 100) = 55.

⇒ Total New Cost = (4 × 140.25) + (5 × 55) = 561 + 275 = 836.

⇒ Percentage Change = [(836 − 800) ÷ 800] × 100 = (36 ÷ 800) × 100 = 4.5%.

∴ The correct answer is Increase by 4.5%.

Additional Information

Percentage Increase and Decrease

To increase a value by x%, multiply it by (100 + x) ÷ 100. To decrease by x%, multiply by (100 − x) ÷ 100.

Weighted Average Concept

Net % change = (w₁c₁ + w₂c₂) ÷ (w₁ + w₂), where w is initial weight and c is % change.

Shortcut Trick

Total Distance = Sum of lengths = 232 + 248 = 480 m

Relative Speed (Opposite) = 51 + 57 = 108 km/hr

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

Time = Distance ÷ Relative Speed = 480 ÷ 30 = 16 seconds

∴ The correct answer is 16 sec.

Alternate Method

Given:

Length of first train (L₁) = 232 m

Length of second train (L₂) = 248 m

Speed of first train (S₁) = 51 km/hr

Speed of second train (S₂) = 57 km/hr

Formula Used:

Time = (Total Length) ÷ (Relative Speed)

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

Calculations:

⇒ Total distance to be covered = L₁ + L₂ = 232 + 248 = 480 m

⇒ Relative speed in opposite direction = S₁ + S₂ = 51 + 57 = 108 km/hr

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

⇒ Time taken = 480 ÷ 30 = 16 seconds

∴ The correct answer is 16 sec.

Additional Information

Relative Speed Concepts

When two objects move in opposite directions, relative speed is (S₁ + S₂). When they move in the same direction, relative speed is |S₁ − S₂|.

Unit Conversion

Always ensure units are consistent. Multiply km/hr by 5/18 to get m/s; multiply m/s by 18/5 to get km/hr.

Train Crossing a Point Object

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

Shortcut Trick

0.36 = (3/5)², 0.216 = (3/5)³, 0.1296 = (3/5)⁴

LHS = [(3/5)²]³ × [(3/5)³]³ ÷ [(3/5)⁴]³

= (3/5)⁶ × (3/5)⁹ ÷ (3/5)¹²

= (3/5)³

RHS = (60/100)^x+1 = (3/5)^x+1

⇒ (3/5)³ = (3/5)^x+1

⇒ x + 1 = 3 ⇒ x = 2

∴ Answer = 2

Alternate Method

Given:

(0.36)³ × (0.216)³ ÷ (0.1296)³ = (60/100)^x+1

Formula used:

a^m × aⁿ ÷ a^p = a^m+n−p

Calculation:

⇒ 0.36 = (3/5)²

⇒ 0.216 = (3/5)³

⇒ 0.1296 = (3/5)⁴

⇒ LHS = [(3/5)²]³ × [(3/5)³]³ ÷ [(3/5)⁴]³

⇒ = (3/5)⁶ × (3/5)⁹ ÷ (3/5)¹²

⇒ = (3/5)⁶⁺⁹⁻¹² = (3/5)³

⇒ RHS = (60/100)^x+1 = (3/5)^x+1

⇒ Equate powers:

⇒ x + 1 = 3

⇒ x = 2

∴ The correct answer is 2.

Additional Information

Exponent Rule

Add/subtract powers for same base

Base Conversion

Convert decimals into fractional powers

Equality Rule

If bases are same, equate exponents

Key Insight

Simplify into single base before solving

Shortcut Trick

Required percentage = (Marks of B / Marks of A) × 100

Required percentage = (50 / 75) × 100 = (2 / 3) × 100

2/3 of 100% is approximately 66.67%.

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by B = 50

Marks obtained by A = 75

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

⇒ Required Percentage = (Marks of B / Marks of A) × 100

⇒ Required Percentage = (50 / 75) × 100

⇒ Required Percentage = (2 / 3) × 100

⇒ Required Percentage = 66.666...

⇒ Rounding off to two decimal places, we get 66.67%.

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

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

Percentage of Total

To find what percentage a score is of the maximum marks, use: (Obtained Marks / Maximum Marks) × 100.

Fraction to Percentage Conversion

Common fractions like 1/3 ≈ 33.33%, 2/3 ≈ 66.67%, and 1/4 = 25% are useful for quick calculations.

The logic followed here is:

For, CPWG → XTTI

For, NILC → IMIE

Now, apply the same to OKZD:

So, OKZD → JOWF

Hence, the correct answer is Option 3.

BODMAS Table: 

Decoding the given information:

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

⇒ 24 × 19 − 84 ÷ 7 + 127 = ?

⇒ 456 − 12 + 127 = ?

⇒ 444 + 127 = ?

⇒ 571

Hence, the correct option is "Option 1".

The logic followed here is:

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

For, (25, 72, 194)

⇒ 25 + 72 = 97

⇒ 97 × 2 = 194

For, (32, 80, 224)

⇒ 32 + 80 = 112

⇒ 112 × 2 = 224

Now, checking the options using the same pattern:

Option 1) (45, 89, 268)

⇒ 45 + 89 = 134

⇒ 134 × 2 = 268

Option 2) (74, 88, 304)

⇒ 74 + 88 = 162

⇒ 162 × 2 = 324 ≠ 304

Option 3) (91, 65, 310)

⇒ 91 + 65 = 156

⇒ 156 × 2 = 312 ≠ 310

Option 4) (56, 98, 312)

⇒ 56 + 98 = 154

⇒ 154 × 2 = 308 ≠ 312

Hence, the correct answer is Option 1.

Given: (Left) @ 3 6 7 & 8 # & + $ 1 ? 9 1 + 7 1 # # < 2 (Right)

Now,

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

So, Needed pattern is: Symbol - Symbol - Number

(Left) @ 3 6 7 & 8 # & + $ 1 ? 9 1 + 7 1 # # < 2 (Right)

So, total valid symbols = 2

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "4" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

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

Only two people sit between F and G when counted from the left of G.

A sits third to the left of E.

C sits to the immediate right of E.

C sits second to the left of F.

B is an immediate neighbour of A.

So, all 5 people  lie between F and D.

Hence, the correct answer is "Option 1".

Given:

OAVY is coded as 56 and SIXT is coded as 47

Logic:  (sum of reverse positions) + 11

Here,

OAVY → O(12) + A(26) + V(5) + Y(2) = 45 → 45 + 11 = 56

SIXT → S(8) + I(18) + X(3) + T(7) = 36 → 36 + 11 = 47

Now,  Apply to TMZB

T(7) + M(14) + Z(1) + B(25) = 47

Code = 47 + 11 = 58

Hence, the correct answer is Option 4.

The logic followed here is:

So, 'TYC95' will replace the question mark in the given series.

Hence, the correct answer is "Option 1"

Given: (Left) # 4 ^ 8 8 < 2 / $7$ \ 1 & ^ 8 4 \ ? \ 8 (Right)

Now,

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

So, Needed pattern is: Number - Number - Symbol

(Left) # 4 ^ 8 8 < 2 / $7$ \ 1 & ^ *8 4 * ? \ 8 (Right)

So, total valid numbers = 2

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "98" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

BODMAS Table: 

Given equation: 12 ÷ 5 + 21 × 3 - 2 = ? + 5

<br>

Now, replacing the symbols:

⇒ 12 × 5 - 21 ÷ 3 + 2 = ? - 5

⇒ 12 × 5 - 21 ÷ 3 + 2 = ? - 5

⇒ 60 - 21 ÷ 3 + 2 = ? - 5

⇒ 60 - 7 + 2 = ? - 5

⇒ 60 - 7 + 2 = ? - 5

⇒ 53 + 2 = ? - 5

⇒ 55 = ? - 5

⇒ ? = 55 + 5

⇒ 60

Hence, the correct answer is "Option 4".