RRB Group D Official Paper (Held On: 11 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is Blue litmus paper will turn red.

Key Points

• Acid rain contains acids such as sulfuric acid (H₂SO₄) and nitric acid (HNO₃), formed when sulfur dioxide (SO₂) and nitrogen oxides (NOₓ) react with water vapor in the atmosphere.
• Blue litmus paper is an indicator used to test the acidity or basicity of a solution.
• When blue litmus paper comes in contact with an acidic substance (pH less than 7), it turns red, indicating the presence of an acid.
• Since acid rain water has a low pH (typically between 4-5), it is acidic in nature. Hence, when a drop of acid rain water is placed on blue litmus paper, the paper turns red.
• This reaction is due to the presence of hydrogen ions (H⁺) in the acid rain water, which interact with the litmus indicator, causing the color change.

Additional Information

• Litmus Paper:
• Litmus paper is a natural indicator made from lichens. It is widely used to test the acidity or alkalinity of substances.
• Blue litmus paper turns red in the presence of an acid, while red litmus paper turns blue in the presence of a base.
• Neutral substances do not cause any color change in litmus paper.
• Acid Rain:
• Acid rain occurs due to the emission of sulfur dioxide (SO₂) and nitrogen oxides (NOₓ) from industrial processes, vehicular emissions, and burning of fossil fuels.
• These gases react with atmospheric water and oxygen to form acids such as sulfuric acid and nitric acid, which precipitate as acid rain.
• Acid rain has harmful effects on plants, aquatic life, buildings, and human health.
• pH Scale:
• The pH scale measures the acidity or alkalinity of a solution, ranging from 0 to 14.
• A pH value less than 7 indicates an acidic solution, while a value greater than 7 indicates a basic solution.
• A pH of 7 indicates a neutral solution, such as pure water.
• Environmental Impact of Acid Rain:
• Acid rain damages crops, forests, and aquatic ecosystems by lowering the pH of soil and water bodies.
• It corrodes buildings, monuments, and infrastructure, especially those made of limestone or marble.
• Efforts to reduce acid rain include minimizing SO₂ and NOₓ emissions through cleaner energy sources and stricter regulations.

Important Points

• Indicators: Indicators like litmus paper, phenolphthalein, and methyl orange help in identifying the nature of substances (acidic, basic, or neutral).
• Reducing Acid Rain: Adoption of renewable energy sources, installation of scrubbers in industries, and promoting afforestation are effective measures to mitigate acid rain.
• Global Efforts: International agreements like the Clean Air Act and protocols aim to reduce emissions causing acid rain.

The correct answer is Menstruation.

Key Points

• Menstruation refers to the periodic shedding of the uterine lining (endometrium) in human females when fertilization of the egg does not occur.
• This process is part of the menstrual cycle, which typically lasts around 28 days, though it can range from 21 to 35 days in different individuals.
• Menstruation occurs due to a drop in the levels of the hormones estrogen and progesterone, leading to the breakdown and discharge of the endometrial lining.
• The discharge during menstruation consists of blood, mucus, and tissue from the uterine lining, and it typically lasts for 3-7 days.
• It is a natural and essential part of the female reproductive cycle, preparing the body for a potential pregnancy each month. If fertilization does not occur, the uterine lining is shed, and the cycle begins again.

Additional Information

• Parturition:
• Parturition refers to the process of giving birth to a baby. It typically occurs at the end of the gestation period (about 9 months in humans).
• This process involves uterine contractions, dilation of the cervix, and the delivery of the baby and placenta.
• Parturition is not related to the menstrual cycle but is a culmination of pregnancy.
• Zygote Formation:
• A zygote is formed when a sperm cell fertilizes an egg cell. This marks the beginning of pregnancy in humans.
• The zygote undergoes multiple cell divisions and develops into an embryo, eventually implanting itself into the uterine lining.
• If fertilization does not occur, zygote formation does not take place, and menstruation begins.
• Embryo Development:
• Embryo development is the process by which a fertilized egg (zygote) grows and differentiates into a multicellular organism.
• This process includes stages like cleavage, blastocyst formation, implantation, and further differentiation.
• Embryo development occurs only when fertilization has taken place, making it irrelevant in the case of menstruation.
• Hormonal Regulation in Menstruation:
• The menstrual cycle is regulated by hormones such as estrogen, progesterone, luteinizing hormone (LH), and follicle-stimulating hormone (FSH).
• FSH stimulates the growth of ovarian follicles, while LH triggers ovulation. Estrogen and progesterone regulate the thickening of the uterine lining.
• When fertilization does not occur, the levels of estrogen and progesterone drop, leading to the shedding of the uterine lining (menstruation).

Important Points

• Duration of Menstrual Cycle: The average menstrual cycle is about 28 days, but it can vary from 21 to 35 days.
• Age of Onset: Menstruation typically begins during puberty, around the age of 11-14 years, and continues until menopause (around 45-55 years).
• Common Symptoms: Symptoms during menstruation can include cramps, mood swings, fatigue, bloating, and headaches.
• Significance: Menstruation is a natural biological process that prepares the body for pregnancy and reflects a healthy reproductive system.

The correct answer is Object height is negative when the object is placed above the principal axis..

Key Points

• Statement 1: Object distance is negative.
• According to the standard sign convention for spherical lenses, distances measured against the direction of the incoming light are taken as negative. Since the object is usually placed to the left of the lens (against the direction of light), the object distance is always negative.
• Hence, this statement is correct.
• Statement 2: Object height is negative when the object is placed above the principal axis.
• In the sign convention for spherical lenses, heights measured above the principal axis are considered positive, while heights measured below are taken as negative. Thus, the object height cannot be negative if the object is placed above the principal axis.
• Hence, this statement is incorrect.
• Statement 3: The focal length of a convex lens is positive.
• A convex lens converges light rays, and its focal point lies on the side opposite to the object. Therefore, the focal length of a convex lens is taken as positive.
• Hence, this statement is correct.
• Statement 4: The focal length of a concave lens is negative.
• A concave lens diverges light rays, and its focal point lies on the same side as the object. Therefore, the focal length of a concave lens is taken as negative.
• Hence, this statement is correct.

Additional Information

• Sign Convention for Spherical Lenses:
• Distances are measured from the optical center of the lens.
• Distances measured in the direction of the incoming light are positive, while distances against the direction of light are negative.
• Heights above the principal axis are positive, and heights below the axis are negative.
• Convex Lens Characteristics:
• A convex lens is a converging lens that focuses light rays to a point on the other side of the lens.
• Focal length is positive, and the lens is used in applications like magnifying glasses, cameras, and telescopes.
• Concave Lens Characteristics:
• A concave lens is a diverging lens that spreads out light rays, appearing to come from a focal point on the same side as the object.
• Focal length is negative, and the lens is used in applications like eyeglasses for myopia (short-sightedness) and peepholes.
• Applications of Spherical Lenses:
• Convex lenses are used in devices like projectors, microscopes, and cameras due to their converging properties.
• Concave lenses are used in optical instruments like binoculars and corrective glasses for certain vision impairments.

Important Points

• Lens Formula:
• The lens formula is given by 1/f = 1/v - 1/u, where:
• f: Focal length of the lens
• v: Image distance
• u: Object distance
• This formula applies to both convex and concave lenses, with the sign convention dictating the values of v, u, and f.
• Magnification:
• The magnification (M) of a lens is given by M = h'/h = -v/u, where:
• h': Height of the image
• h: Height of the object
• A positive magnification indicates an erect image, while a negative magnification indicates an inverted image.

The correct answer is Potential difference across the bulb.

Key Points

• A voltmeter is an instrument specifically designed to measure the potential difference (voltage) between two points in an electrical circuit.

• When connected across the terminals of a glowing bulb, the voltmeter measures the voltage drop across the bulb. This drop indicates the amount of electrical energy being consumed by the bulb per unit charge.

• The voltmeter operates by having a high internal resistance, which ensures that it does not significantly affect the circuit or draw current during measurement. Hence, the voltmeter measures the potential difference across the bulb.

• The voltmeter does not measure the current through the bulb, as current measurement requires an ammeter connected in series with the circuit. Hence, option 2 is incorrect.

• The voltmeter does not directly measure the heat produced by the bulb, as heat is a secondary effect of electrical power dissipation. Power can be calculated using the potential difference and current. Hence, option 3 is incorrect.

• The voltmeter cannot measure the resistance of the bulb. Resistance is calculated indirectly using Ohm's law (R=V/IR=V/IR=V/IR=V/IR=V/IR=V/IR=V/IR=V/I

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         ). *Hence, option 4 is incorrect.*
  

Additional Information

• Voltmeter:

• A voltmeter is connected in parallel to the component whose voltage is to be measured.

• It has a high internal resistance to prevent current from flowing through it, ensuring accurate voltage measurements.

• Modern voltmeters are often digital, providing precise readings of potential difference.

• Example: Multimeters can function as voltmeters to measure voltage in circuits.

• Ohm’s Law:

• This fundamental law states that the current (IIIIIIII

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         ) through a conductor is directly proportional to the voltage (VVVVVVVV
  
         ) across it, provided the temperature remains constant: V=IRV=IRV=IRV=IRV=IRV=IR            .
  

• Using Ohm's Law, we can calculate the resistance of the bulb if both the voltage (VVVVVV ) and current (II ) are known.

• Applications: Used in designing circuits, calculating power, and determining electrical parameters.

• Electrical Power:

• Power consumed by the bulb (PP ) is calculated using the formula: P=VIP=VI , where VVVVVV is voltage and II is current.

• Heat produced by the bulb is a result of electrical power dissipation in the filament.

• Example: For a bulb with a voltage of 12V and current of 2A, the power is P=12×2=24WattsP=12×2=24Watts .

• Resistance of Bulb:

• The resistance (RR ) of the bulb is related to its material and design, and it affects the current flow and brightness.

• It can be calculated indirectly using R=V/IR=V/IR=V/IR=V/IR=V/IR=V/I , where VV is potential difference and II is current.

• Example: A bulb with a voltage of 12V and current of 2A has a resistance of R=12/2=6ΩR=12/2=6ΩR=12/2=6ΩR=12/2=6ΩR=12/2=6ΩR=12/2=6Ω .

Important Points

• Measurement Devices:
• Ammeter: Used to measure current, connected in series in the circuit.
• Voltmeter: Used to measure voltage, connected in parallel.
• Multimeter: A versatile device that measures voltage, current, and resistance.
• Wattmeter: Used to measure electrical power.
• Power and Energy:
• Power indicates the rate at which energy is consumed by the bulb.
• Energy consumption is measured in kilowatt-hours (kWhkWh ), commonly used in electricity bills.
• Example: A 100W bulb used for 10 hours consumes 100×10=1,000Wh=1kWh100×10=1,000Wh=1kWh .
• Applications in Daily Life:
• Voltage measurement is essential in household appliances to ensure safety and functionality.
• Understanding resistance and power helps in choosing energy-efficient devices.
• Electrical measurements are vital in industries, laboratories, and renewable energy systems.

The correct answer is H₂O.

Key Points

• H₂O, commonly known as water, is a molecule of a compound. It is composed of two hydrogen atoms covalently bonded to one oxygen atom.
• A compound is defined as a substance that consists of atoms of two or more different elements chemically bonded together in a fixed ratio. H₂O fits this definition as it contains hydrogen and oxygen atoms.
• Other options like H₂, N₂, and O₂ are molecules of elements, not compounds, as they consist of only one type of element (hydrogen, nitrogen, or oxygen respectively).
• Water is an essential compound in nature, serving as a solvent in various biological and chemical processes. Its molecular structure contributes to its unique properties like high boiling and melting points, surface tension, and ability to dissolve many substances.
• H₂O is formed through chemical reactions such as the combination of hydrogen gas and oxygen gas in the presence of energy (e.g., combustion reaction).

Additional Information

• H₂:
• H₂ is a molecule of the element hydrogen. It consists of two hydrogen atoms bonded covalently.
• It is not a compound because it contains only one type of element.
• Hydrogen gas is commonly used in industries, such as in fuel cells and as a reducing agent in chemical reactions.
• N₂:
• N₂ is a molecule of the element nitrogen. It is composed of two nitrogen atoms bonded by a triple covalent bond, making it highly stable.
• It is not a compound as it contains only one type of element.
• Nitrogen gas makes up about 78% of Earth's atmosphere and is essential for life, especially in the nitrogen cycle.
• O₂:
• O₂ is a molecule of the element oxygen, consisting of two oxygen atoms bonded by a double covalent bond.
• It is not a compound as it contains only oxygen atoms.
• Oxygen gas is crucial for respiration in most living organisms and is vital for combustion processes.
• H₂O:
• H₂O is a compound because it contains two different elements: hydrogen and oxygen.
• Its molecular shape is bent due to the lone pairs on the oxygen atom, resulting in a polar molecule.
• It exhibits unique properties like being a universal solvent, having high specific heat capacity, and supporting life on Earth.
• Examples of its use include drinking water, agriculture, and as a solvent in laboratories.

Important Points

• Difference Between Molecules of Elements and Compounds:
• A molecule of an element contains atoms of only one type of element (e.g., H₂, O₂, N₂).
• A molecule of a compound contains atoms of two or more different elements chemically bonded (e.g., H₂O, CO₂, NaCl).
• Examples of Compounds:
• Water (H₂O): Composed of hydrogen and oxygen.
• Carbon Dioxide (CO₂): Composed of carbon and oxygen.
• Sodium Chloride (NaCl): Composed of sodium and chlorine.
• Properties of Water:
• High boiling point (100°C) and freezing point (0°C) due to hydrogen bonding.
• Excellent solvent for ionic and polar substances.
• Essential for biological processes like photosynthesis and cellular respiration.

The correct answer is Wavelength and Frequency.

Key Points

• The speed of sound (v) is calculated using the formula: v = λ × f, where λ (wavelength) is the distance between two consecutive crests or troughs of a wave, and f (frequency) is the number of vibrations or oscillations per second.
• The wavelength (λ) is typically measured in meters (m), while the frequency (f) is measured in hertz (Hz). Multiplying these two properties gives the speed of sound in meters per second (m/s).
• Sound waves are longitudinal waves, meaning that the particles of the medium through which the sound travels oscillate parallel to the direction of wave propagation.
• The speed of sound depends on the properties of the medium, such as density and elasticity, but the relationship between wavelength and frequency remains constant for a given medium.
• The formula v = λ × f is fundamental in wave mechanics and applies to both sound and electromagnetic waves.

Additional Information

• Amplitude and Frequency: Amplitude refers to the maximum displacement of particles from their mean position in a wave, and frequency is the number of oscillations per second. While both are important wave properties, they do not determine the speed of sound. Instead, amplitude is related to the energy of the wave, and frequency is related to pitch.
• Wavelength and Amplitude: Wavelength is the distance between two consecutive points of similar phase (like crests), while amplitude determines the wave's intensity or loudness. Their product does not give the speed of sound.
• Amplitude and Time Period: Time period (T) is the reciprocal of frequency (T = 1/f). While both amplitude and time period are critical in describing wave behavior, their product does not relate to the speed of sound.
• Physical Factors Affecting Sound Speed: The speed of sound varies with the medium's properties, such as temperature, density, and elasticity. For example:
• In air at 20°C, the speed of sound is approximately 343 m/s.
• In water, the speed of sound is significantly higher, around 1482 m/s, due to the medium's higher density and elasticity.
• Practical Applications:
• In sonar technology, the relationship between wavelength, frequency, and speed is used to detect objects underwater.
• In musical instruments, tuning is based on frequency, which affects the speed and wavelength of sound waves in air.

Important Points

• Frequency (f): It is measured in hertz (Hz) and represents the number of wave cycles per second. For example, a sound wave with a frequency of 100 Hz completes 100 oscillations per second.
• Wavelength (λ): It is the distance between two consecutive crests or troughs of a wave, measured in meters (m). A longer wavelength corresponds to a lower frequency for a given speed of sound.
• Relationship Between Speed, Wavelength, and Frequency: For a wave propagating through a medium, the speed remains constant unless the medium’s properties change. Thus, if the wavelength increases, the frequency decreases proportionally, and vice versa.
• Medium Dependency: In gases, the speed of sound is slower compared to liquids and solids, as the intermolecular forces and density differ across these states of matter.

The correct answer is The particle is slowing down if moving in the positive direction..

Key Points

• A velocity-time graph with a negative slope indicates that the velocity of the particle is decreasing over time.
• This decrease in velocity implies that the particle is slowing down, provided it is moving in the positive direction.
• The slope of the velocity-time graph represents the acceleration. A negative slope corresponds to a negative acceleration (deceleration).
• If the particle is initially moving in the positive direction and the velocity magnitude reduces, it signifies that the particle is slowing down. Hence, the correct option is identified.
• Other options such as the particle being at rest or moving with increasing acceleration are incorrect, as these scenarios would result in different graph characteristics.

Additional Information

• Velocity-Time Graph:
• The velocity-time graph provides insights into how the velocity of a particle changes with time.
• A straight line on the graph indicates uniform acceleration or deceleration.
• If the slope of the line is negative, the particle is decelerating or slowing down.
• If the slope is positive, the particle is accelerating in the direction of motion.
• Acceleration:
• Acceleration is the rate of change of velocity with respect to time.
• A negative acceleration (also called deceleration) implies that the velocity of the particle is decreasing.
• In the context of the graph, the slope of the velocity-time curve directly represents the acceleration.
• Incorrect Options Explanation:
• Option 1: "The particle is at rest." - This would be represented by a horizontal line on the velocity-time graph at velocity = 0.
• Option 2: "The particle is speeding up in the positive direction." - This would require a positive slope on the graph, not a negative one.
• Option 4: "The particle is moving with increasing acceleration." - This would result in a curved line on the velocity-time graph, not a straight line with a negative slope.
• Real-Life Example:
• A car moving forward and applying brakes experiences deceleration, which is depicted as a negative slope on the velocity-time graph.
• Similarly, a ball rolling uphill slows down due to gravity, resulting in a similar graph pattern.

Important Points

• Negative Acceleration (Deceleration):
• When a particle experiences a force opposite to its motion, it undergoes negative acceleration.
• Examples include braking vehicles, friction acting on a sliding object, or an object moving against an inclined plane.
• Analysis of Graphs:
• The slope of a velocity-time graph is a fundamental tool in understanding motion.
• Positive slope = acceleration, negative slope = deceleration, horizontal line = constant velocity, and vertical line = instantaneous change (not physically possible).
• Practical Application:
• Velocity-time graphs are widely used in fields like transportation engineering, physics simulations, and sports analytics to study motion and predict outcomes.

The correct answer is Dendrites, cell body, axon and nerve endings.

Key Points

• Neurons, also called nerve cells, are the fundamental units of the nervous system responsible for transmitting information throughout the body.
• A typical neuron consists of four main components: dendrites, cell body (soma), axon, and nerve endings.
• Dendrites: These are branched projections that receive electrical signals from other neurons and convey them to the cell body.
• Cell Body (Soma): This is the central part of the neuron, containing the nucleus and cytoplasm. It processes incoming signals and generates outgoing signals.
• Axon: A long, thread-like projection that transmits signals away from the cell body to other neurons, muscles, or glands.
• Nerve Endings (Axon Terminals): These are the endpoints of the axon, where the neuron communicates with other cells via neurotransmitter release at synapses.

Additional Information

• Types of Neurons:
• Sensory Neurons: Transmit sensory information (e.g., touch, light, sound) from sensory receptors to the central nervous system (CNS).
• Motor Neurons: Send signals from the CNS to muscles or glands, enabling movement or secretion.
• Interneurons: Connect sensory and motor neurons within the CNS for processing and integration of information.
• Function of Dendrites: Dendrites increase the surface area of neurons for receiving signals and play a crucial role in integrating incoming information.
• Structure of the Axon: Axons are often covered by a myelin sheath, which is made up of glial cells. This sheath increases the speed of electrical signal transmission.
• Synapse: The junction between a nerve ending and another cell (e.g., neuron, muscle cell) where neurotransmitters facilitate communication.
• Neurotransmitters: Chemicals like dopamine, serotonin, and acetylcholine released at synapses to transmit signals between neurons.

Important Points

• Neuron Communication: Neurons transmit signals via electrical impulses (action potentials) and chemical signals (neurotransmitters).
• Myelination: Myelin sheaths are formed by Schwann cells in the peripheral nervous system (PNS) and oligodendrocytes in the central nervous system (CNS).
• Diseases Related to Neurons: Disorders like Alzheimer's disease, Parkinson's disease, and multiple sclerosis involve damage to neurons or their connections.
• Neuroplasticity: The ability of neurons to adapt, reorganize, and form new connections in response to learning or injury.

The correct answer is Behind the mirror.

Key Points

• The centre of curvature of a convex mirror is the center of the sphere from which the mirror is a part.
• In the case of a convex mirror, the reflective surface bulges outward, and the center of the sphere lies behind the mirror.
• The centre of curvature is not located on the reflective side of the convex mirror but rather on the side opposite to the reflection.
• This is because all the normals to the surface of a convex mirror point away from the center of curvature.
• In a convex mirror, the focal point and the center of curvature are both virtual, meaning they cannot be physically located in front of the mirror but are positioned behind it.

Additional Information

• Convex Mirror:
• A convex mirror, also known as a diverging mirror, has a reflective surface that curves outward.
• It always forms virtual, erect, and diminished images, irrespective of the object's position.
• Convex mirrors are widely used in applications like vehicle rearview mirrors due to their ability to provide a wide field of view.
• Centre of Curvature:
• The centre of curvature of a mirror is the center of the sphere of which the mirror surface is a part.
• For a convex mirror, this point lies behind the mirror, making it a virtual point.
• Principal Focus:
• The principal focus of a convex mirror is the point where light rays initially parallel to the principal axis appear to diverge from after reflection.
• It is always located behind the mirror, making it a virtual focus.
• Difference Between Convex and Concave Mirrors:
• While a convex mirror has its center of curvature and focus behind the mirror, a concave mirror has these points located in front of the mirror.
• Convex mirrors always form virtual images, while concave mirrors can form both real and virtual images depending on the object's position.
• Applications of Convex Mirrors:
• Used in vehicle rearview mirrors for a wider field of vision.
• Employed in security mirrors in shopping malls and stores to monitor a larger area.
• Used in automated teller machines (ATMs) to ensure security by providing a broader view of the surroundings.

Important Points

• The focal length of a convex mirror is always positive since the focus lies behind the mirror.
• The mirror formula, given as (1/f = 1/v + 1/u), is applicable for both convex and concave mirrors, where f is the focal length, v is the image distance, and u is the object distance.
• The magnification produced by a convex mirror is always less than 1 because the image is always diminished.
• Convex mirrors are preferred in daily applications requiring a wide field of view because of their unique reflective properties.

The correct answer is Ethanoic acid + Sodium hydroxide → Sodium ethanoate + Water.

Key Points

• Acidic Nature of Ethanoic Acid: The reaction of ethanoic acid with a base (like sodium hydroxide) demonstrates its acidic nature. This is because an acid reacts with a base to form a salt and water, a process known as neutralization.
• Reaction Equation: The reaction is represented as: CH₃COOH + NaOH → CH₃COONa + H₂O. Here, ethanoic acid (CH₃COOH) reacts with sodium hydroxide (NaOH) to produce sodium ethanoate (CH₃COONa) and water (H₂O).
• Acid-Base Neutralization: Ethanoic acid donates a proton (H⁺), and sodium hydroxide provides hydroxide ions (OH^-), which combine to form water, leaving sodium ethanoate as the salt.
• Significance of the Reaction: This reaction is a direct demonstration of the acidic property of ethanoic acid, as acids are characterized by their ability to neutralize bases.
• Why Other Options Are Incorrect: While some other reactions involve ethanoic acid, they do not directly demonstrate its acidic nature. For example, esterification reactions do not involve neutralization with a base.

Additional Information

• Reaction of Ethanoic Acid with Sodium: In this reaction, ethanoic acid reacts with sodium metal to produce sodium ethanoate and hydrogen gas: CH₃COOH + Na → CH₃COONa + H₂. Although this reaction shows the reactivity of ethanoic acid, it is not a direct demonstration of its acidic nature.
• Esterification Reaction: Ethanoic acid reacts with alcohols (e.g., ethanol) in the presence of an acid catalyst to form esters and water. For example, CH₃COOH + C₂H₅OH → CH₃COOC₂H₅ + H₂O. This reaction highlights its chemical versatility but does not emphasize its acidity.
• Reaction with Metals: Ethanoic acid reacts with active metals (like sodium) to release hydrogen gas, a property that confirms it is an acid. However, this reaction is more indicative of general acid behavior rather than a specific demonstration of acidic neutralization.
• Uses of Sodium Ethanoate: Sodium ethanoate, the product of the reaction with sodium hydroxide, is widely used in industries for applications such as food preservation, buffering solutions, and dye production.
• Acidic Properties of Ethanoic Acid: Ethanoic acid is a weak acid, as it partially dissociates in water to release H⁺ ions: CH₃COOH ⇌ CH₃COO^- + H⁺. This property is responsible for its ability to participate in acid-base reactions.

Important Points

• Definition of Acids and Bases: According to the Bronsted-Lowry theory, acids are substances that donate protons (H⁺), while bases accept protons. Ethanoic acid acts as an acid in reactions with bases like sodium hydroxide.
• Neutralization Reaction: This is a chemical reaction in which an acid and a base react quantitatively with each other. In the case of ethanoic acid and sodium hydroxide, the products are sodium ethanoate and water.
• pH of Ethanoic Acid: A dilute solution of ethanoic acid has a pH of around 4-5, indicating its weak acidic nature. It does not fully ionize in water, unlike strong acids such as hydrochloric acid.
• Industrial Applications: Ethanoic acid is used in the production of synthetic fibers, plastics, and food additives. Its reactions with bases and alcohols are foundational in industrial chemistry.
• Distinction Between Weak and Strong Acids: Ethanoic acid is a weak acid because it only partially dissociates in water. Strong acids, like sulfuric acid, fully dissociate and have a much lower pH.

The correct answer is Calcium carbonate.

Key Points

• When carbon dioxide (CO₂) reacts with calcium hydroxide (Ca(OH)₂), the product formed is calcium carbonate (CaCO₃).
• Calcium hydroxide is commonly referred to as lime water, and when CO₂ gas is passed through it, a white precipitate of calcium carbonate is formed.
• The chemical reaction for this process is:
  

Ca(OH)₂ + CO₂ → CaCO₃ + H₂O.

• This reaction is used as a qualitative test for the presence of carbon dioxide gas. The formation of a white precipitate indicates the presence of CO₂.
• If excess CO₂ is passed through lime water, the calcium carbonate reacts further to form calcium bicarbonate (Ca(HCO₃)₂), which is soluble in water.

Additional Information

• Calcium carbonate (CaCO₃):
• It is a naturally occurring substance found in rocks such as limestone, marble, and chalk.
• It is widely used in various industries, including construction (cement production), agriculture (soil pH adjustment), and medicine (antacids).
• It is insoluble in water but reacts with acids to produce carbon dioxide gas.
• Sodium carbonate (Na₂CO₃):
• Also known as washing soda, it is a completely different compound and is not formed in this reaction.
• It is used in cleaning agents, glass manufacturing, and water softening processes.
• Calcium bicarbonate (Ca(HCO₃)₂):
• It forms when excess CO₂ is bubbled through lime water, causing the initial calcium carbonate precipitate to dissolve.
• It is soluble in water and contributes to the hardness of water.
• Calcium oxide (CaO):
• Also known as quicklime, it is formed by heating calcium carbonate (CaCO₃) strongly.
• This compound is not formed in the reaction between CO₂ and lime water.
• It is used in cement production, steel manufacturing, and as a reagent for chemical processes.

Important Points

• Qualitative Test for CO₂:
• The reaction between CO₂ and lime water is a common laboratory experiment to test for the presence of carbon dioxide gas.
• The appearance of a white precipitate (CaCO₃) confirms the presence of CO₂.
• Reactions of CO₂ with Bases:
• Carbon dioxide reacts with bases like calcium hydroxide to form carbonate or bicarbonate compounds.
• This is an example of an acid-base reaction where CO₂ acts as an acidic oxide.
• Applications of Calcium Carbonate:
• Used in the production of chalk, lime, and cement.
• Acts as a filler in paper, rubber, and plastics industries.
• Used in water treatment to neutralize acidic water and adjust pH levels.

The correct answer is A series of compounds having the same general formula, similar chemical properties, and consecutive members differing by a -CH2- group..

Key Points

• Definition of Homologous Series: A homologous series is a group of organic compounds that follow a particular pattern, having the same general formula, similar chemical properties, and a regular difference in their molecular structures by a -CH2- group.
• Consecutive Members Differ by -CH2-: Each member in the homologous series differs from the next by a -CH2- unit. For example, in the alkane series, methane (CH4) and ethane (C2H6) differ by one -CH2- group.
• Similar Chemical Properties: Compounds in a homologous series exhibit similar chemical behavior because they have the same functional group. For instance, all alcohols (R-OH) show similar reactivity due to the hydroxyl group.
• Physical Properties Show Gradation: The physical properties, such as boiling points, melting points, and density, show a gradual change as the molecular size increases.
• Examples of Homologous Series: Common examples include the alkane series (CnH2n+2), alkene series (CnH2n), and alcohol series (CnH2n+1OH).

Additional Information

• Key Characteristics of Homologous Series:
• General Formula: All members of a homologous series can be represented by a single general formula. For example, alkanes can be represented as CnH2n+2.
• Functional Group: The chemical properties of the members are dictated by the presence of a functional group, such as -OH in alcohols or -COOH in carboxylic acids.
• Gradation in Physical Properties: The gradual change in physical properties is due to the increasing molecular mass or size. For instance, boiling points increase in the alkane series as more -CH2- groups are added.
• Option-wise Analysis:
• Option 1: Correct. It accurately describes the characteristics of a homologous series.
• Option 2: Incorrect. A homologous series cannot have different functional groups; this would lead to entirely different chemical properties.
• Option 3: Incorrect. This describes elements with similar physical properties, not a homologous series.
• Option 4: Incorrect. Homologous series can include both straight-chain and branched compounds, but the statement that all members have branched chains is not true.
• Examples of Homologous Series:
• Alkanes: Methane (CH4), ethane (C2H6), propane (C3H8).
• Alkenes: Ethene (C2H4), propene (C3H6), butene (C4H8).
• Alcohols: Methanol (CH3OH), ethanol (C2H5OH), propanol (C3H7OH).

Important Points

• Functional Group: The functional group determines the chemical reactivity of the compounds in the series. For example, -OH in alcohols makes them soluble in water and reactive towards acids.
• Applications: Homologous series help in the systematic study of organic compounds, prediction of properties, and identification of unknown compounds.
• Industrial Relevance: Many homologous series, such as alkanes and alcohols, are crucial in industries, including fuels, pharmaceuticals, and polymers.
• Nomenclature: Compounds in a homologous series follow the IUPAC nomenclature system, ensuring uniform naming.

The correct answer is 1 : 2.

Key Points

• Resistance and its formula: The resistance (R) of a wire is given by the formula: R = ρ × (L / A), where:
• ρ (rho): Resistivity of the material (same for both wires, as they are of the same material).
• L: Length of the wire (same for both wires).
• A: Cross-sectional area of the wire.
• Given condition: Wire A has twice the cross-sectional area of wire B. Let the cross-sectional area of wire B be A, then the cross-sectional area of wire A will be 2A.
• Relation between resistances: Since L and ρ are the same for both wires, the resistance ratio depends only on their cross-sectional areas. Using the formula R ∝ 1/A:
• Resistance of wire A, R_A = ρ × (L / 2A).
• Resistance of wire B, R_B = ρ × (L / A).
• Taking the ratio, R_A : R_B = (L / 2A) : (L / A) = 1 : 2.
• Final result: The ratio of their resistances is 1 : 2.

Additional Information

• Factors affecting resistance: The resistance of a wire depends on:
• Length (L): Resistance increases with an increase in length.
• Cross-sectional area (A): Resistance decreases with an increase in area.
• Material: The resistivity (ρ) of the material determines how much resistance it offers.
• Temperature: For most materials, resistance increases with an increase in temperature.
• Applications of resistance:
• Resistors are used in electrical circuits to control the flow of current.
• High-resistance materials are used as heating elements in devices like toasters and electric heaters.
• Understanding resistivity:
• Resistivity (ρ) is a material property that quantifies how strongly a material opposes the flow of electric current.
• It is measured in ohm-meters (Ω·m).
• Materials with low resistivity (like copper and silver) are good conductors, while materials with high resistivity (like rubber and glass) are insulators.
• Practical implications:
• Thicker wires (larger cross-sectional area) have lower resistance and are used in high-current applications.
• Longer wires have higher resistance, so they are avoided in designs where minimal energy loss is required.

The correct answer is The object’s momentum changes in the direction of the applied force..

Key Points

• Newton’s Second Law of Motion states that the rate of change of momentum of an object is directly proportional to the unbalanced force applied to it and occurs in the direction of the force.
• Momentum is defined as the product of the mass and velocity of an object. It is a vector quantity, meaning it has both magnitude and direction.
• When an unbalanced force is applied to an object, it creates an acceleration according to the equation F = ma (Force = Mass × Acceleration). This acceleration causes a change in the object’s velocity, and since momentum depends on velocity, the momentum of the object also changes.
• The change in momentum occurs in the direction of the applied force, which is why the object’s momentum changes in that specific direction.
• The other options are incorrect because:
• The object does not necessarily move in a circular path unless the force is centripetal (Option 1).
• The object does not come to rest immediately unless the force is applied in the opposite direction and is strong enough to stop the motion (Option 2).
• The momentum cannot remain unchanged when an unbalanced force is applied, as per Newton’s Second Law (Option 3).

Additional Information

• Newton's Laws of Motion:
• First Law: States that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force (Law of Inertia).
• Second Law: Explains how the velocity of an object changes when it is subjected to an external force, mathematically expressed as F = ma.
• Third Law: States that for every action, there is an equal and opposite reaction.
• Momentum:
• It is a measure of the motion of an object and is calculated as p = mv, where p is momentum, m is mass, and v is velocity.
• Momentum is conserved in an isolated system where no external forces act.
• Unbalanced Force:
• An unbalanced force occurs when the forces acting on an object are not equal in magnitude and direction, causing the object to accelerate.
• Example: A car accelerating forward when the engine applies a forward force greater than the resistive forces (friction and air resistance).
• Applications of Newton’s Second Law:
• It is used in designing vehicles to ensure proper acceleration and braking systems.
• It helps in understanding the motion of celestial bodies under gravitational forces.
• Engineers use it to calculate the forces acting on structures and machines.

Important Points

• SI Unit: The SI unit of momentum is kg·m/s, and force is measured in Newtons (N).
• Real-life Example: In sports, such as cricket, a fielder reduces the impact of catching a fast-moving ball by moving their hands backward to increase the time of impact, thereby reducing the force.
• Impulse: The change in momentum of an object is known as impulse, which is equal to the product of force and the time duration of the force applied.
• Rocket Propulsion: Newton’s Second Law is key to explaining how rockets launch, as the expulsion of gas creates a force that changes the rocket’s momentum.

The correct answer is Both A and B.

Key Points

• Statement A: The large spaces between gas particles allow them to be easily squeezed into smaller volumes.
• This statement is correct. Gases have a unique property where their particles are widely spaced compared to solids and liquids. These large intermolecular spaces allow gases to be compressed easily under pressure.
• For example, gases like oxygen and nitrogen can be compressed into cylinders for medical or industrial purposes.
• Statement B: Gas particles have a high average kinetic energy.
• This statement is also correct. The kinetic energy of gas particles is directly proportional to the temperature of the gas (measured in Kelvin). As temperature increases, the particles move faster, indicating higher kinetic energy.
• The kinetic energy of gas particles is a fundamental concept in the kinetic molecular theory of gases.
• Both A and B are correct:
• Both statements align with the properties of gases. Statement A explains why gases are compressible, while statement B explains the relationship between temperature and particle motion in gases.
• These properties are crucial for understanding gas behavior in various applications, such as industrial gas storage, thermodynamics, and fluid dynamics.

Additional Information

• Kinetic Molecular Theory of Gases:
• The theory describes the behavior of gas particles, assuming they are in constant random motion.
• The average kinetic energy of gas particles depends only on the temperature of the gas.
• Gases are assumed to have negligible intermolecular forces, and collisions between particles are perfectly elastic.
• Compressibility of Gases:
• Gases can be compressed due to the large spaces between their particles, as opposed to solids and liquids, where particles are closely packed.
• This property is utilized in various applications, such as compressed natural gas (CNG) and liquefied petroleum gas (LPG) storage.
• Relationship Between Temperature and Kinetic Energy:
• The kinetic energy of gas particles is proportional to the absolute temperature (Kelvin scale).
• For example, doubling the temperature of a gas doubles the average kinetic energy of its particles.
• Applications of Gas Properties:
• Compressibility is crucial for gas storage in industries and medical applications.
• The relationship between temperature and kinetic energy is used in thermodynamics and heat engines.

Important Points

• Boyle’s Law: At constant temperature, the pressure of a gas is inversely proportional to its volume, which explains the compressibility of gases.
• Charles’s Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature, linking temperature and particle motion.
• Real vs. Ideal Gases: While the above properties apply to ideal gases, real gases deviate under high pressure and low temperature due to intermolecular forces.
• Practical Uses: Understanding gas behavior is essential in fields like meteorology, aviation, and chemical engineering.

The correct answer is Using CFC-based refrigerants and spray cans.

Key Points

• Chlorofluorocarbons (CFCs), used in refrigeration, air conditioning, and aerosol spray cans, are one of the primary contributors to ozone layer depletion. These compounds release chlorine atoms upon exposure to ultraviolet (UV) radiation, which then react with ozone (O₃) molecules, breaking them apart.
• The ozone layer is crucial for shielding the Earth from harmful UV-B radiation. Depletion of the ozone layer increases the risk of skin cancer, cataracts, and other health issues, as well as negatively impacting ecosystems and wildlife.
• CFCs were widely used in the 20th century due to their stability, non-flammability, and low toxicity. However, their negative impact on the ozone layer was discovered in the 1970s, leading to global efforts to phase them out.
• The 1987 Montreal Protocol was a landmark international treaty aimed at reducing the production and consumption of ozone-depleting substances, including CFCs. It has been highly effective in curbing ozone depletion.
• Alternative substances, such as hydrofluorocarbons (HFCs), have been developed to replace CFCs. However, HFCs are potent greenhouse gases and contribute to global warming, necessitating further innovation in sustainable refrigerants.

Additional Information

• Ozone Layer:
• The ozone layer is a region in the Earth's stratosphere that contains a high concentration of ozone (O₃) molecules. It absorbs most of the Sun's harmful ultraviolet radiation, particularly UV-B and UV-C rays.
• The majority of ozone is found between 15 and 35 kilometers above the Earth's surface in a region called the stratosphere.
• Ozone depletion results in the formation of the "ozone hole," primarily observed over Antarctica during the spring (September to November).
• Other Ozone-Depleting Substances (ODS):
• In addition to CFCs, other ODS include halons (used in fire extinguishers), carbon tetrachloride, methyl chloroform, and methyl bromide (used in pesticides).
• These substances release halogen atoms (chlorine or bromine) when exposed to UV radiation, which then catalyze ozone destruction.
• Global Efforts to Protect the Ozone Layer:
• The Montreal Protocol (1987) is a legally binding international treaty that has been ratified by 198 countries. It has successfully phased out 99% of ozone-depleting substances globally.
• The Kigali Amendment (2016) to the Montreal Protocol focuses on phasing down hydrofluorocarbons (HFCs), which are potent greenhouse gases.
• Impact of Ozone Depletion:
• Increased exposure to UV radiation can lead to skin cancer, cataracts, and weakened immune systems in humans.
• UV radiation can also harm phytoplankton, the base of marine food chains, and damage crops, leading to reduced agricultural productivity.
• Options Analysis:
• Conserving water: Water conservation does not contribute to ozone depletion. It is an environmentally beneficial practice aimed at reducing water wastage. (Incorrect)
• Composting organic waste: Composting organic waste reduces landfill emissions and promotes soil fertility but has no impact on the ozone layer. (Incorrect)
• Using CFC-based refrigerants and spray cans: This is the primary human activity responsible for ozone layer depletion due to the release of chlorine atoms. (Correct)
• Using solar energy: Solar energy is a clean and renewable energy source that does not contribute to ozone depletion. (Incorrect)

The correct answer is Stratified squamous epithelium.

Key Points

• The outer layer of the skin, also known as the epidermis, is primarily composed of stratified squamous epithelium. This tissue type is specialized to provide protection against mechanical stress, pathogens, and dehydration.
• Stratified squamous epithelium consists of multiple layers of cells, with the outermost layer being keratinized (hardened) to form a tough, protective barrier.
• This tissue type is particularly suited for areas exposed to friction and wear, such as the surface of the skin, due to its ability to regenerate quickly through mitotic activity in the basal layers.
• The keratinized layer of the epidermis is made up of dead cells filled with the protein keratin, which is essential for waterproofing and enhancing the skin's mechanical strength.
• The basal layer of the stratified squamous epithelium contains melanocytes, which produce melanin, contributing to skin pigmentation and protection against ultraviolet (UV) radiation.

Additional Information

• Other Types of Epithelial Tissues:
• Cuboidal epithelium: Composed of cube-shaped cells, this tissue is found in glands, kidney tubules, and the thyroid gland. It is involved in secretion and absorption.
• Columnar epithelium: Made up of tall, column-like cells, it lines the stomach, intestines, and respiratory tract, aiding in absorption and secretion. Some columnar cells may have cilia to move substances.
• Simple squamous epithelium: Consists of a single thin layer of flat cells, enabling easy diffusion of gases and nutrients. It is found in the lungs (alveoli), blood vessels, and glomeruli of kidneys.
• Skin Structure:
• The skin comprises three main layers: the epidermis (outermost), dermis (middle), and hypodermis (innermost).
• The dermis contains connective tissue, blood vessels, sweat glands, and nerve endings.
• The hypodermis, or subcutaneous layer, is primarily made of fat and connective tissue, providing insulation and cushioning.
• Functions of the Skin:
• Protection from environmental damage, including microorganisms, UV radiation, and physical injuries.
• Regulation of body temperature through sweat glands and blood vessel dilation or constriction.
• Sensation of touch, pressure, pain, and temperature through embedded sensory receptors.
• Prevention of water loss through the keratinized layer.
• Keratinization:
• The process by which cells in the epidermis produce keratin, a structural protein, and eventually die to form the tough outer layer of the skin.
• This process is crucial for creating a barrier that is both waterproof and resistant to physical damage.

Important Points

• Melanocytes and Melanin:
• Melanocytes are specialized cells found in the basal layer of the epidermis that produce the pigment melanin.
• Melanin provides skin pigmentation and protects against UV radiation by absorbing harmful rays.
• Skin Disorders:
• Conditions like psoriasis and eczema involve the abnormal functioning of epidermal cells.
• Skin cancer, such as melanoma, arises from uncontrolled growth of melanocytes.

The correct answer is g = GM / R².

Key Points

• Acceleration due to gravity is the force experienced by an object of unit mass due to the gravitational pull of a celestial body (planet, star, etc.). It is denoted by g.
• The formula for calculating acceleration due to gravity on the surface of a planet is derived from Newton's Law of Gravitation.
• The gravitational force is given by the formula: F = GMm / R², where:
• G: Universal gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²).
• M: Mass of the planet.
• m: Mass of the object experiencing the force.
• R: Radius of the planet.
• The acceleration due to gravity, g, is derived by dividing the gravitational force by the mass of the object:
• g = F/m = GM/R².
• Therefore, the correct expression for acceleration due to gravity is g = GM / R².

Additional Information

• Universal Gravitational Constant (G):
• The value of G is constant throughout the universe.
• Its value is 6.674 × 10⁻¹¹ N·m²/kg².
• G was first measured by Henry Cavendish in the late 18th century using the torsion balance experiment.
• Factors Affecting Acceleration Due to Gravity:
• Mass of the planet (M): Higher the mass, greater the gravitational pull and hence the higher the value of g.
• Radius of the planet (R): Larger the radius, lower the value of g due to the inverse square relationship.
• Altitude: As the altitude increases, the value of g decreases because the distance from the center of the planet increases.
• Latitude: Due to the planet's rotation, the value of g is slightly lower at the equator and higher at the poles.
• Applications of Acceleration Due to Gravity:
• Used in calculating the weight of an object (Weight = mass × g).
• Essential in physics experiments such as free-fall and projectile motion.
• Helps in determining escape velocity (velocity required to escape the gravitational field of a planet).
• Variation of g with Depth:
• The value of g decreases as one moves towards the center of the planet.
• At the center of the planet, g = 0 because the gravitational forces cancel out.

Important Points

• Weight vs. Mass:
• Mass is a scalar quantity and remains constant irrespective of location.
• Weight is a vector quantity and depends on the value of g.
• Moon's Gravity:
• The value of g on the Moon is approximately 1/6th of that on Earth.
• It is due to the Moon's smaller mass and radius compared to Earth.
• Gravitational Anomalies:
• Local variations in g may occur due to differences in the Earth's crust density or topographical features.

The correct answer is Fragmentation.

Key Points

• Fragmentation is a type of asexual reproduction in which a new organism develops from a fragment or broken portion of the parent organism.
• This process is commonly observed in certain plants and organisms such as algae, fungi, and some animals like sponges and flatworms.
• The broken fragment of the parent plant has the ability to regenerate and grow into a complete, independent organism under suitable environmental conditions.
• Fragmentation is different from other methods of asexual reproduction like binary fission and spore formation, as it specifically involves the physical splitting of the parent organism.
• Examples of plants that reproduce through fragmentation include hydra, moss, and liverwort. Certain algae, such as Spirogyra, also exhibit this form of reproduction.

Additional Information

• Spore Formation:
• Spore formation is another type of asexual reproduction where organisms produce spores that can grow into new individuals.
• It is common in fungi, bacteria, and certain plants like ferns and mosses.
• Spores are microscopic structures that are resistant to harsh conditions and can remain dormant until favorable conditions arise.
• Fertilization:
• Fertilization is a process of sexual reproduction where the male and female gametes fuse to form a zygote.
• This process is not related to asexual reproduction and involves genetic mixing of two parents.
• Examples include reproduction in humans, animals, and flowering plants.
• Binary Fission:
• Binary fission is a method of asexual reproduction in unicellular organisms like bacteria and amoeba.
• The parent organism splits into two identical daughter organisms, each inheriting the genetic material of the parent.
• This process is faster and commonly observed in prokaryotic organisms.
• Applications of Fragmentation:
• Fragmentation is widely utilized in horticulture and agriculture to propagate plants.
• It ensures genetic uniformity, which is useful for maintaining desired traits in crops or ornamental plants.
• This method is particularly beneficial for reproducing species that do not produce viable seeds.

Important Points

• Comparison of Asexual Reproduction Methods:
• Fragmentation: Involves physical splitting; requires regeneration capabilities.
• Spore Formation: Involves production of spores; resistant to extreme conditions.
• Binary Fission: Quick division of unicellular organisms into two identical cells.
• Budding: Formation of a new organism from a small outgrowth (bud) on the parent.
• Advantages of Fragmentation:
• Allows for rapid reproduction and colonization.
• Ensures genetic uniformity, which is beneficial for stable environments.
• Does not require pollinators or external reproductive agents.
• Environmental Requirements:
• Fragmentation requires adequate moisture, nutrients, and temperature for successful regeneration.
• In aquatic environments, fragmentation is highly efficient due to water’s ability to carry fragments to new locations.

The correct answer is compound.

Key Points

• The formula unit mass of a substance is the sum of the atomic masses of all the atoms present in its formula unit. For compounds, the formula unit represents the simplest ratio of the ions or atoms in the compound.
• This concept is applicable to both ionic compounds and molecular compounds. For example, the formula unit mass of sodium chloride (NaCl) is calculated by adding the atomic masses of sodium (Na) and chlorine (Cl).
• In the case of a compound, the formula unit is representative of the smallest repeating unit that retains the compound’s chemical identity.
• The formula unit mass is expressed in atomic mass units (amu) or unified mass units (u), where 1 amu is defined as one-twelfth of the mass of a carbon-12 atom.
• Hence, the formula unit mass is directly relevant to compounds and not to atoms, mixtures, or individual elements.

Additional Information

• Atomic Mass:
• The mass of a single atom is called its atomic mass. It is measured in atomic mass units (amu).
• For example, the atomic mass of hydrogen is approximately 1 amu, and that of oxygen is approximately 16 amu.
• Molecular Mass:
• The molecular mass of a substance is the sum of the atomic masses of all the atoms in a molecule of the substance.
• For example, the molecular mass of water (H₂O) is calculated as: (2 × atomic mass of H) + (1 × atomic mass of O) = (2 × 1) + (1 × 16) = 18 amu.
• Difference Between a Compound and a Mixture:
• A compound is a pure substance composed of two or more elements chemically combined in a fixed proportion, e.g., sodium chloride (NaCl).
• A mixture is a combination of two or more substances that are not chemically combined, e.g., air or salt water.
• Examples of Formula Unit Mass:
• The formula unit mass of sodium chloride (NaCl): (atomic mass of Na) + (atomic mass of Cl) = 23 + 35.5 = 58.5 amu.
• The formula unit mass of calcium chloride (CaCl₂): (atomic mass of Ca) + 2 × (atomic mass of Cl) = 40 + 2 × 35.5 = 111 amu.

Important Points

• Atoms: Atoms are the smallest units of an element that retain its chemical properties. The concept of formula unit mass does not apply to individual atoms.
• Elements: Elements are pure substances consisting of only one type of atom. The formula unit mass is not relevant to elements as they are not made of multiple atoms in a fixed ratio.
• Mixtures: Mixtures are physical combinations of substances that do not have a definite composition or formula unit. Hence, formula unit mass is not applicable to mixtures.
• Compounds: Compounds are substances formed by the chemical combination of two or more elements in a fixed ratio. The formula unit mass is specifically used for compounds, especially ionic compounds.

The correct answer is 2.

Key Points

• The distributions of electrons in the K, L, and M shells of the chlorine atom are 2, 8, and 7 respectively.
• The chlorine atom has an atomic number of 17, which determines the number of electrons present in a neutral atom.
• Electrons are distributed across shells based on the 2n² rule, where "n" represents the shell number.
• The K shell (n=1) can hold up to 2 electrons, the L shell (n=2) can hold up to 8 electrons, and the M shell (n=3) can hold up to 18 electrons. However, chlorine only needs 7 electrons in the M shell to complete its configuration.
• Thus, the electron configuration for chlorine is 2, 8, 7, aligning with option 2.

Additional Information

• Electron Shell Distribution:
• Electrons are arranged in shells around the nucleus of an atom. The shells are labeled as K, L, M, N, etc., starting from the innermost shell.
• The capacity of each shell is determined by the formula 2n², where "n" represents the shell number.
• For chlorine, the atomic number 17 means it has 17 electrons distributed as 2 in K shell, 8 in L shell, and 7 in M shell.
• Valence Electrons:
• Chlorine has 7 valence electrons in its outermost shell (M shell).
• Valence electrons determine an atom's chemical properties, including its ability to bond with other atoms.
• Chlorine is highly reactive due to its tendency to gain 1 electron to complete its octet.
• Chlorine Atom:
• Chlorine is a chemical element with the symbol Cl and atomic number 17.
• It is a halogen and belongs to Group 17 of the periodic table.
• Chlorine is widely used in industry for water purification, chemical synthesis, and disinfection.
• Electronic Configuration:
• The electronic configuration of chlorine is 1s² 2s² 2p⁶ 3s² 3p⁵, representing the distribution of electrons in its orbitals.
• This configuration highlights chlorine's tendency to form ionic bonds by gaining an electron, resulting in a stable chloride ion (Cl⁻).
• Periodic Table Context:
• Chlorine belongs to the halogen family, which includes elements like fluorine, bromine, iodine, and astatine.
• Halogens are known for their high reactivity due to their need to gain one electron to complete their octet.
• Chlorine's position in the periodic table helps predict its chemical behavior and interactions.

The correct answer is Sexual reproduction.

Key Points

• Sexual reproduction is a biological process in which the fusion of male and female gametes occurs, resulting in the formation of a zygote that develops into a new organism.
• It involves two parent organisms contributing genetic material, ensuring genetic diversity in the offspring.
• This method of reproduction is prevalent in higher plants, animals, and humans.
• Key stages of sexual reproduction include: gamete formation (through meiosis), fertilization (fusion of gametes), and development of the zygote into a new organism.
• Sexual reproduction plays a crucial role in evolution, as it allows for the mixing of genetic material, leading to better adaptability and survival of species.

Additional Information

• Binary fission:
• This is an asexual reproduction method in which a single organism divides into two identical daughter cells.
• It occurs in prokaryotes such as bacteria and some unicellular eukaryotes like amoeba.
• Binary fission does not involve gametes or genetic recombination, resulting in genetically identical offspring.
• Cloning:
• Cloning is an artificial asexual reproduction technique where an exact genetic copy of an organism is created.
• Examples include Dolly the sheep, the first mammal cloned using somatic cell nuclear transfer.
• Clones lack genetic diversity, making them prone to diseases and less adaptable to environmental changes.
• Budding:
• Budding is a form of asexual reproduction where a new organism develops as an outgrowth, or bud, from the parent body.
• It occurs in organisms like yeast, hydra, and some plants.
• The bud eventually detaches and grows into a new individual, genetically identical to the parent.
• Sexual reproduction in detail:
• It involves meiosis, a type of cell division that reduces the chromosome number by half, producing haploid gametes.
• Fertilization restores the diploid chromosome number, forming a zygote.
• Examples include flowering plants (pollination) and humans (internal fertilization).
• Advantages include genetic variation, adaptability, and resilience to environmental changes.

Important Points

• Fertilization in sexual reproduction:
• It can be internal (as in mammals) or external (as in amphibians and fish).
• In internal fertilization, the fusion of gametes takes place inside the female body, ensuring better protection of the zygote.
• Applications of sexual reproduction:
• It is crucial in agriculture for breeding new plant varieties with desirable traits.
• In medical science, understanding sexual reproduction aids in fertility treatments and addressing genetic disorders.
• Evolutionary significance:
• Sexual reproduction fosters natural selection by allowing beneficial traits to spread in a population.
• It helps species adapt to changing environments and increases their chances of survival.

The correct answer is It must be at rest.

Key Points

• Kinetic Energy (KE) of a particle is given by the formula: KE = (1/2)mv², where m is the mass of the particle and v is its velocity. If kinetic energy is zero, the velocity must also be zero.
• A particle with zero velocity implies that it is not in motion. Hence, the particle must be at rest.
• Momentum of a particle is given by the formula: p = mv. Since velocity (v) is zero, the momentum of the particle is also zero.
• The particle cannot have maximum momentum (as suggested in Option 1) or any motion-related property like acceleration (as suggested in Option 2) because both require movement.
• If the particle is at rest, it cannot move with a uniform velocity (as suggested in Option 4) or any velocity other than zero.
• Conclusion: A particle with zero kinetic energy is necessarily at rest.

Additional Information

• Kinetic Energy (KE):
• Kinetic energy is a type of mechanical energy associated with the motion of an object.
• It depends on two factors: the mass of the object and the square of its velocity. Hence, even a small velocity contributes significantly to the particle's kinetic energy.
• Units: The SI unit of kinetic energy is the joule (J).
• Momentum:
• Momentum is a vector quantity defined as the product of an object's mass and velocity.
• If the velocity is zero, the momentum of the particle is also zero, irrespective of its mass.
• Units: The SI unit of momentum is kilogram meter per second (kg·m/s).
• Rest and Motion:
• An object is said to be at rest when its position does not change with respect to a reference point over time.
• An object is in motion when its position changes with respect to a reference point over time.
• Applications of Kinetic Energy:
• Kinetic energy plays a crucial role in various fields such as transportation, sports, and mechanical systems.
• Examples: Moving vehicles, flying aircraft, spinning turbines, and flowing rivers all possess kinetic energy.

Important Points

• Relation Between Kinetic Energy and Momentum:
• Kinetic energy and momentum are related by the equation: KE = p² / 2m, where p is momentum and m is mass.
• If kinetic energy is zero, momentum must also be zero.
• Zero Kinetic Energy:
• An object with zero kinetic energy cannot move until acted upon by an external force, as per Newton's First Law of Motion.
• It signifies the absence of motion in the reference frame being observed.
• Conservation of Energy:
• In an isolated system, the total energy (potential + kinetic) remains constant. If kinetic energy is zero, the energy might be in another form, like potential energy.

The correct answer is Nuclear membrane is a non-porous and rigid structure..

Key Points

• The nucleus plays a central role in cellular reproduction: The nucleus is responsible for regulating the cell cycle, mitosis, and meiosis. It contains genetic information essential for cell division, ensuring the accurate transmission of DNA to daughter cells. Hence, this statement is correct.
• The nucleus contains chromosomes: Chromosomes are thread-like structures made of DNA and proteins found within the nucleus. They carry genetic information required for cellular functions and reproduction. Hence, this statement is correct.
• The nuclear membrane is a double-layered structure: The nuclear envelope consists of two lipid bilayers—an inner membrane and an outer membrane. This structure separates the nucleus from the cytoplasm. Hence, this statement is correct.
• Nuclear membrane is a non-porous and rigid structure: This is incorrect because the nuclear membrane is porous, allowing the exchange of materials like RNA and proteins between the nucleus and cytoplasm. The pores are part of the nuclear pore complex, which facilitates selective transport. The membrane is also flexible, adapting to changes during cell division. Hence, this statement is incorrect.

Additional Information

• Nuclear Membrane:
• The nuclear membrane, also known as the nuclear envelope, is a double-layered structure that separates the nucleus from the cytoplasm.
• It contains nuclear pores, which are complexes facilitating selective transport of molecules like RNA and proteins.
• The nuclear envelope disassembles during cell division (mitosis) and reassembles afterward.
• Chromosomes:
• Chromosomes are composed of DNA and histone proteins, forming a structure called chromatin.
• In humans, there are 46 chromosomes (23 pairs), which include autosomes and sex chromosomes.
• They are essential for carrying genetic information and guiding protein synthesis through gene expression.
• Role of Nucleus in Cellular Reproduction:
• The nucleus orchestrates both mitosis (cell division for growth and repair) and meiosis (cell division for reproduction).
• It ensures the accurate replication and distribution of genetic material through processes like DNA replication and chromosome segregation.
• Nuclear Pore Complex:
• The nuclear pores are large protein complexes embedded within the nuclear envelope.
• They regulate the selective transport of molecules between the nucleus and the cytoplasm, allowing the exchange of RNA, ribosomal subunits, and proteins.
• These pores play a crucial role in maintaining nuclear-cytoplasmic communication.

Important Points

• Nuclear Membrane Dynamics:
• The nuclear membrane breaks down during mitosis to allow chromosome segregation.
• It reforms after mitosis to encapsulate the genetic material within the daughter nuclei.
• Nucleus as the Control Center:
• The nucleus regulates gene expression by controlling transcription (RNA synthesis).
• It stores genetic material in the form of DNA, which directs all cellular activities.

The correct answer is River lift system.

Key Points

• The River lift system is ideal for areas with irregular water supply from reservoirs or canals, as it utilizes available water from nearby rivers by pumping it to the fields for irrigation.
• This system ensures a consistent water supply for irrigation, making it suitable for regions where the canal system is unreliable or intermittent.
• The system involves lifting water from rivers to higher elevations using pumps and supplying it to agricultural lands, ensuring proper irrigation during dry spells.
• It helps in achieving better water management and is particularly effective in areas where farmers live near rivers but face challenges with reservoir-fed canal systems.
• The Government of India promotes such systems under various agricultural and irrigation schemes to enhance productivity and water efficiency.

Additional Information

• Sprinklers:
• Sprinkler irrigation involves spraying water uniformly across the field, mimicking rainfall.
• It is suitable for areas with adequate water pressure but not ideal for regions with irregular water supply as it requires consistent water flow.
• Sprinklers are commonly used for crops like vegetables, fruits, and flowers where water conservation is essential.
• Wells:
• Wells are a traditional source of irrigation, drawing groundwater for agricultural use.
• They are beneficial in areas with significant groundwater reserves but may not be sustainable in regions with low water tables.
• Excessive reliance on wells can lead to groundwater depletion, making them unsuitable for areas with irregular water supply.
• Flood irrigation:
• Flood irrigation involves directly flooding the fields with water, which is less efficient and results in water wastage.
• It is suitable for regions with abundant water resources but not ideal for areas with irregular reservoir-fed canal systems.
• This method can lead to soil erosion and nutrient leaching, reducing agricultural productivity.
• River lift system:
• The system is particularly effective in areas near rivers where farmers face challenges in accessing regular canal water supply.
• It ensures equitable water distribution, supports irrigation during dry periods, and minimizes dependency on reservoirs.
• It is an important tool for regions with inconsistent canal water supply, enhancing agricultural output and reducing water scarcity issues.

Important Points

• Government Initiatives:
• The Pradhan Mantri Krishi Sinchai Yojana (PMKSY) promotes efficient water management systems like river lift irrigation to ensure "Har Khet Ko Pani" (Water for Every Field).
• Under PMKSY, financial assistance is provided to implement sustainable irrigation systems, including pumps and lift irrigation setups.
• Examples:
• In states like Maharashtra and Andhra Pradesh, river lift systems have been successfully implemented to address water scarcity and improve agricultural productivity.
• Such systems help farmers achieve better yields even in regions with unpredictable water supply from reservoirs.

Shortcut Trick

Relative Speed in opposite direction = (Sum of lengths) ÷ Time

Relative Speed = (239 + 161) ÷ 20 = 400 ÷ 20 = 20 m/s

Convert m/s to km/hr: 20 × (18/5) = 72 km/hr

Speed of second train = Relative Speed − Speed of first train = 72 − 43 = 29 km/hr

∴ The correct answer is 29 km/hr.

Alternate Method

Given:

Length of first train (L₁) = 239 m

Length of second train (L₂) = 161 m

Speed of first train (S₁) = 43 km/hr

Time taken to cross each other = 20 seconds

Formula Used:

Relative Speed (Opposite Direction) = S₁ + S₂

Distance = Relative Speed × Time

Conversion: 1 m/s = 18/5 km/hr

Calculations:

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

⇒ Total distance to be covered = L₁ + L₂

⇒ Distance = 239 + 161 = 400 m

⇒ Relative Speed in m/s = Total Distance ÷ Time

⇒ Relative Speed = 400 ÷ 20 = 20 m/s

⇒ Relative Speed in km/hr = 20 × (18/5)

⇒ Relative Speed = 4 × 18 = 72 km/hr

⇒ Since they are moving in opposite directions, Relative Speed = S₁ + S₂

⇒ 72 = 43 + S₂

⇒ S₂ = 72 − 43

⇒ S₂ = 29 km/hr

∴ The correct answer is 29 km/hr.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds (S1 + S2). If they move in the same direction, it is the difference (S1 − S2).

Crossing Distance

To cross each other completely, the total distance covered is always the sum of the lengths of both trains (L1 + L2), regardless of direction.

Unit Conversion

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

Shortcut Trick

Net percentage change for 2 years at 20% = 20 + 20 + (20 × 20) ÷ 100 = 44%

Total amount percentage = 100% (Principal) + 44% (Interest) = 144%

Required Amount = 4400 × 144%

Amount = 44 × 144 = 6336

∴ The correct answer is ₹6,336.

Alternate Method

Given:

Principal (P) = ₹4,400

Rate (R) = 20% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R ÷ 100)ⁿ

Calculations:

⇒ A = 4400 × (1 + 20 ÷ 100)²

⇒ A = 4400 × (1 + 1 ÷ 5)²

⇒ A = 4400 × (6 ÷ 5)²

⇒ A = 4400 × (36 ÷ 25)

⇒ A = 176 × 36

⇒ A = 6336

∴ The correct answer is ₹6,336.

Additional Information

Effective Interest Rate

For a rate of r% for 2 years, the effective compound interest rate is calculated as r + r + (r × r) ÷ 100.

CI vs SI Difference

The difference between Compound Interest and Simple Interest for 2 years is given by P × (R ÷ 100)².

Compounding Periods

If interest is compounded half-yearly, the rate becomes R ÷ 2 and the time period becomes 2n in the formula A = P(1 + R ÷ 100)ⁿ.

Shortcut Trick

For numbers a, b, c, d to be in proportion by adding x, x = (ad − bc) / ((b + c) − (a + d))

x = (10 × 28 − 14 × 38) / ((14 + 38) − (10 + 28)) = (280 − 532) / (52 − 38) = −252 / 14 = −18

Given 3x : y :: y : (5x − 6) ⇒ y² = 3x(5x − 6)

y² = 3(−18) × [5(−18) − 6] = −54 × (−90 − 6) = −54 × −96 = 5184

y = √5184 = 72

∴ The correct answer is 72.

Alternate Method

Given numbers: 10, 14, 38, 28

Condition: (10 + x), (14 + x), (38 + x), (28 + x) are in proportion.

From the proportion:

⇒ (10 + x) / (14 + x) = (38 + x) / (28 + x)

⇒ (10 + x)(28 + x) = (14 + x)(38 + x)

⇒ 280 + 10x + 28x + x² = 532 + 14x + 38x + x²

⇒ 280 + 38x = 532 + 52x

⇒ 52x − 38x = 280 − 532

⇒ 14x = −252

⇒ x = −18

Now, given 3x : y :: y : (5x − 6):

⇒ y² = 3x × (5x − 6)

⇒ y² = 3(−18) × [5(−18) − 6]

⇒ y² = −54 × (−90 − 6)

⇒ y² = −54 × −96

⇒ y² = 5184

⇒ y = √5184

⇒ y = 72 (Since y > 0)

∴ The correct answer is 72.

Additional Information

Mean Proportional

If a, b, c are in continued proportion, then b is the mean proportional where b² = ac or b = √ac.

Fourth Proportional

If a, b, c, d are in proportion, then d is called the fourth proportional, and a/b = c/d or ad = bc.

Third Proportional

In the continued proportion a : b :: b : c, c is the third proportional to a and b, given by c = b² / a.

Shortcut Trick

Cost Price (CP) is taken as 100%.

Discount = 28% of CP.

Selling Price (SP) = CP − Discount% = 100% − 28% = 72%.

SP = 7,400 × 72/100 = 74 × 72 = 5,328.

∴ The correct answer is ₹5,328.

Alternate Method

Given: Cost Price (CP) = ₹7,400 and Discount% = 28%.

Formula Used: Selling Price = Cost Price − (Discount% × Cost Price)

⇒ Discount Amount = 28% of 7,400

⇒ Discount Amount = (28 ÷ 100) × 7,400

⇒ Discount Amount = 28 × 74 = 2,072

⇒ Selling Price = 7,400 − 2,072

⇒ Selling Price = 5,328

∴ The correct answer is ₹5,328.

Additional Information

Standard Discount Formula

Generally, discount is offered on the Marked Price (MP). SP = MP − Discount.

Successive Discounts

If two discounts d₁% and d₂% are given, effective discount = (d₁ + d₂ − (d₁ × d₂)/100)%.

Profit and Loss Percentage

Profit % = (Profit/CP) × 100 and Loss % = (Loss/CP) × 100.

Shortcut Trick

In a right-angled triangle, if the angle of elevation/depression is 45°, then the Perpendicular and Base are always equal (Ratio 1 : 1).

Height of building (Perpendicular) = 15 m

Horizontal distance (Base) = Height of building

∴ Horizontal distance = 15 m

∴ The correct answer is 15 m.

Given:

Height of the building (AB) = 15 m

Angle of depression from top A to point C = 45°

Angle of elevation from point C to top A = 45° (Alternate interior angles)

Formula Used:

tan θ = Perpendicular / Base

In right-angled triangle ABC:

⇒ tan 45° = AB / BC

⇒ 1 = 15 / BC

⇒ BC = 15 m

∴ The correct answer is 15 m.

Standard Trigonometric Ratios

For 30°, tan 30° = 1/√3; For 60°, tan 60° = √3. These are commonly used to find heights and distances.

Angle of Depression vs Elevation

The angle of depression from an observer to an object is numerically equal to the angle of elevation from the object to the observer due to parallel horizontal lines.

Pythagoras Theorem

In any right-angled triangle, Hypotenuse² = Perpendicular² + Base². In this case, AC = √(15² + 15²) = 15√2 m.

Shortcut Trick

Assume the base average for all students is 95.

Group 1 (22 students) has an average of 85, which is 10 less than 95.

Total deviation = 22 × (−10) = −220.

Average deviation for the whole class = −220 ÷ 75 ≈ −2.93.

Whole class average = 95 − 2.93 = 92.066... ≈ 92.1.

∴ The correct answer is 92.1.

Alternate Method

Given: Total students = 75 | Group 1 = 22 students | Average 1 = 85 | Average 2 = 95

Formula: Combined Average = (n₁ × x₁ + n₂ × x₂) ÷ (n₁ + n₂)

⇒ Number of students in Group 2 = 75 − 22 = 53 ⇒ Sum of scores for Group 1 = 22 × 85 = 1870 ⇒ Sum of scores for Group 2 = 53 × 95 = 5035 ⇒ Total sum of scores = 1870 + 5035 = 6905 ⇒ Combined Average = 6905 ÷ 75 ⇒ Combined Average = 92.0666... ⇒ Rounded Average ≈ 92.1

∴ The correct answer is 92.1.

Additional Information

Weighted Average

When groups have different sizes, the combined average is the sum of products of group sizes and their averages, divided by the total count.

Deviation Method

Assume a temporary mean to simplify calculations. The true mean = Assumed Mean + (Sum of deviations ÷ Total count).

Rounding Rule

To round to one decimal place, look at the second decimal digit; if it is 5 or more, increase the first decimal digit by 1.

Shortcut Trick

Net ratio: Profit 25% (4 → 5) and Loss 10% (10 → 9)

Combined multiplier = (5/4) × (9/10) = 45/40 = 9/8

Final Selling Price = ₹9,900 ⇒ 9 units = 9,900

1 unit = 1,100

Initial CP = 8 units = 8 × 1,100 = ₹8,800

∴ The correct answer is ₹8,800.

Alternate Method

Given: Profit% = 25%, Loss% = 10%, Final Selling Price = ₹9,900

Formula: Final SP = CP × (100 + Gain%)/100 × (100 − Loss%)/100

⇒ Let the cost price for the first trader be ₹x.

⇒ First selling price = x × (125/100) = 1.25x

⇒ Final selling price = 1.25x × (90/100)

⇒ 1.25x × 0.9 = 9,900

⇒ 1.125x = 9,900

⇒ x = 9,900 / 1.125

⇒ x = 9,900 ÷ (9/8)

⇒ x = 1,100 × 8

⇒ x = 8,800

∴ The correct answer is ₹8,800.

Additional Information

Successive Percentage Change

When a value changes by a% and then b%, the net change is given by [a + b + (ab/100)]%.

Effective Multiplying Factor

For chain transactions, the final value = Initial Value × mf₁ × mf₂ × ... × mf_n, where mf is the multiplying factor (1 ± r/100).

Profit and Loss Basics

Profit = SP − CP; Loss = CP − SP; SP = CP × (100 + P%)/100 or CP × (100 − L%)/100.

Shortcut Trick

Ratio of Volume to CSA = (πr²h) / (2πrh) = r/2

⇒ r/2 = 1695.6 ÷ 565.2 = 3

⇒ Radius (r) = 6 m

⇒ TSA = CSA + 2 × π × r²

⇒ TSA = 565.2 + 2 × 3.14 × 36 = 565.2 + 226.08 = 791.28 m²

∴ The correct answer is 791.28 m².

Alternate Method

Given: Curved Surface Area (CSA) = 565.2 m², Volume (V) = 1695.6 m³, π = 3.14

Formulae: Volume = πr²h | CSA = 2πrh | TSA = CSA + 2πr²

⇒ Volume / CSA = (πr²h) / (2πrh) = 1695.6 / 565.2

⇒ r / 2 = 3

⇒ r = 6 m

⇒ TSA = CSA + 2 × π × r²

⇒ TSA = 565.2 + 2 × 3.14 × (6)²

⇒ TSA = 565.2 + 6.28 × 36

⇒ TSA = 565.2 + 226.08

⇒ TSA = 791.28 m²

∴ The correct answer is 791.28 m².

Additional Information

Total Surface Area of Cylinder

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

Volume and Radius Relationship

For any cylinder, the radius can be directly found if Volume (V) and Curved Surface Area (CSA) are known: r = 2V ÷ CSA.

Hollow Cylinder Volume

If the pipe has thickness, the volume of material is given by V = πh(R² − r²), where R is outer and r is inner radius.

Shortcut Trick

Cross Product Method:

Units difference = (5 × 4) − (8 × 1) = 20 − 8 = 12 units

Value difference = ₹72,000 × (4 − 1) = ₹2,16,000

12 units = ₹2,16,000 ⇒ 1 unit = ₹18,000

David's Income = 5 units × 18,000 = ₹90,000

∴ The correct answer is ₹90,000.

Alternate Method

Given:

Income ratio (David : Ajay) = 5 : 8

Expenditure ratio (David : Ajay) = 1 : 4

Savings for each = ₹72,000

Formula Used:

Income − Savings = Expenditure

Let David's income be 5x and Ajay's income be 8x.

⇒ (5x − 72000) / (8x − 72000) = 1 / 4

⇒ 4 × (5x − 72000) = 1 × (8x − 72000)

⇒ 20x − 288000 = 8x − 72000

⇒ 20x − 8x = 288000 − 72000

⇒ 12x = 216000

⇒ x = 216000 ÷ 12 = 18000

⇒ David's monthly income = 5x = 5 × 18000 = 90000

∴ The correct answer is ₹90,000.

Additional Information

Income, Expenditure & Savings Relation

The fundamental relation is Income = Expenditure + Savings. If savings are constant, the difference in income units corresponds to the difference in expenditure units after scaling.

Cross-Multiplication Method

For two ratios a:b and c:d where a fixed value k is added/subtracted: 1 unit = |(b×c − a×d)| / |k×(c − d)|.

Equating Ratio Differences

When savings are equal, ensure the difference between income and expenditure units is the same for both persons by multiplying the ratios with suitable constants.

Shortcut Trick

B = Sum of first three even natural numbers = 2 + 4 + 6 = 12.

Third Proportional C = B² ÷ First Term.

C = 12² ÷ 59 = 144 ÷ 59.

C ≈ 2.44.

∴ The correct answer is 2.44.

Alternate Method

Given: First term (a) = 59, Second term (b) = B.

B = First three even natural numbers (2, 4, 6).

Formula: If a, b, and C are in continued proportion, then a : b :: b : C.

⇒ B = 2 + 4 + 6 = 12

⇒ For third proportional: 59 ÷ 12 = 12 ÷ C

⇒ 59 × C = 12 × 12

⇒ C = 144 ÷ 59

⇒ C = 2.44067...

⇒ C ≈ 2.44 (rounded to two decimal places)

∴ The correct answer is 2.44.

Additional Information

Mean Proportional

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

Fourth Proportional

If four numbers a, b, c, and d are in proportion (a : b :: c : d), then the fourth proportional d = (b × c) ÷ a.

Sum of First n Even Numbers

The sum of the first n even natural numbers is calculated using the formula n × (n + 1).

Continued Proportion

If a, b, c are in continued proportion, then a/b = b/c, which implies b² = ac.

Shortcut Trick

Rate of filling pipe = 1/11 per minute.

Rate of emptying pipe = 1/22 per minute.

Net rate of both pipes = 1/11 − 1/22 = 1/22 per minute.

Time for full tank = 22 minutes.

Time for half tank = 22 ÷ 2 = 11 minutes.

∴ The correct answer is 11.

Alternate Method

Given: Fill time = 11 min, Empty time = 22 min.

Let the total capacity of the tank be the LCM of 11 and 22.

⇒ Total capacity = 22 units.

Combined efficiency of (A + B) = 2 − 1 = 1 unit/min.

⇒ Target to fill = 1/2 × Total Capacity

⇒ Work = 1/2 × 22 = 11 units.

⇒ Time required = Work ÷ Combined Efficiency

⇒ Time = 11 ÷ 1 = 11 minutes.

∴ The correct answer is 11.

Additional Information

Negative Work Concept

In pipes and cisterns, a filling pipe has positive efficiency, while an outlet or emptying pipe has negative efficiency.

Total Work Formula

Time required = Total Work ÷ Net Efficiency. Total work is usually taken as the LCM of time taken by individual pipes.

Partial Filling

If only a fraction of the tank needs to be filled, multiply the total capacity by that fraction before dividing by efficiency.

Shortcut Trick

Property: In quadrilateral PQRS, the angle between bisectors of ∠R and ∠S is ∠STR = (∠P + ∠Q) / 2.

∠P + ∠Q = 2 × 42.5° = 85°.

Ratio P : Q = 8 : 9, Sum of ratios = 8 + 9 = 17 units.

17 units = 85° → 1 unit = 5°.

Difference between ∠Q and ∠P = (9 − 8) units = 1 unit = 5°.

∴ The correct answer is 5°.

Alternate Method

Given: ∠STR = 42.5°, Ratio ∠P : ∠Q = 8 : 9.

Formula Used: Sum of angles of a triangle = 180°; Sum of angles of a quadrilateral = 360°.

⇒ In ΔSTR, ∠STR + ∠TSR + ∠TRS = 180°

⇒ 42.5° + ∠S/2 + ∠R/2 = 180°

⇒ (∠S + ∠R)/2 = 180° − 42.5° = 137.5°

⇒ ∠S + ∠R = 137.5° × 2 = 275°

⇒ In quadrilateral PQRS, ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 275° = 360°

⇒ ∠P + ∠Q = 85°

⇒ Let ∠P = 8x and ∠Q = 9x

⇒ 8x + 9x = 85° ⇒ 17x = 85° ⇒ x = 5°

⇒ Difference = ∠Q − ∠P = 9x − 8x = x = 5°

∴ The correct answer is 5°.

Additional Information

Angle Sum Property

The sum of all interior angles of any quadrilateral is always 360°.

Adjacent Angle Bisectors

The angle formed by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the other two angles: ∠T = (∠P + ∠Q)/2.

Opposite Angle Bisectors

The acute angle formed by the intersection of bisectors of two opposite angles is equal to half the difference of the other two angles.

Shortcut Trick

Volume of Cylinder (V) ∝ r² × h

Radius decrease factor = 1 − 0.15 = 0.85

Height increase factor = 1 + 1.49 = 2.49

New Volume Factor = (0.85)² × 2.49 = 0.7225 × 2.49

New Volume Factor = 1.799025

Percentage increase = (1.799025 − 1) × 100 ≈ 80%

∴ The correct answer is 80%.

Alternate Method

Given:

Decrease in radius (r) = 15%

Increase in height (h) = 149%

Formula Used:

Volume of cylinder = πr²h

Let initial radius (r₁) = 100 units and initial height (h₁) = 100 units

⇒ Initial Volume (V₁) = π × (100)² × 100 = 1,000,000π cubic units

⇒ New radius (r₂) = 100 − 15% of 100 = 85 units

⇒ New height (h₂) = 100 + 149% of 100 = 249 units

⇒ New Volume (V₂) = π × (85)² × 249

⇒ V₂ = π × 7225 × 249 = 1,799,025π cubic units

⇒ Increase in Volume = 1,799,025π − 1,000,000π = 799,025π

⇒ Percentage Increase = (799,025π / 1,000,000π) × 100

⇒ Percentage Increase = 79.9025% ≈ 80%

∴ The correct answer is 80%.

Additional Information

Curved Surface Area (CSA) of Cylinder

The area of the side surface of the cylinder is given by 2 × π × r × h.

Total Surface Area (TSA) of Cylinder

The sum of the CSA and the area of the two circular bases: 2πr(h + r).

Successive Percentage Formula

For two successive changes x% and y%, the net change is [x + y + (xy/100)]%.

Volume proportionality

If radius changes by factor 'a' and height by factor 'b', volume changes by factor a² × b.

Shortcut Trick

Time Ratio = 36 : 39 → 12 : 13

Since work is constant, Men Ratio is inverse of Time Ratio = 13 : 12

Difference in ratio parts = 13 − 12 = 1 unit

Given, 1 unit = 3 people

Value of X (Original Men) = 13 units = 13 × 3 = 39

∴ The correct answer is 39.

Alternate Method

Given:

Initial number of people = X

Initial time taken = 36 days

Number of people after 3 people didn't turn up = X − 3

Actual time taken = 39 days

Formula Used: M₁ × D₁ = M₂ × D₂

Calculations:

⇒ X × 36 = (X − 3) × 39

⇒ 36X = 39X − 117

⇒ 39X − 36X = 117

⇒ 3X = 117

⇒ X = 117 ÷ 3

⇒ X = 39

∴ The correct answer is 39.

Additional Information

Chain Rule Formula

For work problems involving men (M), days (D), and hours (H), the formula is (M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂.

Inverse Proportion in Work

The number of people is inversely proportional to the time taken to complete a constant amount of work (M ∝ 1/D).

Unitary Method for Time & Work

If one person can do a job in 'n' days, then their work rate is 1/n per day.

Shortcut Trick

Numbers between 309 and 320 are 310, 311, 312, 313, 314, 315, 316, 317, 318, 319.

Exclude evens (310, 312, 314, 316, 318) and multiples of 5 (315).

Check remaining: 311 (Prime), 313 (Prime), 317 (Prime), 319 (11 × 29).

Total prime numbers = 3.

∴ The correct answer is 3.

Alternate Method

Given: Range of numbers between 309 and 320.

Method Used: Primality testing of numbers within the specified interval.

⇒ Numbers to check: {310, 311, 312, 313, 314, 315, 316, 317, 318, 319}

⇒ For a number n to be prime, it must not be divisible by any prime number ≤ √n.

⇒ √320 ≈ 17.88. Check divisibility by 2, 3, 5, 7, 11, 13, 17.

⇒ 311: Not div. by 2, 3 (sum=5), 5, 7, 11, 13, 17 → Prime

⇒ 313: Not div. by 2, 3 (sum=7), 5, 7, 11, 13, 17 → Prime

⇒ 315: Divisible by 5 (ends in 5) → Composite

⇒ 317: Not div. by 2, 3 (sum=11), 5, 7, 11, 13, 17 → Prime

⇒ 319: Divisible by 11 (3 − 1 + 9 = 11) → Composite (11 × 29)

⇒ Total Primes = 3

∴ The correct answer is 3.

Additional Information

Prime Number Definition

A natural number greater than 1 that has no positive divisors other than 1 and itself.

Primality Test

To check if N is prime, test its divisibility by all prime numbers up to √N.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and the sum of digits at even positions is 0 or a multiple of 11.

Shortcut Trick

Teacher’s age = New Average + (Initial number of students × Increase in average)

Increase in average = 14 − 13 = 1 year

Teacher’s age = 14 + (43 × 1)

Teacher’s age = 14 + 43 = 57 years

∴ The correct answer is 57.

Alternate Method

Given:

Number of students = 43 | Initial Average = 13 years | New Average (with teacher) = 14 years

Formula Used:

Total Age = Average × Number of People

⇒ Total age of 43 students = 43 × 13

⇒ Total age of 43 students = 559 years

⇒ Total people including teacher = 43 + 1 = 44

⇒ Total age of 44 people = 44 × 14

⇒ Total age of 44 people = 616 years

⇒ Age of teacher = Total age of group − Total age of students

⇒ Age of teacher = 616 − 559 = 57 years

∴ The correct answer is 57.

Additional Information

Average Formula

Average = (Sum of all observations) ÷ (Total number of observations).

Inclusion Shortcut

When one person joins and average increases: New person age = Old Average + [New total members × Increase in Avg].

Exclusion Shortcut

When one person leaves and average changes: Age of person who left = Old Average ± [New total members × Change in Avg].

Weighted Average

Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n is count and A is average.

Shortcut Trick

Using algebraic identities, identify the standard reduction forms.

(a³ − b³) / (a² + ab + b²) = a − b

(a³ + b³) / (a² − ab + b²) = a + b

The expression simplifies to: (a − b) + (a + b) = 2a

∴ The correct answer is 2a.

Alternate Method

Given:

Expression: [(a³ − b³) / (a² + ab + b²)] + [(a³ + b³) / (a² − ab + b²)]

Formula Used:

1. a³ − b³ = (a − b)(a² + ab + b²)
2. a³ + b³ = (a + b)(a² − ab + b²)

Calculation:

⇒ [(a − b)(a² + ab + b²) / (a² + ab + b²)] + [(a + b)(a² − ab + b²) / (a² − ab + b²)]

⇒ (a − b) + (a + b)

⇒ a − b + a + b

⇒ 2a

∴ The correct answer is 2a.

Additional Information

Sum of Cubes Identity

The sum of cubes can be factored as a³ + b³ = (a + b)(a² − ab + b²).

Difference of Cubes Identity

The difference of cubes can be factored as a³ − b³ = (a − b)(a² + ab + b²).

Difference of Squares

One of the most used identities is a² − b² = (a + b)(a − b).

BODMAS rule

Always solve expressions in the order: Brackets, Orders, Division/Multiplication, and Addition/Subtraction.

Shortcut Trick

Let initial Income = 100, then Savings = 20 and Expenditure = 80.

Change in Income (ΔI) = 100 × 36% = 36 units.

Change in Expenditure (ΔE) = 80 × 40% = 32 units.

Change in Savings (ΔS) = ΔI − ΔE = 36 − 32 = 4 units.

Value of 100 units = ₹43,800 ⇒ Value of 1 unit = ₹438.

Savings increase = 4 units × ₹438 = ₹1,752.

∴ The correct answer is increase by ₹1,752.

Alternate Method

Given: Initial Income = ₹43,800 | Savings % = 20% | Income increase = 36% | Expenditure increase = 40%

Formula: Income = Expenditure + Savings

⇒ Initial Savings = 20% of 43,800 = 0.20 × 43,800 = ₹8,760

⇒ Initial Expenditure = 43,800 − 8,760 = ₹35,040

⇒ Increase in Income = 36% of 43,800 = 0.36 × 43,800 = ₹15,768

⇒ Increase in Expenditure = 40% of 35,040 = 0.40 × 35,040 = ₹14,016

⇒ Change in Savings = Increase in Income − Increase in Expenditure

⇒ Change in Savings = 15,768 − 14,016 = ₹1,752

⇒ Since the result is positive, savings increase by ₹1,752.

∴ The correct answer is increase by ₹1,752.

Additional Information

Income-Expenditure Relationship

The fundamental relation is Income = Expenditure + Savings. Any change in two components directly determines the change in the third.

Percentage Change Formula

Percentage change = (Final Value − Initial Value) ÷ Initial Value × 100. This is useful when the question asks for the percentage increase or decrease of savings.

Alligation Method for Income-Expenditure

This topic can also be solved using alligation where the ratio of Expenditure to Savings (80:20 or 4:1) is placed at the bottom, and the growth rates are placed at the top.

Shortcut Trick

Check the sum of digits for divisibility by 3.

Sum of digits of 51 = 5 + 1 = 6.

Since 6 is divisible by 3, 51 is also divisible by 3 (51 = 3 × 17).

Numbers 31, 67, and 79 cannot be divided by any number other than 1 and themselves.

∴ The correct answer is 51.

Alternate Method

Given: Numbers provided are 51, 31, 79, and 67.

Rule: A prime number is a natural number greater than 1 that has exactly two factors: 1 and the number itself.

⇒ For 31: Factors are 1 and 31 (Prime).

⇒ For 67: Factors are 1 and 67 (Prime).

⇒ For 79: Factors are 1 and 79 (Prime).

⇒ For 51: Divisors are 1, 3, 17, and 51.

⇒ Since 51 has more than two factors, it is a composite number.

∴ The correct answer is 51.

Additional Information

Prime Numbers

A number that has only two factors, 1 and itself. Examples: 2, 3, 5, 7, 11, 13, etc.

Composite Numbers

A number that has more than two factors. The number 1 is neither prime nor composite.

Divisibility Rule of 3

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

Smallest Prime Number

The number 2 is the smallest prime number and the only even prime number.

Relative Speed (Opposite Direction) = 45 + 63 = 108 km/hr

Convert to m/s: 108 × (5/18) = 30 m/s

Total distance to cover = Sum of train lengths = 170 + 220 = 390 m

Time = Distance ÷ Speed = 390 ÷ 30 = 13 seconds

∴ The correct answer is 13 seconds.

Alternate Method

Given:

Length of first train (L₁) = 170 m

Length of second train (L₂) = 220 m

Speed of first train (S₁) = 45 km/hr

Speed of second train (S₂) = 63 km/hr

Formula Used:

Relative Speed (Opposite) = S₁ + S₂

Time = (L₁ + L₂) ÷ (Relative Speed)

Calculations:

⇒ Total distance to be covered = 170 + 220 = 390 m

⇒ Relative speed = 45 + 63 = 108 km/hr

⇒ Speed in m/s = 108 × (5/18) m/s

⇒ Speed in m/s = 6 × 5 = 30 m/s

⇒ Time taken to cross each other = 390 ÷ 30

⇒ Time = 13 seconds

∴ The correct answer is 13 seconds.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction, Relative Speed = |S₁ − S₂|.

Unit Conversion

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

Crossing a Stationary Object

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

Shortcut Trick

Solve inner small bracket: (14 − 24 ÷ 4) = 14 − 6 = 8

Solve nested parenthesis: (81 ÷ 9 − 8 ÷ 8) = 9 − 1 = 8

Solve square bracket: [49 − 8] = 41

Final calculation: 72 − 41 = 31

∴ The correct answer is 31.

Alternate Method

Given: 72 − [49 − (81 ÷ 9 − (14 − 24 ÷ 4) ÷ 8)]

Formula Used: VBODMAS rule (Vinculum, Bracket, Of, Division, Multiplication, Addition, Subtraction)

⇒ 72 − [49 − (81 ÷ 9 − (14 − 6) ÷ 8)]

⇒ 72 − [49 − (81 ÷ 9 − 8 ÷ 8)]

⇒ 72 − [49 − (9 − 1)]

⇒ 72 − [49 − 8]

⇒ 72 − 41

⇒ 31

∴ The correct answer is 31.

Additional Information

VBODMAS Rule

Simplification must follow the sequence: Vinculum, Brackets, Of, Division, Multiplication, Addition, and finally Subtraction.

Priority of Brackets

Solve internal expressions in the specific order: Parentheses ( ), then Curly Braces { }, and lastly Square Brackets [ ].

Sign Rules

Multiplying two negative signs yields a positive result (− × − = +), while a negative outside a bracket reverses the signs of all terms inside.

Shortcut Trick

Extra cost for 2 more glasses = 330 − 270 = ₹60.

Cost of 1 glass = 60 ÷ 2 = ₹30.

Cost of 4 glasses = 4 × 30 = ₹120.

Cost of 3 cups = 270 − 120 = ₹150.

Cost of 1 cup = 150 ÷ 3 = ₹50.

∴ The correct answer is ₹50.

Alternate Method

Given:

3 cups + 4 glasses = ₹270

3 cups + 6 glasses = ₹330

Formula Used:

Method of Elimination for Linear Equations in two variables.

Let the cost of one cup be C and the cost of one glass be G.

⇒ 3C + 4G = 270 → (1)

⇒ 3C + 6G = 330 → (2)

⇒ Subtracting equation (1) from equation (2):

⇒ (3C + 6G) − (3C + 4G) = 330 − 270

⇒ 2G = 60

⇒ G = 60 ÷ 2 = 30

⇒ Substituting G = 30 in equation (1):

⇒ 3C + 4 × 30 = 270

⇒ 3C + 120 = 270

⇒ 3C = 270 − 120

⇒ 3C = 150

⇒ C = 150 ÷ 3 = 50

∴ The correct answer is ₹50.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c represent a relationship between two quantities. They are solved using substitution or elimination methods.

Elimination Method

In this method, one variable is eliminated by adding or subtracting the equations after making the coefficients of that variable equal.

Unitary Method

Finding the value of a single unit (item) first to calculate the total value of multiple units or vice versa. For example, Cost of 1 unit = Total Cost ÷ Total Quantity.

Shortcut Trick

Ratio of Ages R : S : T based on given conditions (LCM of 2 and 3 = 6):

R : S = 2 : 1 → 6 : 3

R : T = 3 : 1 → 6 : 2

Combined Ratio R : S : T = 6 : 3 : 2

Difference between S and T = 3 − 2 = 1 unit = 5 years.

Required difference between R and T = 6 − 2 = 4 units = 4 × 5 = 20 years.

∴ The correct answer is 20.

Alternate Method

Given:

R = 2 × S ⇒ S = R/2

R = 3 × T ⇒ T = R/3

S − T = 5 years

Substituting the values of S and T in the difference equation:

⇒ R/2 − R/3 = 5

⇒ (3R − 2R) / 6 = 5

⇒ R / 6 = 5

⇒ R = 30 years

⇒ T = 30 / 3 = 10 years

⇒ Difference R − T = 30 − 10 = 20 years

∴ The correct answer is 20.

Additional Information

Ratio of Ages

When comparing three or more quantities, find a common base (LCM) to create a single ratio chain (A:B:C).

Constant Difference Property

The numerical difference between the ages of two people always remains constant regardless of the number of years passed or future time.

Average Age

Average age = (Sum of ages of all persons) ÷ (Total number of persons).

Shortcut Trick

Profit % = 35% → Multiplying Factor = 1 + 35/100 = 1.35

Income = Expenditure × 1.35

Income = 660 × 1.35 = 891

∴ The correct answer is ₹891 crore.

Alternate Method

Given: Expenditure = ₹660 crore, Profit % = 35%

Formula: Income = Expenditure × (100 + Profit %)/100

⇒ Profit = 35% of ₹660 crore

⇒ Profit = (35 ÷ 100) × 660

⇒ Profit = 0.35 × 660 = 231

⇒ Income = Expenditure + Profit

⇒ Income = 660 + 231

⇒ Income = 891

∴ The correct answer is ₹891 crore.

Additional Information

Profit Percentage Formula

Profit % = [(Income − Expenditure) ÷ Expenditure] × 100. This is the standard formula used for company financial performance.

Loss Percentage Formula

Loss % = [(Expenditure − Income) ÷ Expenditure] × 100. It occurs when expenditure exceeds total income.

Direct Income Relation

Income = Expenditure × [(100 + Profit %) ÷ 100]. This shortcut avoids calculating profit separately.

Shortcut Trick

Price after 1st discount of 5% = 21000 × 0.95 = ₹19950

Total discount amount required from this price = 19950 − 11172 = ₹8778

Second discount percentage Z = (8778 ÷ 19950) × 100

Z = 44%

∴ The correct answer is 44.

Alternate Method

Given: Marked Price (MP) = ₹21,000, Final Selling Price (SP) = ₹11,172, Discount 1 = 5%, Discount 2 = Z%.

Formula: Final SP = MP × (100 − d₁)/100 × (100 − d₂)/100

⇒ 11172 = 21000 × (100 − 5)/100 × (100 − Z)/100

⇒ 11172 = 21000 × (95/100) × (100 − Z)/100

⇒ 11172 = 210 × 95 × (100 − Z)/100

⇒ 11172 = 19950 × (100 − Z)/100

⇒ 11172 ÷ 199.5 = 100 − Z

⇒ 56 = 100 − Z

⇒ Z = 100 − 56

⇒ Z = 44

∴ The correct answer is 44.

Additional Information

Successive Discount Formula

When two discounts a% and b% are given successively, the equivalent single discount is given by [a + b − (ab/100)]%.

Effective Price Calculation

The selling price after multiple discounts can be found using the multiplying factor method: SP = MP × (1 − d₁) × (1 − d₂) × ... × (1 − d_n).

Discount Percentage

Discount % is always calculated on the Marked Price (MP) unless specified otherwise. Formula: Discount % = (MP − SP)/MP × 100.

According to the given information,

<br>

To reach point A again, he should drive 26 - 18 = 8 km in East direction.

Hence, the correct answer is “Option 2”.

Pattern followed here is,

Set 1: DIVE - DEVI - EVID

Set 2: BEAR - BRAE - RAEB

Similarly,

Option (1): COLD - CDLO - DLOC

Hence, the correct answer is “Option 1”.

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

1. Only three people sit between D and L, when counted from the right of L.
2. Only three people sit between K and I, when counted from the left of K.
3. B sits to the immediate right of D.
4. J is an immediate neighbour of L as well as I. 

J sits second to the left of C.

Hence, the correct answer is “Option 1”.

Logic followed here is,

12 + 16 = 28

28 + 17 = 45

45 + 18 = 63

63 + 19 = 82

82 + 20 = 102

Hence, the correct answer is “Option 1”.

Given:

Rakesh is 13th from the top.

Rakesh is 18th from the bottom.

Total = Rank from Top + Rank from Bottom - 1

Number of students = 13 + 18 - 1

= 31 - 1

= 30 students.

Hence, the correct answer is "Option 2".

Given: (Left) 515 758 761 726 579 (Right)

<br>

Highest number = 761

Second digit of the highest number = 6

Lowest number = 515 

Second digit of the lowest number = 1

Now, 6 + 1 = 7

Hence, the correct answer is “Option 4”.

‘royal snow blue’ is coded as 'tl zw cr'

And ‘

blue two web’ is coded as ‘rg iq zw’. 

‘blue’ coded in the given language as "zw".

Hence, the correct answer is “Option 2”.

Given series:

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

Condition to be checked:

Even digit ← Even digit → Even digit

Checking the series:

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

There is only one such digit that satisfies the given condition.

Hence, the correct answer is "Option 4".

According to the given information,

<br>

To reach point A again, he should drive 77 - 65 = 12 km in West direction.

Hence, the correct answer is “Option 1”.

Given series:

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

Condition to be checked:

Even digit ← Even digit → Even digit

Checking the series:

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

There are four such digits that satisfy the given condition.

Hence, the correct answer is "Option 1".

<br> <br>

Pattern followed here is,

Hence, the correct answer is “Option 2”.

<br> <br>

(1) BL-WR

(2) HR-CX

(3) QA-KG

(4) UE-PK

Hence, the correct answer is “Option 3”.

BODMAS Table: 

Given equation: 21 − 4 ÷ 371 + 111 × 3 = ?

<br>

Now, replacing the symbols:

⇒ 21 × 4 + 371 − 111 ÷ 3 = ?

⇒ 21 × 4 + 371 − 111 ÷ 3 = ?

⇒ 84 + 371 − 37 = ?

⇒ 84 + 371 − 37 = ?

⇒ 455 − 37 = ?

⇒ 418.

Hence, the correct answer is "Option 4".

Logic followed here is,\

18 + 4 = 22

22 + 8 = 30

30 + 12 = 42

42 + 16 = 58

58 + 20 = 78

Hence, the correct answer is “Option 4”.

<br> <br>

(1) WT-YU

(2) UR-WS

(3) SP-UQ

(4) QN-SP

Hence, the correct answer is “Option 4”.

Least possible Venn diagram,

Conclusions:

<br>

 (I) Some samosas are pizzas → Does not follow (There is no direct relation given between samosas and pizzas. so, it is possible not definite).

<br>

(II) All pizzas are burgers → Does not follow (As some burgers are pizzas. so, All pizzas are burger is not true).

Neither conclusion (I) nor (II) follows.

Hence, the correct answer is “Option 4”.

<br> <br>

Pattern followed here is,

Hence, the correct answer is “Option 4”.

Pattern followed here is,

Set 1: RAMP - PRMA - RPAM

Set 2: SEND - DSNE - SDEN

Similarly,

Option (4): QUAD - DQAU - QDUA

Hence, the correct answer is “Option 4”.

<br> <br>

CEID is related to JLPK,

EGKF is related to LNRM,

Similarly,

GIMH related to,

 

Hence, the correct answer is “Option 4”.

Given:

1 is added to each even digit and 2 is subtracted from each odd digit in the number 54286379.

<br>

The new number formed is 35397157.

Now, counting digits that appear more than once:

• 3 appears 2 times
• 5 appears 2 times
• 7 appears 2 times

<br>

Total digits repeated = 3

Hence, the correct answer is "Option 4".

‘dirt is there’ is coded as ‘mp rc gp’ And ‘

‘clean the dirt’ is coded as ‘hk mp bo’. 

‘dirt’ coded in the given language as "mp".

Hence, the correct answer is “Option 3”.

Logic followed here is,

Hence, the correct answer is “Option 1”.

The logic followed here is:

Logic: 1st number + 8 = 2nd number

2nd number × 2 + 5 = 3rd number.

Now,

For, (12, 20, 45)

⇒ 12 + 8 = 20

⇒ (20 × 2) + 5 = 40 + 5 = 45.

For, (18, 26, 57)

⇒ 18 + 8 = 26

⇒ (26 × 2) + 5 = 52 + 5 = 57.

Similarly, for (13, 21, 47)

⇒ 13 + 8 = 21

⇒ (21 × 2) + 5 = 42 + 5 = 47.

Hence, the correct answer is "Option 1".