RRB Group D Official Paper (Held On: 22 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is All a, b and c.

Key Points

• Carbon has the unique ability to form bonds with other atoms of carbon, giving rise to large molecules. This property is called catenation, which allows for the formation of long chains, branched chains, and ring structures. Hence, statement a is correct.
• Carbon atoms may be linked by single, double, or triple bonds. Compounds of carbon, which are linked by only single bonds between the carbon atoms are called saturated compounds, while compounds of carbon having double or triple bonds between their carbon atoms are called unsaturated compounds. Hence, statement b is correct.
• Since carbon has a valency of four, it is capable of bonding with four other atoms of carbon or atoms of some other mono-valent elements. Carbon forms bonds with atoms of oxygen, hydrogen, nitrogen, sulphur, chlorine, and many other elements, giving rise to compounds with specific properties. Hence, statement c is correct.
• The versatile nature of carbon is essentially a combination of its tetravalency and its power of catenation. These two factors together lead to the existence of a very large number of organic compounds.
• The bonds that carbon forms with most other elements are very strong, making these compounds exceptionally stable. One reason for the formation of strong bonds by carbon is its small size, which enables the nucleus to hold on to the shared pairs of electrons strongly.

Additional Information

• Tetravalency of Carbon:
• Carbon has an atomic number of 6 and its electronic configuration is 2, 4. It has four valence electrons in its outermost shell (L-shell).
• To achieve a noble gas configuration, it shares these four electrons with other atoms, forming covalent bonds. This ability to form four bonds is known as tetravalency.
• Catenation and Bond Strength:
• While other elements like Silicon (Group 14) also exhibit catenation, they form chains that are much more reactive and less stable.
• Carbon-carbon bonds are very strong and stable, which allows for the creation of macromolecules like DNA, proteins, and polymers.
• Functional Groups:
• When carbon bonds with heteroatoms like oxygen or nitrogen, it forms functional groups (such as alcohols, aldehydes, and carboxylic acids).
• These functional groups dictate the chemical properties of the carbon compound, regardless of the length of the carbon chain.
• Isomerism:
• Carbon compounds also show isomerism, where compounds with the same molecular formula possess different structural arrangements.
• This structural diversity contributes significantly to the millions of carbon-based substances found in nature, ranging from hydrocarbons to complex biomolecules.

The correct answer is 2.02 x 10^20 N.

Key Points

• The gravitational force between the Earth and the Moon is calculated using Newton's Law of Universal Gravitation. This law states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The formula used for this calculation is F = G (m1m2 / r^2), where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two bodies, and r is the distance between them.
• Based on accepted scientific data, the mass of the Earth (m1) is approximately 5.97 x 10^24 kg, and the mass of the Moon (m2) is approximately 7.35 x 10^22 kg.
• The average distance (r) between the centers of the Earth and the Moon is 3.84 x 10^8 meters (or 384,400 km).
• The Universal Gravitational Constant (G) is a fixed value of 6.67 x 10^-11 N m^2/kg^2.
• By substituting these values into the formula: F = (6.67 x 10^-11) * [(5.97 x 10^24 * 7.35 x 10^22) / (3.84 x 10^8)^2], the resulting force is approximately 2.02 x 10^20 Newtons (N).

Additional Information

• Nature of Gravitational Force: Gravity is a non-contact force and is always attractive in nature. It acts along the line joining the centers of the two masses, making it a central force.
• Orbital Stability: This massive gravitational pull of 2.02 x 10^20 N provides the centripetal force required to keep the Moon in its elliptical orbit around the Earth. Without this force, the Moon would move in a straight line into outer space due to its inertia.
• Tidal Effects: While the Earth exerts this force on the Moon, the Moon exerts an equal and opposite force on the Earth (consistent with Newton's Third Law). This lunar pull is the primary cause of ocean tides on Earth, creating tidal bulges in the hydrosphere.
• Inverse Square Law: The force of gravity follows the inverse square law. If the distance between the Earth and the Moon were to double, the gravitational force would drop to one-fourth of its current value (approximately 0.505 x 10^20 N).
• Comparison with Surface Gravity: It is important not to confuse the gravitational force between two celestial bodies with the acceleration due to gravity (g). On Earth's surface, g is 9.8 m/s^2, which represents the acceleration of an object, whereas the 2.02 x 10^20 N is the total magnitude of force acting between the two planetary bodies.

Important Points

• Henry Cavendish: The value of the Gravitational Constant (G) was first experimentally determined by Henry Cavendish in 1798 using a torsion balance.
• Barycenter: The Earth and Moon actually orbit around their common center of mass, known as the barycenter. Because the Earth is so much more massive, the barycenter is located about 4,670 km from the Earth's center, which is still inside the Earth's interior (about 1,700 km below the surface).
• Unit of Force: The Newton (N) is the SI unit of force, named after Sir Isaac Newton. One Newton is the force needed to accelerate one kilogram of mass at the rate of one meter per second squared.
• Mass of the Moon: The Moon's mass is about 1.2% of the Earth's mass. This significant mass ratio (compared to other planet-satellite systems in our solar system) is why the gravitational interaction between the two is so powerful.

The correct answer is 1 joule of work is done to move a charge of 1 coulomb.

Key Points

• Potential Difference between two points in an electric circuit is defined as the amount of work done in moving a unit charge from one point to the other.
• The SI unit of electric potential difference is the Volt (V), named after the Italian physicist Alessandro Volta.
• One Volt is formally defined as the potential difference between two points in a current-carrying conductor when 1 Joule of work is done to move a charge of 1 Coulomb from one point to the other. Hence, the statement in option 3 is correct.
• Mathematically, it is expressed as V = W / Q, where V is the potential difference, W is the work done in Joules, and Q is the quantity of charge in Coulombs.
• Option 1 mentions 1 joule of energy dissipated in 1 second, which actually defines 1 Watt (W), the unit of Power. Hence, statement 1 is incorrect.
• Option 2 mentions 1 coulomb of charge flowing in 1 second, which defines 1 Ampere (A), the SI unit of Electric Current. Hence, statement 2 is incorrect.
• Option 4 mentions 1 milliampere of current through 1 Ohm of resistance; according to Ohm's Law (V = I x R), this results in 0.001 Volts or 1 millivolt (mV). Hence, statement 4 is incorrect.

Additional Information

• Instrument for Measurement: The potential difference is measured by an instrument called the Voltmeter. A voltmeter is always connected in parallel across the points between which the potential difference is to be measured.
• Voltmeter Resistance: To ensure that it does not significantly alter the current in the circuit, a voltmeter is designed to have very high resistance.
• Ohm's Law Relationship: The potential difference (V) across the ends of a given metallic wire in an electric circuit is directly proportional to the current (I) flowing through it, provided its temperature remains the same. This is expressed as V = I x R, where R is the Resistance.
• Electric Potential vs. Potential Difference: While Electric Potential at a point is the work done to bring a unit positive charge from infinity to that point, Potential Difference specifically refers to the work done between two defined points in a circuit.

Important Points

• Scalar Quantity: Electric potential and potential difference are scalar quantities, meaning they have magnitude but no specific direction in space.
• Standard Values: In India, the standard domestic supply voltage is 220 V with a frequency of 50 Hz. Commercial batteries, such as AA or AAA alkaline cells, typically provide a potential difference of 1.5 V.
• Work and Energy: Since V = W / Q, the work done (or energy transferred) can be calculated as W = V x Q. This relationship is fundamental in understanding electrical energy consumption.
• High Voltage Transmission: Long-distance power transmission is usually done at high voltages (e.g., 11 kV, 33 kV, or 400 kV) to minimize energy loss due to heating in the transmission lines, as higher voltage allows for lower current for the same power level.

The correct answer is 0.2 m/s2.

Key Points

• Acceleration is defined as the rate of change of velocity with respect to time. Mathematically, it is expressed as a = (v - u) / t, where v is the final velocity, u is the initial velocity, and t is the time taken.
• In the given problem, Meenu starts from rest, which means her initial velocity (u) is 0 m/s.
• The final velocity (v) reached by Meenu is 5 m/s after a duration of 25 seconds.
• By substituting these values into the formula: a = (5 m/s - 0 m/s) / 25 s.
• The calculation results in 5 / 25, which simplifies to 1/5 or 0.2 m/s2. Thus, the acceleration of the bicycle is 0.2 m/s2.

Additional Information

• SI Unit of Acceleration: The standard unit for measuring acceleration in the International System of Units (SI) is meter per second squared (m/s2). It represents how many meters per second the velocity changes every second.
• Vector Quantity: Acceleration is a vector quantity, meaning it has both magnitude and direction. If an object slows down, the acceleration is in the opposite direction of the motion, often referred to as deceleration or retardation.
• Equations of Motion: This calculation is based on the first equation of motion, which is v = u + at. These equations, derived by Sir Isaac Newton, describe the relationship between velocity, acceleration, time, and displacement for objects moving with uniform acceleration.
• Uniform vs. Non-uniform Acceleration: If the velocity of an object changes by equal amounts in equal intervals of time, it is called uniform acceleration. If the rate of change of velocity is not constant, it is non-uniform acceleration.

Important Points

• Initial State: Whenever a problem mentions that an object starts from rest, the initial velocity (u) must always be taken as zero.
• Final State: Conversely, if an object comes to a halt or stops, the final velocity (v) is considered 0 m/s.
• Graphical Representation: On a Velocity-Time graph, the slope of the line represents the acceleration of the object. A straight line indicates constant acceleration, while a curved line indicates changing acceleration.
• Average Acceleration: It is the total change in velocity divided by the total time interval. In this specific case, since the change is linear, the acceleration calculated is the average acceleration over the 25-second period.

The correct answer is A stone dropping freely toward the ground.

Key Points

• Newton’s Third Law of Motion states that for every action, there is always an equal and opposite reaction. These forces always act on two different bodies.
• A stone dropping freely toward the ground is primarily an illustration of Newton’s Second Law of Motion and the Universal Law of Gravitation. The motion is defined by the force of gravity acting on a single mass, causing it to accelerate at 9.8 m/s². While a reaction force technically exists (the stone pulls the Earth), the scenario is the standard example for F = ma. Hence, statement 1 is not an example of the Third Law.
• A gun moving backward after firing a bullet is a classic example of the Third Law. When the gun exerts a forward force on the bullet (action), the bullet exerts an equal and opposite backward force on the gun (reaction), known as recoil. Hence, statement 2 is an example of the Third Law.
• A rocket rising upward functions on the principle of action and reaction. The rocket engine expels exhaust gases downward at very high velocity (action), and these gases exert an equal upward thrust on the rocket (reaction), allowing it to overcome Earth's gravity. Hence, statement 3 is an example of the Third Law.
• A horse pulling a loaded cart involves the horse pushing the ground backward with its hooves (action), and the ground pushing the horse forward with an equal force (reaction). This interaction allows the horse to move the cart forward. Hence, statement 4 is an example of the Third Law.

Additional Information

• Newton's First Law of Motion:
• Also known as the Law of Inertia, it states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external unbalanced force.
• This law defines the concept of inertia, which is the inherent property of an object to resist changes in its state of motion.
• Newton's Second Law of Motion:
• It states that the magnitude of the force is the product of mass and acceleration (F = ma).
• It provides a quantitative measure of force and relates it to the rate of change of momentum.
• Law of Conservation of Momentum:
• Closely linked to the Third Law, it states that the total momentum of an isolated system remains constant if no external forces act on it.
• This principle is used to calculate recoil velocity in firearms and propulsion in jet engines.
• Historical Context:
• Sir Isaac Newton formulated these three laws of motion in his monumental work, Philosophiae Naturalis Principia Mathematica, first published in 1687.
• These laws form the foundation of Classical Mechanics, which describes the motion of macroscopic objects.

Important Points

• The action and reaction forces mentioned in the Third Law never cancel each other out because they act on different objects, not the same one.
• Universal Law of Gravitation: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• In the case of a falling stone, the acceleration of the Earth toward the stone is negligible due to the Earth's massive mass (approximately 5.97 × 10^24 kg), which is why we only observe the stone moving.

The correct answer is Carbon has four electrons in its outer shell and shares electrons to complete its octet.

Key Points

• Carbon is a chemical element with the atomic number 6 and the symbol C. It is a non-metal and belongs to group 14 of the periodic table.
• The electronic configuration of carbon is 2, 4. This means it has four electrons in its outermost shell (valence shell).
• According to the Octet Rule, atoms tend to gain, lose, or share electrons to achieve a stable configuration of eight electrons in their valence shell, similar to noble gases.
• For carbon, gaining four electrons to form a C4- anion is difficult because a nucleus with only 6 protons cannot easily hold onto 10 electrons. Hence, statement 1 is incorrect.
• Conversely, losing four electrons to form a C4+ cation requires an extremely large amount of ionization energy to overcome the electrostatic attraction of the nucleus. Hence, statements 2 and 4 are incorrect.
• To achieve stability, carbon resolves this by sharing its four valence electrons with other atoms (such as Hydrogen, Oxygen, or other Carbon atoms), resulting in the formation of covalent bonds. Hence, statement 3 is correct.

Additional Information

• Tetravalency of Carbon:
• Since carbon has four valence electrons, it is capable of forming four covalent bonds with other atoms. This property is known as tetravalency.
• This allows carbon to form a vast variety of complex molecules, which is the basis of Organic Chemistry.
• Catenation:
• Carbon has the unique ability to form strong covalent bonds with other carbon atoms, giving rise to long chains and rings. This property is called catenation.
• The C-C bond energy is very high (approx 348 kJ/mol), making these structures exceptionally stable.
• Nature of Organic Compounds:
• Most organic compounds are poor conductors of electricity because they do not have free electrons or ions to carry charge.
• They generally have low melting and boiling points compared to ionic compounds because the intermolecular forces (Van der Waals forces) are weaker than ionic bonds.

Important Points

• Covalent Bond Types:
• Carbon can form single bonds (sharing one pair of electrons), double bonds (sharing two pairs), or triple bonds (sharing three pairs).
• Example: In Methane (CH4), carbon forms four single covalent bonds. In Ethene (C2H4), it forms a double bond, and in Ethyne (C2H2), it forms a triple bond.
• Allotropes of Carbon:
• Carbon exists in different physical forms called allotropes, such as Diamond, Graphite, and Fullerenes.
• In Diamond, each carbon atom is bonded to four others in a rigid 3D tetrahedral structure, making it the hardest natural substance.
• In Graphite, each carbon atom is bonded to three others in hexagonal layers, leaving one free electron, which makes graphite a good conductor of electricity.

The correct answer is internal fertilisation.

Key Points

• Fertilisation is the process of the fusion of a male gamete (sperm) with a female gamete (ovum/egg) to form a zygote.
• When this fusion occurs inside the female body, it is known as internal fertilisation. This process ensures that the embryo is protected from the external environment during its initial stages of development.
• Internal fertilisation is a characteristic feature of many terrestrial animals, including mammals (such as humans, cows, and dogs), birds (such as hens), and reptiles.
• In this method, the male deposits sperm into the female reproductive tract, where one sperm eventually fuses with the egg.
• This type of fertilisation has a higher success rate for the survival of the offspring compared to external fertilisation because the zygote develops in a controlled, safe environment within the mother's body.

Additional Information

• External Fertilisation:
• This process occurs when the fusion of male and female gametes takes place outside the female body, typically in an aquatic medium like water.
• It is common in aquatic animals such as fish, starfish, and amphibians like frogs.
• During the breeding season (often spring or monsoon), the female lays hundreds of eggs in the water, and the male releases sperm over them to facilitate fertilisation.
• In Vitro Fertilisation (IVF):
• IVF is a medical technique where fertilisation is carried out outside the body in a laboratory (often in a glass container or petri dish).
• It is primarily used for couples who face issues like blocked oviducts, which prevent natural internal fertilisation.
• The resulting zygote is allowed to develop for about one week in a controlled environment before being implanted into the mother's uterus for further growth.
• Test-Tube Babies:
• The term "test-tube babies" refers to infants born through the IVF technique.
• It is a misnomer because the baby does not actually grow in a test tube; only the initial fertilisation happens in a laboratory, while the rest of the gestation occurs in the mother's womb.
• Zygote and Embryo:
• The zygote is the single cell formed immediately after fertilisation.
• As the zygote divides repeatedly, it forms a ball of cells called an embryo, which then travels to the uterus to get embedded in the uterine wall for further development into a foetus.

The correct answer is Testosterone.

Key Points

• Testosterone is the primary male sex hormone and an androgen, which plays a critical role in the sexual maturation of boys during puberty.
• It is primarily produced in the testes by the Leydig cells, under the influence of the Luteinizing Hormone (LH) secreted by the anterior pituitary gland.
• This hormone triggers the development of secondary sexual characteristics, such as the growth of facial, axillary, and pubic hair, and the deepening of the voice due to the enlargement of the larynx.
• Testosterone is also responsible for the increase in muscle mass, bone density, and the stimulation of spermatogenesis (the production of sperm).
• While Estrogen and Progesterone are found in males in very small quantities, they are not the primary hormones for male maturation. Hence, the correct answer is Testosterone.

Additional Information

• Estrogen:
• It is the primary female sex hormone responsible for the development of female secondary sexual characteristics, such as breast development and the regulation of the menstrual cycle.
• In females, it is mainly produced in the ovaries.
• Progesterone:
• This hormone is primarily involved in the female reproductive system, playing a vital role in maintaining the lining of the uterus (endometrium) during pregnancy.
• It is secreted by the corpus luteum in the ovary after ovulation occurs.
• Oxytocin:
• Often referred to as the "bonding hormone" or "love hormone," it is produced in the hypothalamus and released by the posterior pituitary gland.
• Its primary functions include stimulating uterine contractions during childbirth and the let-down reflex during breastfeeding.
• Puberty in Boys:
• The process of sexual maturation typically begins between the ages of 9 and 14 years.
• The Hypothalamus releases Gonadotropin-Releasing Hormone (GnRH), which signals the pituitary gland to release FSH (Follicle-Stimulating Hormone) and LH, eventually leading to the production of Testosterone.

Important Points

• Chemical Nature: Testosterone, Estrogen, and Progesterone are all steroid hormones derived from cholesterol.
• Anabolic Effects: Testosterone has significant anabolic effects, meaning it promotes the synthesis of proteins and the growth of tissues with androgen receptors.
• Regulatory Mechanism: The production of sex hormones is regulated via a negative feedback loop involving the Hypothalamic-Pituitary-Gonadal (HPG) axis. When testosterone levels are high, the brain reduces the signals to produce more.
• Secondary Characteristics: Other male changes include increased sebaceous gland activity (often leading to acne) and a change in body fat distribution toward a more android (apple-shaped) pattern.

The correct answer is 4R.

Key Points

• The resistance (R) of a uniform metallic conductor is directly proportional to its length (L) and inversely proportional to its cross-sectional area (A). This is expressed by the formula R = ρ (L/A), where ρ (rho) is the electrical resistivity of the material.
• When a wire is stretched, its volume (V) remains constant. The volume of a cylindrical wire is calculated as the product of its cross-sectional area and its length (V = A × L).
• If the wire is stretched to twice its length, the new length L' becomes 2L. Because the volume is constant, the new area A' must decrease to maintain the equation A × L = A' × (2L). This mathematical relationship implies that the new area A' = A/2. Hence, the change in area is a critical factor in determining the final resistance.
• By substituting the new dimensions into the resistance formula, the new resistance R' becomes ρ (2L / (A/2)). This simplifies to ρ (4L / A), which is 4 × R. Hence, the new resistance is four times the original resistance.
• This result demonstrates that stretching a wire increases resistance not just by increasing length, but also by making the wire thinner (reducing the area), both of which impede the flow of electrons. Hence, the correct answer is 4R.

Additional Information

• Electrical Resistivity (ρ): This is an intrinsic property of a material that quantifies how strongly it opposes the flow of electric current. It does not depend on the dimensions (length or area) of the wire but depends on the nature of the material and the temperature. The SI unit of resistivity is the Ohm-meter (Ω·m).
• Ohm's Law: Formulated by Georg Simon Ohm in 1827, it states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, provided physical conditions like temperature remain constant (V = IR).
• Conductivity (σ): This is the inverse of resistivity (σ = 1/ρ). Materials with high conductivity, such as silver and copper, are excellent conductors of electricity and are commonly used in electrical wiring.
• Factors Affecting Resistance:
• Nature of Material: Different materials have different densities of free electrons.
• Temperature: For most metals, resistance increases as the temperature rises because ions vibrate more vigorously, causing more frequent collisions with moving electrons.
• Dimensions: As seen in this problem, length and thickness (area) directly dictate the numerical value of resistance for a specific object.

Important Points

• Relation with Radius: Since the area (A) of a wire is πr² (where r is the radius), the resistance is inversely proportional to the fourth power of the radius when the wire is stretched. For example, if the radius is halved by stretching, the resistance would increase by 16 times.
• Stretching vs. Connecting: It is important to distinguish between stretching a wire and connecting another wire in series. If you simply add another identical wire of length L to the original, the length doubles but the area stays the same, resulting in a resistance of only 2R. Stretching is different because the conservation of volume forces the wire to become thinner.
• Numerical Application: This principle is often used in strain gauges, which are sensors used to measure the strain on an object. When the object is deformed, the electrical resistance of the foil in the gauge changes, allowing for precise measurement of mechanical stress.

The correct answer is They slow down or become imbalanced.

Key Points

• Thyroxin (also known as T4) is the primary hormone secreted by the thyroid gland, which is located in the neck region. It is essential for the regulation of the Basal Metabolic Rate (BMR) in the body.
• When thyroxin is absent or deficient, the chemical reactions that convert oxygen and nutrients into energy (metabolism) are severely impacted. This results in the metabolic processes to slow down or become imbalanced as the body cannot maintain its normal physiological functions at the required speed.
• The hormone acts on nearly every cell in the body to stimulate cellular respiration and the oxidation of glucose. Without this stimulation, energy production drops, leading to symptoms like excessive fatigue, weight gain, and mental sluggishness.
• Thyroxin also maintains the balance of carbohydrates, proteins, and fats metabolism. An imbalance in these areas can lead to physical growth issues and improper development of the nervous system.
• Hence, the correct answer is They slow down or become imbalanced.

Additional Information

• Hypothyroidism: This is a clinical condition characterized by the underproduction of thyroid hormones. In adults, severe hypothyroidism is known as Myxedema, which causes swelling of the face and hands. In infants, a total lack of thyroxin can lead to Cretinism, characterized by stunted physical and mental growth.
• Hyperthyroidism: This occurs when the thyroid gland is overactive and produces too much thyroxin. In this case, metabolic activities speed up uncontrollably, leading to rapid weight loss, increased heart rate, and protruding eyeballs (a condition known as Exophthalmic Goitre).
• Role of Iodine: The thyroid gland requires Iodine to synthesize thyroxin. A deficiency of iodine in the diet leads to Goitre, where the thyroid gland enlarges significantly to compensate for the hormone shortage.
• Basal Metabolic Rate (BMR): It refers to the minimum amount of energy required by the body to maintain vital functions while at complete rest. Thyroxin is the chief regulator of this rate.
• Feedback Mechanism: The production of thyroxin is regulated by the Thyroid Stimulating Hormone (TSH), which is secreted by the anterior pituitary gland. This ensures that the levels of thyroxin in the blood remain within a narrow, healthy range.

Important Points

• Thermoregulation: Thyroxin is vital for maintaining body temperature. By increasing metabolic activity, it generates heat as a byproduct, helping the body stay warm in cold environments.
• Types of Thyroid Hormones: The gland produces two main hormones: Triiodothyronine (T3) and Thyroxine (T4). Although T4 is produced in larger quantities, T3 is the more biologically active form.
• Calcitonin: Besides metabolic hormones, the thyroid also secretes Calcitonin, which helps in regulating calcium levels in the blood by promoting the deposition of calcium into the bones.
• Atomic Fact: Iodine, with the atomic number 53, is the critical element found in thyroxin. Without sufficient intake of iodized salt or iodine-rich foods, thyroxin levels cannot be maintained.

The correct answer is It is converted into ethanoic acid.

Key Points

• The oxidation of ethanol (CH₃CH₂OH) is a chemical reaction where it is transformed into ethanoic acid (CH₃COOH), also known as acetic acid.
• This process is carried out using strong oxidizing agents such as alkaline potassium permanganate (KMnO₄) or acidified potassium dichromate (K₂Cr₂O₇).
• When ethanol is heated with these reagents, the hydroxyl group (-OH) is converted into a carboxyl group (-COOH). Hence, the statement that it is converted into ethanoic acid is correct.
• In this reaction, the addition of oxygen and the removal of hydrogen take place, which is the fundamental definition of oxidation in organic chemistry.
• The reaction typically proceeds through an intermediate called ethanal (acetaldehyde), which is rapidly further oxidized to ethanoic acid under strong oxidizing conditions.

Additional Information

• Esterification Reaction:
• When ethanol reacts with ethanoic acid in the presence of an acid catalyst like concentrated sulphuric acid, it forms an ester called ethyl ethanoate.
• Esters are sweet-smelling substances used in perfumes and as flavoring agents.
• Dehydration of Ethanol:
• Heating ethanol at 443 K with excess concentrated sulphuric acid results in the dehydration of ethanol to form ethene (C₂H₄).
• In this reaction, sulphuric acid acts as a dehydrating agent by removing a water molecule.
• Combustion:
• Ethanol is a highly flammable liquid that burns in air with a blue flame to produce carbon dioxide, water, and a significant amount of heat and light.
• Due to its clean-burning properties, it is often used as a biofuel or added to petrol.
• Ethanoic Acid Properties:
• Ethanoic acid belongs to a group of acids called carboxylic acids.
• A 5-8% solution of ethanoic acid in water is known as vinegar and is widely used as a preservative in pickles.
• The melting point of pure ethanoic acid is 290 K, and it often freezes during winter in cold climates, leading to the name glacial acetic acid.

Important Points

• Catalysts vs. Reagents: In the oxidation process, KMnO₄ and K₂Cr₂O₇ are consumed or changed, acting as oxidizing agents, whereas in dehydration, H₂SO₄ acts as a catalyst.
• Physical State: Ethanol is a liquid at room temperature with a boiling point of 351 K. It is the active ingredient in all alcoholic drinks and is a good solvent used in medicines like tincture iodine, cough syrups, and many tonics.
• Conversion to Methane: Conversion of ethanol to methane is not a standard direct chemical reaction. Methane is usually produced by the decarboxylation of sodium ethanoate using soda lime.

The correct answer is The number of protons in the nucleus of an atom.

Key Points

• The atomic number of an element is defined as the total number of protons present in the nucleus of an atom. It is traditionally represented by the symbol 'Z'. Hence, statement 1 is correct.
• The atomic number is a unique identifier for each chemical element. For instance, all Hydrogen atoms have an atomic number of 1, while all Carbon atoms have an atomic number of 6.
• In the Modern Periodic Table, elements are arranged in increasing order of their atomic numbers, as this property determines the chemical identity and characteristics of the element.
• Since an atom is electrically neutral in its ground state, the number of protons (positive charge) is exactly equal to the number of electrons (negative charge). However, the definition of the atomic number specifically refers to the protons because the number of electrons can change during chemical reactions or ionization.
• Hence, the correct answer is The number of protons in the nucleus of an atom.

Additional Information

• Valence Electrons:
• The number of electrons in the outermost shell of an atom is known as the valence electrons. These electrons participate in chemical bonding and determine the valency of the element. Hence, statement 2 is incorrect.
• Neutrons and Isotopes:
• The number of neutrons in the nucleus can vary even among atoms of the same element. Atoms with the same atomic number but different numbers of neutrons are called isotopes (e.g., Protium, Deuterium, and Tritium for Hydrogen). Hence, statement 3 is incorrect.
• Mass Number (A):
• The total number of protons and neutrons in the nucleus is called the mass number, denoted by 'A'. These subatomic particles are collectively known as nucleons. Hence, statement 4 is incorrect.
• The relationship between these values is given by the formula: Mass Number (A) = Atomic Number (Z) + Number of Neutrons (N).

Important Points

• Henry Moseley: In 1913, the English physicist Henry Moseley provided the physical basis for the atomic number through X-ray spectroscopy experiments. He demonstrated that the atomic number is a more fundamental property of an element than its atomic weight.
• Periodic Law: The Modern Periodic Law states that "the physical and chemical properties of elements are periodic functions of their atomic numbers." This replaced Mendeleev's earlier concept based on atomic mass.
• Representation: In chemical notation, the atomic number is usually written as a subscript on the left side of the element's symbol, while the mass number is written as a superscript. For example, in ¹²₆C, 6 is the atomic number and 12 is the mass number.
• Nuclear Charge: The atomic number also represents the magnitude of the positive charge in the nucleus, as each proton carries one unit of positive charge.

The correct answer is Sound requires a medium for propagation.

Key Points

• Sound is a mechanical wave that results from the back-and-forth vibration of the particles of the medium through which it moves.
• Statement 1 suggests that sound travels faster in gases than in solids. In reality, the speed of sound depends on the density and elasticity of the medium. Particles in solids are packed more tightly than in liquids or gases, allowing sound to travel fastest in solids (approximately 5000 m/s in steel) and slowest in gases (about 343 m/s in air). Hence, statement 1 is incorrect.
• Statement 2 mentions that sound can travel in a vacuum. Since sound is a mechanical wave, it requires a material medium (solid, liquid, or gas) to propagate. In a vacuum, such as outer space, there are no molecules to vibrate, meaning sound cannot travel. Hence, statement 2 is incorrect.
• Statement 3 states that sound travels as transverse waves in air. In fluids like air and water, sound travels as longitudinal waves, where the particles of the medium vibrate parallel to the direction of energy transport. Transverse waves are those where particles move perpendicular to the wave direction, which is characteristic of light or waves on a string. Hence, statement 3 is incorrect.
• Statement 4 asserts that sound requires a medium for propagation. This is a fundamental property of mechanical waves. Sound energy is transferred through the collisions of atoms and molecules within a substance. Hence, statement 4 is correct.
• Hence, the correct answer is Sound requires a medium for propagation.

Additional Information

• Nature of Sound Waves:
• Sound waves are categorized as longitudinal waves because the displacement of the medium is in the same direction as the wave propagation.
• They consist of compressions (regions of high pressure and density) and rarefactions (regions of low pressure and density).
• Speed of Sound Factors:
• Temperature: The speed of sound increases with an increase in temperature. For every 1 degree Celsius rise in temperature, the speed of sound in air increases by about 0.6 m/s.
• Humidity: Sound travels faster in humid air than in dry air because moist air is less dense than dry air.
• State of Matter: Sound travels through steel about 15 times faster than through air.
• Human Audibility:
• The audible range for the average human ear is between 20 Hz and 20,000 Hz (20 kHz).
• Sounds below 20 Hz are termed infrasonic, while sounds above 20,000 Hz are termed ultrasonic.

Important Points

• Bell Jar Experiment: This is a classic experiment used to prove that sound needs a medium. As the air is removed from a sealed jar using a vacuum pump, the sound of an electric bell inside becomes fainter and eventually disappears, even though the hammer is still seen hitting the bell.
• Reflection of Sound: Like light, sound follows the laws of reflection. This phenomenon results in echoes and is used in technologies like SONAR (Sound Navigation and Ranging) and ultrasound imaging.
• Characteristics of Sound: Sound is defined by its amplitude (which determines loudness), frequency (which determines pitch), and timbre (the quality or tone).

The correct answer is Convex lens.

Key Points

• Hypermetropia, also known as long-sightedness or farsightedness, is a common vision defect where a person can see distant objects clearly but cannot see nearby objects distinctly.
• In a hypermetropic eye, the image of a nearby object is formed behind the retina instead of on it. This happens because the light rays do not converge enough by the time they reach the retina.
• This defect arises either because the focal length of the eye lens is too long or the eyeball has become too small.
• To correct this, a convex lens (converging lens) of appropriate focal length is used. The convex lens provides the additional focusing power needed to converge the light rays, ensuring the image is formed exactly on the retina.
• Hence, the correct answer is Convex lens.

Additional Information

• Myopia (Near-sightedness):
• In this defect, a person can see nearby objects clearly but cannot see distant objects distinctly. The image is formed in front of the retina.
• It is corrected by using a concave lens (diverging lens), which diverges the incoming light rays so that the eye lens can focus them properly on the retina.
• Presbyopia:
• This is a vision defect that usually occurs with aging. The power of accommodation of the eye decreases as the ciliary muscles weaken and the eye lens loses its flexibility.
• People suffering from this often require bifocal lenses, where the upper part is a concave lens (for distant vision) and the lower part is a convex lens (for near vision).
• Astigmatism:
• This defect occurs when the cornea is not spherical in shape, leading to light being focused at different points. It results in blurred vision for both near and far objects.
• It is corrected by using cylindrical lenses, which compensate for the uneven curvature of the cornea or lens.
• Power of a Lens:
• The power (P) of a lens is defined as the reciprocal of its focal length (f) in meters (P = 1/f).
• The SI unit of power is the Dioptre (D). A convex lens has positive power, while a concave lens has negative power.

Important Points

• The near point for a normal human eye is approximately 25 cm. For a hypermetropic eye, the near point is farther away than 25 cm.
• The far point for a normal human eye is infinity. In myopic eyes, the far point is closer than infinity.
• The retina acts as a light-sensitive screen where the image is formed. It contains specialized cells called rods and cones that convert light into electrical signals sent to the brain via the optic nerve.
• The Iris controls the amount of light entering the eye by adjusting the size of the pupil.
• The human eye lens is a fibrous, jelly-like material that changes its shape to adjust focal length, a process known as accommodation.

The correct answer is It increases.

Key Points

• According to Gay-Lussac’s Law, the pressure of a given mass of gas is directly proportional to its absolute temperature, provided the volume remains constant.
• When the temperature of a gas increases, the average kinetic energy of the gas molecules also increases. This causes the molecules to move at higher velocities.
• As the molecules move faster, they collide with the walls of the container more frequently and with greater force. Since pressure is defined as the force exerted per unit area, these intensified collisions result in an increase in pressure.
• This relationship can be mathematically expressed as P/T = k (where k is a constant) or P1/T1 = P2/T2. This implies that if the temperature (T) increases, the pressure (P) must also increase to maintain the constant ratio. Hence, the statement that pressure increases is correct.
• The temperature in these gas law calculations must always be expressed in Kelvin (absolute temperature), where 0 Kelvin represents absolute zero, the point at which molecular motion theoretically stops.

Additional Information

• Gay-Lussac's Law:
• This law was formulated by the French chemist Joseph Louis Gay-Lussac in the year 1802.
• It describes the Pressure-Temperature relationship. A practical example is a pressure cooker; as the heat (temperature) increases inside the sealed pot (constant volume), the steam pressure rises rapidly to cook food faster.
• Another example is automobile tires; during long drives in summer, the friction and road heat increase the temperature of the air inside the tires, leading to an increase in tire pressure, which can sometimes cause a blowout.
• Ideal Gas Law:
• The Ideal Gas Law is represented by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature.
• This law combines Boyle's Law, Charles's Law, and Avogadro's Law into a single equation that describes the behavior of a hypothetical ideal gas.
• Kinetic Molecular Theory (KMT):
• The Kinetic Molecular Theory explains the macroscopic properties of gases, such as pressure and temperature, by considering their molecular composition and motion.
• It assumes that gas particles are in constant, random motion and that the temperature of the gas is a measure of the average kinetic energy of these particles.

Important Points

• Isochoric Process: A thermodynamic process that occurs at a constant volume is known as an isochoric or isometric process. In such a process, the work done by the gas is zero because there is no change in volume.
• Standard Units: In scientific contexts, pressure is often measured in Pascals (Pa), Atmospheres (atm), or Bar, while temperature is measured in Kelvin (K). To convert Celsius to Kelvin, you add 273.15 to the Celsius value.
• Boyle's Law vs. Charles's Law:
• Boyle's Law states that pressure is inversely proportional to volume at a constant temperature (P1V1 = P2V2).
• Charles's Law states that volume is directly proportional to temperature at a constant pressure (V1/T1 = V2/T2).

The correct answer is Plant.

Key Points

• Movement refers to the act of changing the position of one or more parts of the body relative to the whole organism, whereas locomotion is the movement of the entire organism from one place to another, resulting in a change of location.
• Plants are generally stationary or fixed in the soil by their roots; therefore, they do not exhibit locomotion. However, they show various types of movements such as the opening and closing of flowers, the bending of stems toward light, and the closing of leaves in response to touch. Hence, statement 3 is correct.
• A Dog is a multicellular animal that uses its limbs to move from one place to another for food, shelter, or protection. This represents both movement and locomotion. Hence, statement 1 is incorrect.
• A Bird utilizes wings for flying and feet for walking or perching, which allows it to change its geographical position entirely. This is a clear example of locomotion. Hence, statement 2 is incorrect.
• A Human uses skeletal muscles and bones to walk, run, jump, or swim. Since humans can move their whole body from point A to point B, they exhibit locomotion. Hence, statement 4 is incorrect.

Additional Information

• Types of Plant Movements:
• Tropic Movements (Tropisms): These are growth-dependent movements in response to a stimulus. Examples include Phototropism (response to light), Geotropism (response to gravity), and Hydrotropism (response to water).
• Nastic Movements: These are non-directional movements that do not depend on growth. A classic example is Thigmonasty, seen in the Mimosa pudica (touch-me-not plant), where leaves fold inward when touched.
• Purpose of Locomotion in Animals:
• Locomotion is essential for survival in the animal kingdom. Animals move to find nutrition, seek a mate, escape from predators, or find suitable climatic conditions.
• In humans and many vertebrates, locomotion is achieved through the musculoskeletal system, coordinated by the nervous system.
• Distinction between Movement and Locomotion:
• All locomotion is movement, but all movements are not locomotion. For instance, the beating of the heart or breathing are movements occurring within the body, but they do not cause the organism to travel.

Important Points

• Cellular Movement: Even within stationary organisms like plants, cytoplasmic streaming (cyclosis) occurs, which is the movement of the fluid substance inside a living cell.
• Evolutionary Context: Plants evolved to be autotrophs, meaning they produce their own food via photosynthesis using sunlight. Since their "food" comes to them in the form of light and minerals from the soil, they do not need locomotion.
• Anatomical Features: Animals possess specialized organs of locomotion like cilia, flagella, fins, or legs, which are absent in higher plants.

The correct answer is Potassium.

Key Points

• Potassium (K) is classified as a macronutrient because it is required by plants in relatively large amounts for their growth and development.
• Macronutrients are essential elements that are generally present in plant tissues in concentrations exceeding 10 mmole/kg of dry matter.
• Potassium is absorbed by plants in the form of potassium ions (K+). It is found in abundance within meristematic tissues, buds, leaves, and root tips. Hence, the correct option is Potassium.
• The primary functions of Potassium include maintaining the anion-cation balance in cells, protein synthesis, the opening and closing of stomata, and the activation of enzymes involved in photosynthesis and respiration.
• Along with Nitrogen (N) and Phosphorus (P), Potassium is considered a Primary Macronutrient often found in commercial fertilizers.

Additional Information

• Micronutrients: These are elements required by plants in very small quantities, specifically less than 10 mmole/kg of dry matter. These are also known as trace elements.
• Copper (Cu): It is a micronutrient absorbed as cupric ions (Cu2+). It is essential for the overall metabolism in plants and is associated with enzymes involved in redox reactions. Hence, statement 1 is incorrect.
• Zinc (Zn): It is a micronutrient absorbed as Zn2+ ions. It is necessary for the synthesis of auxin (a plant hormone) and activates various enzymes like carboxylases. Hence, statement 2 is incorrect.
• Iron (Fe): Although required in larger amounts than other micronutrients, it is still classified as a micronutrient. It is a key constituent of ferredoxin and cytochromes and is essential for the formation of chlorophyll. Hence, statement 4 is incorrect.

Important Points

• There are 17 essential elements required for plant growth, which are categorized based on their quantitative requirements.
• The 9 Macronutrients include: Carbon (C), Hydrogen (H), Oxygen (O), Nitrogen (N), Phosphorus (P), Potassium (K), Calcium (Ca), Magnesium (Mg), and Sulphur (S).
• The 8 Micronutrients include: Iron (Fe), Manganese (Mn), Copper (Cu), Molybdenum (Mo), Zinc (Zn), Boron (B), Chlorine (Cl), and Nickel (Ni).
• Carbon, Hydrogen, and Oxygen are mainly obtained from CO2 and H2O, while the others are absorbed from the soil as mineral nutrients.
• Deficiency of these nutrients can lead to various physiological disorders such as Chlorosis (yellowing of leaves), Necrosis (death of tissues), and stunted growth.

The correct answer is They protect against sexually transmitted infections..

Key Points

• Contraceptive pills, often referred to as oral contraceptive pills (OCPs), primarily function by using synthetic hormones such as estrogen and progesterone to suppress the release of eggs from the ovaries, a process known as ovulation. Hence, statement 2 is correct.
• These pills are considered a highly effective method of birth control; when taken consistently at the same time every day, they have a perfect-use efficacy rate of over 99%. Hence, statement 3 is correct.
• While generally safe, hormonal contraceptives can cause side effects in some users, including nausea, breast tenderness, headaches, and irregular bleeding or spotting between periods. Hence, statement 1 is correct.
• It is a common medical fact that oral contraceptive pills do not provide any protection against Sexually Transmitted Infections (STIs) or HIV/AIDS, as they do not act as a physical barrier to prevent the exchange of bodily fluids. Hence, statement 4 is incorrect.
• Because they do not prevent STIs, health organizations often recommend "dual protection," which involves using both the pill for pregnancy prevention and condoms to reduce infection risks. Hence, the incorrect statement is 4.

Additional Information

• Mechanism of Action:
• Beyond inhibiting ovulation, hormonal pills work by thickening the cervical mucus, which makes it difficult for sperm to enter the uterus and reach an egg.
• They also thin the lining of the uterus (the endometrium), making it less likely for a fertilized egg to implant itself.
• Types of Pills:
• Combined Oral Contraceptives (COCs): These contain both estrogen and progestin. They are the most common type and come in various dosage formats.
• Progestogen-only pills (POPs): Also known as the mini-pill, these do not contain estrogen and are often prescribed for women who are breastfeeding or have specific health risks like high blood pressure.
• Non-Contraceptive Benefits:
• Medical practitioners often prescribe these pills to manage conditions like Polycystic Ovary Syndrome (PCOS), endometriosis, and severe menstrual cramps.
• Long-term use of the pill has also been associated with a reduced risk of ovarian and endometrial cancers.

Important Points

• Effectiveness Data: According to the World Health Organization (WHO), the typical-use failure rate is approximately 7%, often due to missed doses or inconsistent timing.
• STI Protection: Only barrier methods such as male condoms and female condoms are clinically proven to significantly reduce the transmission of STIs like syphilis, gonorrhea, and chlamydia.
• Age and Health Factors: Women over the age of 35 who smoke are generally advised against using combined pills due to an increased risk of cardiovascular issues or blood clots.
• Emergency Contraception: It is important to distinguish regular daily pills from emergency contraceptive pills (the "morning-after" pill), which are intended for use only after unprotected intercourse and are not for routine use.

The correct answer is 100 J.

Key Points

• Work done (W) is defined as the product of the magnitude of the force applied and the displacement of the object in the direction of that force.
• The mathematical formula for work is W = F × d × cos(θ), where F represents the constant force, d is the displacement, and θ is the angle between the force and the direction of motion.
• In this specific case, the constant force (F) is given as 20 N and the displacement (d) is 5 m.
• Since the object moves in the same direction as the force, the angle (θ) is 0 degrees. The value of cos(0°) is 1.
• Substituting the values into the formula: W = 20 N × 5 m × 1 = 100 J. Thus, the work done is 100 Joules.

Additional Information

• Nature of Work: Work is a scalar quantity, which means it has only magnitude and no direction, even though it is derived from two vector quantities (force and displacement).
• SI Unit: The Standard International (SI) unit of work is the Joule (J). One Joule is defined as the work performed when a force of 1 Newton (N) displaces an object by 1 meter (m).
• Positive, Negative, and Zero Work:
• Positive Work occurs when the force and displacement are in the same direction (0° ≤ θ < 90°).
• Negative Work occurs when the force acts in the opposite direction to the displacement (90° < θ ≤ 180°), such as the force of friction acting against a moving car.
• Zero Work occurs when the force is perpendicular to the displacement (θ = 90°), or if either the force or displacement is zero.
• Relationship with Energy: Work is closely related to energy. When work is done on an object, energy is transferred to that object. For example, lifting an object involves doing work against gravity, which increases the object's potential energy.

Important Points

• Dimensional Formula: The dimensional formula for work is [ML²T⁻²]. This is identical to the dimensions of kinetic energy and potential energy, confirming the Work-Energy Theorem.
• Work-Energy Theorem: This theorem states that the net work done by all forces acting on a body is equal to the change in its kinetic energy.
• CGS Unit: In the Centimetre-Gram-Second (CGS) system, the unit of work is the erg. One Joule is equal to 10,000,000 (10⁷) ergs.
• Power and Work: Power is defined as the rate of doing work. It is calculated by dividing the work done by the time (t) taken to complete it (P = W / t). The SI unit for power is the Watt (W).

The correct answer is They have striations.

Key Points

• Smooth muscles are also known as non-striated or unstriped muscles because they do not exhibit the transverse bands or striations seen in skeletal or cardiac muscles. This lack of striations is due to the absence of highly organized sarcomeres. Hence, statement 4 is NOT true.
• These muscle cells are uninucleate, meaning each individual cell contains only one single, centrally located nucleus. Hence, statement 1 is correct.
• Smooth muscles control involuntary movements, meaning they function automatically without conscious effort. Their activity is regulated by the autonomic nervous system and hormones. Hence, statement 2 is correct.
• The structure of smooth muscle cells is spindle-shaped (also called fusiform), where the cells are thick in the middle and tapered at both ends. Hence, statement 3 is correct.
• Hence, the correct answer is They have striations.

Additional Information

• Location and Distribution:
• Smooth muscles are found in the walls of hollow visceral organs. Examples include the stomach, intestines, urinary bladder, uterus, and gallbladder.
• They are also present in the walls of blood vessels (arteries and veins), where they regulate blood pressure by contracting or relaxing to change the vessel diameter.
• They are found in the respiratory tract, iris of the eye (controlling pupil size), and the arrector pili muscles of the skin.
• Mechanism of Contraction:
• Unlike skeletal muscles, smooth muscles contract slowly but can maintain the contraction for a longer period without fatigue.
• They use actin and myosin filaments for contraction, but these are arranged in a loose network rather than the rigid sarcomere structure, allowing the muscle to shorten significantly more than skeletal muscle.
• Comparison with Other Muscle Types:
• Skeletal Muscle: These are striated, multinucleated, and voluntary muscles attached to the skeleton.
• Cardiac Muscle: These are striated, typically uninucleate, and involuntary muscles found only in the heart. They possess intercalated discs for rapid signal transmission.
• Key Biological Functions:
• In the digestive system, smooth muscles facilitate peristalsis, the wave-like contraction that moves food through the tract.
• In the urinary system, they assist in the expulsion of urine from the body.
• They play a vital role in homeostasis by controlling the flow of fluids and the internal environment of the body.

The correct answer is Both A and R are true, and R is the correct explanation of A.

Key Points

• Assertion (A) states that if a substance loses oxygen during a chemical reaction, it is said to be reduced. In classical chemistry, the removal of oxygen from a compound is one of the primary identifiers of a reduction reaction. Hence, statement A is correct.
• Reason (R) provides the definition of reduction, stating it is the gain of hydrogen or the loss of oxygen. This is the fundamental classical definition used to identify reduction in redox reactions. Hence, statement R is correct.
• In the context of the given statements, the Assertion describes a specific scenario (loss of oxygen), and the Reason provides the general rule or definition that encompasses that scenario.
• Because the definition in Reason (R) directly confirms why the process mentioned in Assertion (A) is classified as reduction, it serves as the correct explanation for the assertion.
• An example of this is the reaction between Copper Oxide (CuO) and Hydrogen (H2), where CuO loses oxygen to become metallic Copper (Cu); here, the copper oxide is reduced.

Additional Information

• Oxidation: This is the chemical process that acts as the opposite of reduction. It is defined as the gain of oxygen or the loss of hydrogen by a substance. In a reaction like 2Mg + O2 → 2MgO, Magnesium undergoes oxidation as it gains oxygen.
• Redox Reactions: The term Redox is a portmanteau of Reduction and Oxidation. In most chemical reactions involving these processes, they occur simultaneously. If one substance is reduced (loses oxygen), another must be oxidized (gains that oxygen).
• Reducing and Oxidizing Agents:
• A Reducing Agent is a substance that facilitates reduction by losing hydrogen or gaining oxygen (it gets oxidized itself).
• An Oxidizing Agent is a substance that facilitates oxidation by providing oxygen or accepting hydrogen (it gets reduced itself).
• Modern Electronic Concept: While the classical definition focuses on oxygen and hydrogen, the modern definition focuses on electrons. Oxidation is the loss of electrons (OIL: Oxidation Is Loss), while Reduction is the gain of electrons (RIG: Reduction Is Gain).
• Practical Examples: The corrosion of metals (like rusting of iron) and rancidity of fats/oils are common everyday examples of oxidation processes. In rusting, iron reacts with oxygen in the presence of moisture to form Hydrated Iron (III) Oxide.

Important Points

• Oxidation State: During reduction, the oxidation state (or oxidation number) of an element decreases (becomes more negative or less positive). During oxidation, the oxidation state increases.
• Electronegativity: In a broader sense, oxidation is the addition of an electronegative element, and reduction is the addition of an electropositive element.
• Historical Context: The term reduction originally comes from the metallurgical process of "reducing" a metal ore (like Iron Oxide) to its pure metallic form by removing oxygen, which results in a "reduction" in the mass of the ore.

The correct answer is The sky appears blue because air absorbs blue light..

Key Points

• Rayleigh Scattering states that the amount of scattering is inversely proportional to the fourth power of the wavelength of light. This means that shorter wavelengths (such as blue and violet) are scattered much more efficiently than longer wavelengths (such as red). Hence, statement 1 is correct.
• The blue appearance of the sky is a result of scattering, not absorption. When sunlight enters the Earth's atmosphere, it strikes gas molecules and scatters in all directions. Because blue light has a shorter wavelength, it is scattered more widely than other colors, making the sky look blue to our eyes. Hence, statement 2 is incorrect.
• The Tyndall effect is the process of light scattering by colloidal particles or particles in a very fine suspension. It is a reliable method used in laboratories to detect colloids and distinguish them from true solutions, where the particles are too small to scatter light. Hence, statement 3 is correct.
• Scattering is fundamentally dependent on the wavelength of the incident light. The interaction between light and atmospheric particles (like Nitrogen and Oxygen molecules) changes based on the light's color and frequency. Hence, statement 4 is correct.
• Hence, the correct answer is "The sky appears blue because air absorbs blue light" as it is the only incorrect statement.

Additional Information

• The Visible Spectrum: The human eye can perceive light within the visible spectrum, which ranges from approximately 400 nm (violet) to 700 nm (red). Blue light typically has a wavelength between 450 and 495 nm, while red light ranges from 620 to 750 nm.
• Mechanism of Tyndall Effect: In a colloid, the particle size ranges between 1 nanometer and 1000 nanometers. These particles are large enough to deflect light rays, making the path of a beam (like a laser or flashlight) visible when passed through the mixture.
• Atmospheric Phenomena: At sunrise and sunset, sunlight must travel through a much thicker layer of the atmosphere. During this long journey, most of the shorter wavelength blue light is scattered away, leaving only the longer wavelength red and orange light to reach our eyes.
• Lord Rayleigh's Contribution: The Rayleigh Scattering Law was formulated in the 19th century. It explains why the sky is blue and why the sun appears yellow/red. It specifically applies when the scattering particles are much smaller than the wavelength of the light, such as individual molecules.

Important Points

• Color of the Sun: Outside the atmosphere, the Sun appears white because all wavelengths reach the eye without being scattered. The yellowish hue we see from Earth is due to the removal of blue light from the direct beam by the atmosphere.
• Effect of Particle Size: If the Earth had no atmosphere, the sky would appear completely black, as seen by astronauts in space. Larger particles like dust and water droplets (in clouds) scatter all wavelengths of light almost equally, which is why clouds appear white (Mie scattering).
• Violet vs. Blue: Although violet light has an even shorter wavelength than blue light and scatters more, the sky appears blue because the Sun emits more blue light than violet, and the human eye is more sensitive to blue.

The correct answer is The sum of the atomic masses of the ions in an ionic compound as represented by its formula.

Key Points

• The formula unit mass of a substance is the sum of the atomic masses of all atoms in a formula unit of a compound.
• It is specifically used for substances whose constituent particles are ions. For example, in Sodium Chloride (NaCl), the particles are Na+ and Cl- ions rather than discrete molecules. Hence, statement 3 is correct.
• The calculation method is exactly the same as that for molecular mass; however, we use the term "formula unit" for those substances where the basic building blocks are ions.
• It represents the mass of one formula unit of an ionic compound relative to the mass of a Carbon-12 atom.
• The unit for formula unit mass is the unified atomic mass unit, denoted by 'u'.

Additional Information

• Atomic Mass:
• It is the mass of an individual atom of a chemical element. Hence, statement 1 is incorrect as it defines atomic mass.
• It is measured in atomic mass units (amu) or unified mass (u), where 1u is defined as 1/12th of the mass of a Carbon-12 atom.
• Molecular Mass:
• The molecular mass of a substance is the sum of atomic masses of all the atoms in a molecule of the substance. Hence, statement 2 is incorrect as it defines molecular mass.
• It is typically used for covalent compounds (like H2O or CH4) where atoms are bonded together to form distinct molecules.
• Molar Mass:
• The molar mass is the mass of one mole of a substance (element or compound). Hence, statement 4 is incorrect as it defines molar mass.
• It is numerically equal to the atomic or molecular mass but its units are grams per mole (g/mol).
• One mole contains exactly 6.02214076 x 10^23 elementary entities, known as Avogadro's number.

Important Points

• Calculation Example (NaCl): To find the formula unit mass of Sodium Chloride (NaCl), we add the atomic mass of Sodium (23 u) and Chlorine (35.5 u), resulting in 58.5 u.
• Calculation Example (CaCl2): For Calcium Chloride, the formula unit mass is 1 x 40 u (Ca) + 2 x 35.5 u (Cl) = 40 + 71 = 111 u.
• Ionic vs. Covalent: In ionic compounds, there is no single "molecule" because the ions are arranged in a three-dimensional crystal lattice. Therefore, the simplest ratio of ions (the formula unit) is used for mass calculations.
• History of Unit: The Unified Atomic Mass Unit (u) was adopted by the IUPAC in 1961 to replace the older oxygen-based scales.

The correct answer is Zinc.

Key Points

• Zinc (Zn) is a moderately reactive metal that reacts readily with dilute Hydrochloric acid (HCl) to produce Zinc Chloride (ZnCl₂) and evolve Hydrogen gas (H₂).
• This reaction is a classic example of a single displacement reaction, where a more reactive metal displaces hydrogen from an acid. The chemical equation for this process is: Zn + 2HCl → ZnCl₂ + H₂. Hence, the option Zinc is correct.
• In the reactivity series of metals, Zinc is positioned above Hydrogen. This higher position indicates that it has a greater tendency to lose electrons and undergo oxidation compared to hydrogen.
• During the reaction, Zinc atoms lose two electrons to form Zn²⁺ ions (oxidation), while Hydrogen ions (H⁺) from the acid gain these electrons to form Hydrogen gas (reduction).
• In contrast, metals like Copper, Silver, and Gold are placed below hydrogen in the reactivity series. Because they are less reactive than hydrogen, they cannot displace it from dilute acids. Hence, these options are incorrect.

Additional Information

• The Reactivity Series:
• The reactivity series is a list of metals arranged in descending order of their chemical reactivity. Metals at the top, such as Potassium (K) and Sodium (Na), are highly reactive and can react even with cold water.
• Metals located above Hydrogen in the series (like Magnesium, Aluminum, Zinc, and Iron) can displace hydrogen from dilute acids like HCl or H₂SO₄.
• Properties of Hydrogen Gas:
• The Hydrogen gas produced during the reaction is colorless and odorless. It can be identified using the "Pop" test: when a burning splinter is brought near the mouth of the test tube containing the gas, it burns with a characteristic pop sound.
• Characteristics of Noble Metals:
• Gold (Au) and Silver (Ag) are known as noble metals because they are chemically inert and do not react with ordinary dilute acids. Gold only dissolves in a special mixture called Aqua Regia (a 3:1 mixture of concentrated HCl and HNO₃).
• Copper (Cu) is also relatively unreactive and does not produce hydrogen with dilute HCl, although it may react with oxidizing acids like concentrated Nitric acid.

Important Points

• Zinc Metal: It has the atomic number 30 and is commonly used in galvanization to prevent the rusting of iron. In the laboratory, granulated zinc is preferred for producing hydrogen gas because its larger surface area increases the rate of reaction.
• Acid Strength: While Zinc reacts with dilute HCl and H₂SO₄ to give hydrogen, it does not typically produce hydrogen gas with Nitric acid (HNO₃). This is because HNO₃ is a strong oxidizing agent that oxidizes the produced hydrogen into water (H₂O) and itself gets reduced to nitrogen oxides.
• Displacement Logic: The ability of a metal to react with an acid depends on its electrode potential. Metals with a negative reduction potential (relative to the standard hydrogen electrode) are capable of liberating hydrogen gas from acids.

The correct answer is Sclerenchyma.

Key Points

• Sclerenchyma is a simple permanent plant tissue characterized by cells that are dead at maturity. These cells lose their protoplasm during development, leaving behind only the cell wall. Hence, option 4 is correct.
• The cell walls of sclerenchyma are significantly thickened due to the deposition of lignin, a complex polymer that acts as a waterproof cementing material.
• Due to the extreme thickness of the lignified walls, there are no intercellular spaces between the cells, making the tissue very dense and rigid.
• The primary function of sclerenchyma is to provide mechanical strength and structural rigidity to the plant, allowing it to withstand environmental stresses like wind and weight.
• This tissue is commonly found in the stems (around vascular bundles), in the veins of leaves, and in the hard coverings of seeds and nuts, such as the shell of a coconut.

Additional Information

• Parenchyma:
• This is the most fundamental and abundant simple tissue in plants. The cells are living with thin cell walls made of cellulose.
• They possess large central vacuoles and are involved in photosynthesis, storage of food, and secretion. Hence, option 2 is incorrect.
• Collenchyma:
• This tissue consists of living cells that are elongated and have unevenly thickened cell walls, particularly at the corners.
• The thickening is due to the deposition of cellulose, hemicellulose, and pectin. It provides flexibility and mechanical support to young growing parts like the petiole of a leaf. Hence, option 3 is incorrect.
• Aerenchyma:
• It is a specialized type of parenchyma found in hydrophytes (aquatic plants).
• It contains large air cavities or spaces that facilitate the exchange of gases and provide buoyancy to the plant, helping it float. Since it is a parenchyma variant, its cells are living. Hence, option 1 is incorrect.

Important Points

• Types of Sclerenchyma:
• Fibers: These are long, narrow, and thick-walled cells with tapering ends. They usually occur in bundles and are used commercially in making ropes and textiles (e.g., jute and hemp).
• Sclereids: Also known as stone cells, these are spherical or oval cells with extremely thick walls and very narrow lumens. They are responsible for the gritty texture found in fruits like pears and sapota.
• Chemical Composition:
• The presence of lignin is a key distinguishing feature of sclerenchyma compared to collenchyma (which uses pectin) and parenchyma (which uses thin cellulose). Lignification is the process that eventually leads to the death of the protoplast.
• Comparison:
• While Xylem (a complex tissue) also contains dead cells like tracheids and vessels for water transport, sclerenchyma is the primary simple permanent tissue dedicated solely to mechanical protection.

Shortcut Trick

Total List Price = 625 × 94 = 58,750.

Net Multiplier = (80 ÷ 100) × (93 ÷ 100) × (80 ÷ 100) = 0.8 × 0.93 × 0.8 = 0.5952.

Cash Payment = 58,750 × 0.5952 = 34,968.

∴ The correct answer is ₹34,968.

Alternate Method

Given: Quantity = 625, Rate = ₹94, Discounts = 20%, 7%, 20%. Formulas: Total Value = Quantity × Rate | SP = MP × (100 − D₁)% × (100 − D₂)% × (100 − D₃)%

⇒ Total List Price = 625 × 94 = ₹58,750. ⇒ After 20% discount = 58,750 × 0.80 = 47,000. ⇒ After 7% discount = 47,000 × 0.93 = 43,710. ⇒ Final Cash Payment (after 20% further discount) = 43,710 × 0.80 = ₹34,968.

∴ The correct answer is ₹34,968.

Additional Information

Successive Discounts

When multiple discounts are applied one after another, the net selling price is calculated on the reducing balance at each step.

Cash Discount

A cash discount is an incentive offered by a seller to a buyer in return for paying a bill before the scheduled due date.

Shortcut Trick

Age of new person (Teacher) = New Average + (Original Strength × Increase in Average)

Teacher age = 31 + (31 × (31 − 30))

⇒ 31 + (31 × 1) = 31 + 31 = 62 years.

Alternatively, Teacher age = Old Average + (New Strength × Increase in Average)

⇒ 30 + (32 × 1) = 30 + 32 = 62 years.

∴ The correct answer is 62 years.

Alternate Method

Given:

Number of students = 31

Initial average age = 30 years

New average after including teacher = 31 years

Total members (Students + Teacher) = 31 + 1 = 32

Formula Used:

Average = Total Sum of Ages ÷ Number of Persons

Total Sum = Average × Number of Persons

Calculations:

⇒ Total age of 31 students = 31 × 30 = 930 years

⇒ Total age of group including teacher = 32 × 31

⇒ Total age (32 members) = 992 years

⇒ Age of the teacher = Total age with teacher − Total age of students

⇒ Age of the teacher = 992 − 930

⇒ Age of the teacher = 62 years

∴ The correct answer is 62 years.

Additional Information

Basic Average Formula

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

Exclusion Case Shortcut

Age of excluded person = Old Average + (Old Strength − 1) × (Decrease in Average).

Replacement Case Shortcut

Value of new person = Value of replaced person + (Total number × Increase in Average).

Weighted Average

Combined Average = (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂), where n is quantity and A is average.

Shortcut Trick

Net multiplier = (1 − 0.45) × (1 − 0.35) × (1 − 0.30) × (1 + 0.45)

Net multiplier = 0.55 × 0.65 × 0.70 × 1.45

Net multiplier = 0.3575 × 1.015 = 0.3628625

Net Change % = (0.3628625 − 1) × 100 = −63.71375%

∴ The correct answer is Decreases by 63.71%.

Alternate Method

Given: Successive changes of −45%, −35%, −30% and +45%.

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

⇒ Let the Original Price of fuel be 100.

⇒ Price after 1^st month = 100 × (100 − 45)/100 = 55

⇒ Price after 2^nd month = 55 × (100 − 35)/100 = 55 × 0.65 = 35.75

⇒ Price after 3^rd month = 35.75 × (100 − 30)/100 = 35.75 × 0.7 = 25.025

⇒ Price after 4^th month = 25.025 × (100 + 45)/100 = 25.025 × 1.45 = 36.28625

⇒ Total change = 36.28625 − 100 = −63.71375

⇒ Percentage Decrease ≈ 63.71%

∴ The correct answer is Decreases by 63.71%.

Additional Information

Successive Percentage Change

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

Multiplier Method

Final value = Initial value × (1 + r/100)ⁿ. For different rates, multiply each factor (1 ± r/100) consecutively.

Percentage Decrease

It is calculated as [(Initial Value − Final Value) / Initial Value] × 100.

Shortcut Trick

Apply the algebraic identity: (a + b)² + (a − b)² = 2(a² + b²).

Here, a = 4 and b = √3.

Sum of areas = 2 × [4² + (√3)²]

⇒ 2 × (16 + 3) = 2 × 19 = 38 cm².

∴ The correct answer is 38 cm².

Alternate Method

Given:

Side of the first square (s₁) = (4 + √3) cm

Side of the second square (s₂) = (4 − √3) cm

Formula Used:

Area of a square = (Side)²

Calculations:

⇒ Area of first square = (4 + √3)²

⇒ Area₁ = 4² + (√3)² + 2 × 4 × √3

⇒ Area₁ = 16 + 3 + 8√3 = (19 + 8√3) cm²

⇒ Area of second square = (4 − √3)²

⇒ Area₂ = 4² + (√3)² − 2 × 4 × √3

⇒ Area₂ = 16 + 3 − 8√3 = (19 − 8√3) cm²

⇒ Sum of areas = Area₁ + Area₂

⇒ Sum = (19 + 8√3) + (19 − 8√3)

⇒ Sum = 19 + 19 + 8√3 − 8√3 = 38 cm²

∴ The correct answer is 38 cm².

Additional Information

Algebraic Identities

(a + b)² = a² + b² + 2ab and (a − b)² = a² + b² − 2ab. These are fundamental for solving area-based problems involving irrational numbers.

Perimeter of a Square

The perimeter of a square is 4 × side. For the first square, perimeter = 4(4 + √3) = 16 + 4√3 cm.

Diagonal of a Square

The diagonal of a square with side 's' is given by s√2. Area can also be calculated as (Diagonal)² ÷ 2.

Shortcut Trick

In any quadrilateral PQRS, the angle formed by the bisectors of two adjacent angles (say, ∠R and ∠S) meeting at point T is always equal to half the sum of the other two angles (∠P and ∠Q):

∠STR = (∠P + ∠Q) ÷ 2

⇒ 22.5° = (∠P + ∠Q) ÷ 2

⇒ ∠P + ∠Q = 45°

Since the ratio of ∠P to ∠Q is 4 : 5, the difference between them is:

Difference = (5 − 4) ÷ (4 + 5) × 45° = 1/9 × 45° = 5°

∴ The correct answer is 5°.

Alternate Method

Given:

• In quadrilateral PQRS, the bisectors of ∠R and ∠S meet at T.

• ∠STR = 22.5°

• Ratio of ∠P to ∠Q = 4 : 5

Formula Used:

• Sum of angles in a triangle = 180°

• Sum of angles in a quadrilateral = 360°

Calculations:

In triangle STR:

⇒ ∠TSR + ∠TRS + ∠STR = 180°

⇒ ∠TSR + ∠TRS + 22.5° = 180°

⇒ ∠TSR + ∠TRS = 157.5°

Since ST and RT are the angle bisectors of ∠S and ∠R respectively:

⇒ ∠TSR = ∠S ÷ 2

⇒ ∠TRS = ∠R ÷ 2

⇒ (∠S ÷ 2) + (∠R ÷ 2) = 157.5°

⇒ ∠S + ∠R = 2 × 157.5° = 315°

In quadrilateral PQRS, the sum of all interior angles is 360°:

⇒ ∠P + ∠Q + ∠R + ∠S = 360°

⇒ ∠P + ∠Q + 315° = 360°

⇒ ∠P + ∠Q = 45°

Given the ratio of ∠P to ∠Q is 4 : 5:

⇒ Let ∠P = 4x and ∠Q = 5x

⇒ 4x + 5x = 45°

⇒ 9x = 45°

⇒ x = 5°

The difference between the measures of ∠Q and ∠P is:

⇒ ∠Q − ∠P = 5x − 4x = x

⇒ Difference = 5°

Explanation of Options:

• Option (1) is incorrect because a difference of 1° would mean the sum of angles is not 45° under the 4 : 5 ratio.

• Option (2) is incorrect because a difference of 2° does not satisfy the ratio and sum conditions.

• Option (3) is correct as the calculated difference is exactly 5°.

• Option (4) is incorrect because a difference of 6° is mathematically inconsistent with the given angle bisector value.

∴ The correct answer is 5°.

Additional Information

Angle Bisector Theorem in Quadrilaterals

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

Sum of Interior Angles of a Polygon

The sum of the interior angles of an n-sided polygon is (n − 2) × 180°. For any quadrilateral, this sum is always 360°.

Opposite Angle Bisectors

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

Shortcut Trick

For a rate of 20% and time of 2 years, the effective compound interest rate is calculated as: 20 + 20 + (20 × 20)/100 = 44%.

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

Amount = 4,200 × 144/100 = 42 × 144 = ₹6,048.

∴ The correct answer is ₹6,048.

Alternate Method

Given:

Principal (P) = ₹4,200

Rate (R) = 20% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 4200 × (1 + 20/100)²

⇒ A = 4200 × (1 + 1/5)²

⇒ A = 4200 × (6/5)²

⇒ A = 4200 × (36/25)

⇒ A = 168 × 36

⇒ A = 6048

∴ The correct answer is ₹6,048.

Additional Information

Effective Rate for Compound Interest

For 2 years, the net effective interest rate is given by the formula: x + y + (xy/100), where x and y are the rates for the first and second years respectively.

Difference between CI and SI

The difference (D) between compound interest and simple interest for 2 years is D = P(R/100)².

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R/2) and the time is doubled (2n).

Shortcut Trick

Total distance covered = Speed × Time

Total distance = 63 × 28 = 1764 m

Train length (X) = Total distance − Platform length

X = 1764 − 213 = 1551 m

∴ The correct answer is 1551 m.

Alternate Method

Given:

Speed of train = 63 m/s

Time taken = 28 sec

Platform length = 213 m

Train length = X m

Formula Used:

Speed = (Length of Train + Length of Platform) ÷ Time

Calculations:

⇒ 63 = (X + 213) ÷ 28

⇒ X + 213 = 63 × 28

⇒ X + 213 = 1764

⇒ X = 1764 − 213

⇒ X = 1551

∴ The correct answer is 1551 m.

Additional Information

Crossing Objects with Length

When a train crosses a platform, bridge, or another train, the total distance covered is the sum of the lengths of the train and the object.

Relative Speed

If two objects move in the same direction, relative speed is S₁ − S₂. In opposite directions, it is S₁ + S₂.

Speed Conversion

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

Shortcut Trick Given that Cost Price (CP) is 80% of Selling Price (SP). Ratio of CP : SP = 80 : 100 = 4 : 5. Profit = SP − CP = 5 − 4 = 1 unit. Profit% = (Profit ÷ CP) × 100 = (1 ÷ 4) × 100 = 25%. ∴ The correct answer is 25%.

Alternate Method Given: Selling Price (SP) = ₹100, Cost Price (CP) = 80% of SP. Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ Cost Price (CP) = 80% of 100 = (80 ÷ 100) × 100 = ₹80. ⇒ Selling Price (SP) = ₹100. ⇒ Profit = SP − CP = 100 − 80 = ₹20. ⇒ Profit Percentage = (20 ÷ 80) × 100 = 1/4 × 100 = 25%. ∴ The correct answer is 25%.

Additional Information Profit and Loss on CP Unless otherwise stated, Profit% and Loss% are always calculated on the Cost Price. Relation between CP and SP If SP > CP, it is a Profit. If CP > SP, it is a Loss.

Shortcut Trick

Number of oranges = (Total − Difference) ÷ 2

⇒ Oranges = (42 − 6) ÷ 2

⇒ Oranges = 36 ÷ 2 = 18

∴ The correct answer is 18.

Alternate Method

Given: Total fruits = 42, Apple (A) − Orange (O) = 6

Let the number of oranges be x.

Then, the number of apples = x + 6.

⇒ (x + 6) + x = 42

⇒ 2x + 6 = 42

⇒ 2x = 42 − 6

⇒ 2x = 36

⇒ x = 18

∴ The correct answer is 18.

Additional Information

Sum and Difference Rule

If the sum (S) and difference (D) of two numbers are known, the larger number is (S + D) ÷ 2 and the smaller number is (S − D) ÷ 2.

Linear Equations in One Variable

Equations where the highest power of the variable is 1, used to find unknown quantities by establishing relationships based on given conditions.

Shortcut Trick

Combined efficiency of A, B, and C = 4 + 1 + 1 = 6 units/day (taking Total Work as 12 units).

Twice the work = 12 × 2 = 24 units.

Time taken = 24 ÷ 6 = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given:

Time taken by A = 3 days

Time taken by B = 12 days

Time taken by C = 12 days

Formula Used:

Total Work = LCM of individual days

Efficiency = Total Work ÷ Time Taken

Time Taken = Total Work ÷ Combined Efficiency

Calculations:

⇒ Total Work = LCM of 3, 12, 12 = 12 units

⇒ Combined Efficiency of A, B, and C = 4 + 1 + 1 = 6 units/day

⇒ Twice the Work = 2 × 12 = 24 units

⇒ Time taken to complete twice the work = 24 ÷ 6 = 4 days

∴ The correct answer is 4 days.

Additional Information

Total Work Concept

Total work is assumed as the LCM of individual times to make efficiency calculations integer-based and simpler.

Work and Efficiency Relation

Efficiency is inversely proportional to time taken when the total work is constant.

Shortcut Trick

Efficiency ratio A : B = 100 : 125 = 4 : 5.

Total Work = Efficiency of A × Time of A = 4 × 10 = 40 units.

Work to be completed = 60% of 40 = 24 units.

Combined efficiency = 4 + 5 = 9 units/day.

Time required = 24 ÷ 9 = 8/3 = 2(2/3) days.

∴ The correct answer is 2(2/3) days.

Alternate Method

Given: Time taken by A = 10 days; B is 25% more efficient than A.

Formula Used: Efficiency × Time = Work; Efficiency ∝ 1/Time

⇒ Efficiency of B = 4 + 25% of 4 = 4 + 1 = 5 units/day.

⇒ Total work = 10 × 4 = 40 units.

⇒ 60% of work = 40 × 0.6 = 24 units.

⇒ Combined Efficiency of (A + B) = 4 + 5 = 9 units/day.

⇒ Time taken = 24 ÷ 9 = 8/3 days = 2(2/3) days.

∴ The correct answer is 2(2/3) days.

Additional Information

Efficiency and Time Relationship

Efficiency is inversely proportional to the time taken to complete a task. If efficiency ratio is x:y, time ratio is y:x.

Work and Wages

Wages are distributed in the ratio of the actual work done by each person, which simplifies to the ratio of their efficiencies if they work for the same duration.

Combined Work Formula

If A takes 'a' days and B takes 'b' days, time taken together is (a × b) ÷ (a + b) days.

Shortcut Trick

Total number of observations (n) = 11 (Odd).

Median is the 6^th observation in ascending order.

Replacing a value lower than the median (41 → 43, both < 58) and a value higher than the median (92 → 99, both > 58) does not change the middle element.

Since the relative position of the 6^th term remains the same, the median is unchanged.

∴ The correct answer is 58.

Alternate Method

Given observations: 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33

Step 1: Arrange data in ascending order.

33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92

Number of observations (n) = 11

Median = ((n + 1) ÷ 2)^th term = 6^th term = 58

Step 2: Apply the changes (Replace 92 with 99 and 41 with 43).

Step 3: New ascending order:

33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99

⇒ New n = 11

⇒ New Median = 6^th term

⇒ New Median = 58

∴ The correct answer is 58.

Additional Information

Median for Even Observations

If n is even, Median = Average of (n/2)^th and (n/2 + 1)^th observations.

Empirical Relationship

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

Outlier Resistance

Median is a measure of central tendency that is not affected by extreme outliers, unlike the arithmetic mean.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 40 + 50 + 60 = 150 km

Total Time = (40/60) + (50/50) + (60/40) = 2/3 + 1 + 3/2 = 19/6 hours

Average Speed = 150 ÷ (19/6) = 900/19 ≈ 47.4 km/hr

∴ The correct answer is 47.4 km/hr.

Alternate Method

Given:

Distance 1 (d₁) = 40 km, Speed 1 (s₁) = 60 km/hr

Distance 2 (d₂) = 50 km, Speed 2 (s₂) = 50 km/hr

Distance 3 (d₃) = 60 km, Speed 3 (s₃) = 40 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = (Total Distance) ÷ (Total Time)

Calculations:

⇒ Time t₁ = 40 ÷ 60 = 2/3 hours

⇒ Time t₂ = 50 ÷ 50 = 1 hour

⇒ Time t₃ = 60 ÷ 40 = 3/2 hours

⇒ Total Time = 2/3 + 1 + 3/2 = (4 + 6 + 9) ÷ 6 = 19/6 hours

⇒ Total Distance = 40 + 50 + 60 = 150 km

⇒ Average Speed = 150 ÷ (19/6) = (150 × 6) ÷ 19

⇒ Average Speed = 900 ÷ 19 ≈ 47.368...

∴ The correct answer is 47.4 km/hr.

Additional Information

Equal Distance Case

If two equal distances are covered at speeds u and v, then Average Speed = (2 × u × v) ÷ (u + v).

Average Speed Concept

Average speed is NOT the simple arithmetic mean of speeds unless the time spent at each speed is identical.

Relative Speed

Same direction: S₁ − S₂ | Opposite direction: S₁ + S₂.

Shortcut Trick Sum of odd-place digits: 7 + x + 9 + 5 = 21 + x Sum of even-place digits: 4 + 8 + 3 + y = 15 + y Difference (D) = (21 + x) − (15 + y) = 6 + x − y For divisibility by 11, D must be 0, 11, or −11. If D = 0, y − x = 6 ⇒ (x, y) = (1, 7), (2, 8), (3, 9) [3 pairs] If D = 11, x − y = 5 ⇒ (x, y) = (5, 0) is not allowed as y is positive, so (6, 1), (7, 2), (8, 3), (9, 4) [4 pairs] ∴ The correct answer is 7. Alternate Method Given: Eight digit number 74x8935y is divisible by 11. x, y > 0. Formula Used: Divisibility rule of 11: Sum of digits at odd places − Sum of digits at even places = 0 or multiple of 11.

⇒ Odd place sum S1 = 7 + x + 9 + 5 = 21 + x ⇒ Even place sum S2 = 4 + 8 + 3 + y = 15 + y ⇒ Difference D = S1 − S2 = (21 + x) − (15 + y) = 6 + x − y ⇒ Case 1: 6 + x − y = 0 ⇒ y − x = 6. Since x, y ≥ 1: (1, 7), (2, 8), (3, 9) ⇒ Case 2: 6 + x − y = 11 ⇒ x − y = 5. Since x, y ≥ 1: (6, 1), (7, 2), (8, 3), (9, 4) ⇒ Case 3: 6 + x − y = −11 ⇒ y − x = 17 (Not possible as max difference is 8 for non-zero integers) ⇒ Total pairs = 3 + 4 = 7 ∴ The correct answer is 7. Additional Information Divisibility by 11 A number is divisible by 11 if the alternating sum of its digits is divisible by 11. Positive Integers constraint Since x and y are positive integers, they must be from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. Zero is not included. Ordered Pairs An ordered pair (x, y) counts unique combinations where the order of x and y matters relative to their positions in the number.

Shortcut Trick

Discount% = [Free Articles ÷ (Paid Articles + Free Articles)] × 100

Discount% = [2 ÷ (6 + 2)] × 100

Discount% = (2 ÷ 8) × 100 = (1 ÷ 4) × 100 = 25%

∴ The correct answer is 25%.

Alternate Method

Given: Number of articles purchased = 6, Number of free articles = 2. Formulas: Discount% = (Discount ÷ MP) × 100 | Total MP = MP of all articles (Bought + Free).

⇒ Let the Marked Price (MP) of 1 article be 1 unit. ⇒ Total MP of 8 articles (6 + 2) = 8 units. ⇒ The Selling Price (SP) is equal to the price of 6 articles = 6 units. ⇒ Discount = MP − SP = 8 − 6 = 2 units. ⇒ Discount% = (2 ÷ 8) × 100 = 25%.

∴ The correct answer is 25%.

Additional Information

Marked Price & Discount

The price printed on the label is the Marked Price. Discount is always calculated on this value.

Free Item Concept

In 'Buy X Get Y Free', the total quantity (X+Y) represents the Marked Price equivalent, and Y represents the Discount.

Shortcut Trick

Numbers divisible by 5 = 1000 ÷ 5 = 200

Numbers divisible by 7 = 1000 ÷ 7 = 142 (integer part)

Numbers divisible by both (LCM of 5 and 7 = 35) = 1000 ÷ 35 = 28 (integer part)

Total numbers = 200 + 142 − 28 = 314

∴ The correct answer is 314.

Alternate Method

Given: Total limit = 1000, Divisors = 5, 7

Formula Used: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

⇒ Let A be the set of numbers divisible by 5: n(A) = floor(1000/5) = 200

⇒ Let B be the set of numbers divisible by 7: n(B) = floor(1000/7) = 142

⇒ Let A ∩ B be numbers divisible by LCM(5, 7) = 35: n(A ∩ B) = floor(1000/35) = 28

⇒ Total numbers divisible by 5 or 7 = 200 + 142 − 28

⇒ Total = 342 − 28 = 314

∴ The correct answer is 314.

Additional Information

Inclusion-Exclusion Principle

For two sets, the number of elements in their union is found by adding the individual sizes and subtracting the size of their intersection.

Divisibility in a Range

The number of integers divisible by 'k' in the range [1, N] is given by the floor function ⌊N/k⌋.

Least Common Multiple (LCM)

To find numbers divisible by both 'a' and 'b', we must find numbers divisible by their LCM.

Shortcut Trick

Since Surface Area is a 2-dimensional property (depends on side²), use the Successive Percentage formula:

Net Increase = x + x + (x × x)/100

Here, increase (x) = 10%

Total Increase = 10 + 10 + (10 × 10)/100 = 20 + 1 = 21%

∴ The correct answer is 21%.

Alternate Method

Given: Each edge of a cube is increased by 10%.

Formula Used: Total Surface Area of a cube = 6 × (edge)²

⇒ Let the original edge (a) = 10 units.

⇒ Original Surface Area (S₁) = 6 × (10)² = 6 × 100 = 600 sq. units.

⇒ New edge (a’) = 10 + (10% of 10) = 11 units.

⇒ New Surface Area (S₂) = 6 × (11)² = 6 × 121 = 726 sq. units.

⇒ Increase in Surface Area = 726 − 600 = 126 sq. units.

⇒ Percentage Increase = (126 ÷ 600) × 100 = 21%.

∴ The correct answer is 21%.

Additional Information

Volume Percentage Change

Since Volume = side³, an increase of x% leads to a volume increase of [ (1 + x/100)³ − 1 ] × 100%. For 10%, it is 33.1%.

Ratio Method for Area

If side ratio is 10 : 11, then Area ratio is 10² : 11² = 100 : 121. The increase is directly 21 over 100, which is 21%.

Applicability to Other Shapes

This rule applies to all 3D regular figures. If the linear dimension of a sphere or cone increases by x%, its surface area always increases by 2x + x²/100 %.

Shortcut Trick

Sum of present ages = 56 years.

Sum after 8 years = 56 + (8 × 2) = 72 years.

Ratio after 8 years (Mother : Daughter) = 3 : 1.

Daughter's age after 8 years = 72 × (1/4) = 18 years.

Daughter's present age = 18 − 8 = 10 years.

∴ The correct answer is 10 years.

Alternate Method

Given: Sum of present ages of Mother (M) and Daughter (D) is 56.

Formula Used: Linear equation in one or two variables.

⇒ Let Daughter's present age be D and Mother's present age be M.

⇒ M + D = 56 → (i)

⇒ After 8 years: (M + 8) = 3 × (D + 8)

⇒ M + 8 = 3D + 24

⇒ M − 3D = 16 → (ii)

⇒ Subtracting (ii) from (i): (M + D) − (M − 3D) = 56 − 16

⇒ 4D = 40

⇒ D = 10 years.

∴ The correct answer is 10 years.

Additional Information

Sum of Ages in Future/Past

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

Age Ratio Comparison

The difference between the ages of two people remains constant throughout their lives, regardless of the years passed.

Linear Equations in Ages

Always assume variables for present ages to avoid confusion while moving back or forward in time.

Shortcut Trick

In a 30°-60°-90° triangle, the ratio of sides is 1 : √3 : 2.

The side opposite to 30° (Base) = 30 m.

The side opposite to 60° (Height) = Base × √3.

Height = 30 × 1.73 = 51.9 m.

∴ The correct answer is 51.9 m.

Alternate Method

Given:

Distance from the base (Base) = 30 m

Angle of elevation (θ) = 60°

Value of √3 = 1.73

Formula Used:

tan θ = Perpendicular / Base

Calculations:

⇒ tan 60° = Height / 30

⇒ √3 = Height / 30

⇒ Height = 30 × √3

⇒ Height = 30 × 1.73

⇒ Height = 51.9 m

∴ The correct answer is 51.9 m.

Additional Information

Trigonometric Ratios

Standard values for calculations: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

Angle of Elevation vs Depression

The angle of elevation is measured upwards from the horizontal, while the angle of depression is measured downwards; both are numerically equal for the same line of sight due to alternate interior angles.

Pythagoras Theorem

In any right-angled triangle, Hypotenuse² = Base² + Perpendicular².

Shortcut Trick

Required percentage = (Value ÷ Base Value) × 100

Required percentage = (1/30 ÷ 1/90) × 100

Required percentage = (90 ÷ 30) × 100 = 3 × 100 = 300%

∴ The correct answer is 300%.

Alternate Method

Given:

Target value = 1/30

Base value = 1/90

Formula Used:

Percentage = (A / B) × 100

Calculations:

⇒ Percentage = [(1/30) / (1/90)] × 100

⇒ Percentage = (1/30) × (90/1) × 100

⇒ Percentage = 3 × 100

⇒ Percentage = 300%

∴ The correct answer is 300%.

Additional Information

Fraction to Percentage

To convert any fraction x/y into a percentage, multiply it by 100 and add the % symbol.

Reciprocal Comparison

When comparing unit fractions (1/n), the ratio of 1/a to 1/b is equal to b/a.

Percentage Change

Percentage change = [(New value − Old value) / Old value] × 100.

Shortcut Trick

Apply the BODMAS rule to evaluate multiplication and division first.

First part: 0.7 × 72 ÷ 14 = 50.4 ÷ 14 = 3.6

Second part: 8.34 × 0.15 = 1.251

Substitute the parts back into the equation: 3.6 + 6.25 − 1.251 + 5.95

Add the positive terms together and subtract: 15.80 − 1.251 = 14.549

∴ The correct answer is 14.549.

Alternate Method

Given:

Expression: 0.7 × 72 ÷ 14 + 6.25 − 8.34 × 0.15 + 5.95

Concept Used:

BODMAS Rule: Brackets, Order, Division, Multiplication, Addition, and Subtraction.

Calculations:

⇒ 0.7 × 72 ÷ 14 + 6.25 − 8.34 × 0.15 + 5.95

⇒ (0.7 × 72) ÷ 14 + 6.25 − (8.34 × 0.15) + 5.95

⇒ 50.4 ÷ 14 + 6.25 − 1.251 + 5.95

⇒ 3.6 + 6.25 − 1.251 + 5.95

Group all the positive values together first to make the addition easier:

⇒ (3.6 + 6.25 + 5.95) − 1.251

⇒ 15.80 − 1.251

⇒ 14.549

∴ The correct answer is 14.549.

Additional Information

BODMAS Rule

It specifies the sequence of mathematical operations: Brackets first, then Orders (powers/roots), followed by Division and Multiplication (left to right), and finally Addition and Subtraction (left to right).

Decimal Addition and Subtraction

When adding or subtracting decimal numbers, always align the decimal points precisely before proceeding with the calculation to prevent place-value errors.

Decimal Multiplication

Multiply the numbers as whole integers, then place the decimal point in the resulting product so that it has the exact total number of decimal places present in both of the original factors.

Shortcut Trick

Income ratio (David : Ashish) = 5 : 8

Expenditure ratio = 2 : 4 = 1 : 2

Since savings are equal, make the difference between income and expenditure parts equal.

Multiply expenditure ratio (1:2) by 3 → 3 : 6. Now, Savings = Income − Expenditure = 5−3 = 2 units and 8−6 = 2 units.

2 units = ₹81,000 ⇒ 1 unit = ₹40,500.

David income = 5 units = 5 × 40,500 = ₹2,02,500.

∴ The correct answer is ₹2,02,500.

Alternate Method

Given: Ratio of Income = 5 : 8, Ratio of Expenditure = 2 : 4, Savings of each = ₹81,000.

Formula: Income − Savings = Expenditure

⇒ Let David income = 5x and Ashish income = 8x.

⇒ According to question: (5x − 81,000) / (8x − 81,000) = 2 / 4

⇒ (5x − 81,000) / (8x − 81,000) = 1 / 2

⇒ 2(5x − 81,000) = 1(8x − 81,000)

⇒ 10x − 1,62,000 = 8x − 81,000

⇒ 2x = 1,62,000 − 81,000

⇒ 2x = 81,000

⇒ x = 40,500

⇒ David income = 5x = 5 × 40,500 = 2,02,500

∴ The correct answer is ₹2,02,500.

Additional Information

Concept of Savings

Savings is the surplus of income over expenditure: Savings = Income − Expenditure.

Ratio and Proportion Application

If two ratios are given and a constant amount is added/subtracted, we use the cross-multiplication method or x-variable method to solve for the original values.

Income-Expenditure Relationship

Income is always equal to the sum of Expenditure and Savings (I = E + S).

Shortcut Trick

Total Surface Area of n equal parts formed by m diametrical planes passing through a common axis = 4πr² + 2mπr².

Each diametrical cut adds 2 circular faces (each πr²) to the total area.

Given m = 3 planes and r = 14 cm.

Total Surface Area = 4πr² + 2(3)πr² = 10πr².

Calculation ⇒ 10 × (22/7) × 14 × 14 = 6160 cm².

∴ The correct answer is 6160 cm².

Alternate Method

Given:

Radius of sphere (r) = 14 cm

Number of diametrical planes = 3

Number of equal parts = 6

Formula Used:

Surface area of sphere = 4πr²

Area of one circular cross-section = πr²

Calculations:

⇒ Original surface area of sphere = 4 × (22/7) × 14 × 14

⇒ Original area = 4 × 22 × 2 × 14 = 2464 cm²

⇒ Each diametrical cut creates 2 new circular faces.

⇒ Total new faces created by 3 planes = 3 × 2 = 6 faces.

⇒ Total area of new faces = 6 × πr²

⇒ New faces area = 6 × (22/7) × 14 × 14 = 6 × 22 × 2 × 14

⇒ New faces area = 3696 cm²

⇒ Total surface area of all pieces = Original area + New faces area

⇒ Total surface area = 2464 + 3696 = 6160 cm²

∴ The correct answer is 6160 cm².

Additional Information

Surface Area of Hemisphere

Curved surface area (CSA) = 2πr² and Total surface area (TSA) = 3πr².

Volume of Sphere

The volume of a sphere is given by V = (4/3)πr³.

Cutting Rule for Solids

Every time a solid is cut, the total volume remains constant, but the total surface area increases by the area of the newly exposed surfaces.

Shortcut Trick

Total Deduction = 65 + 39 + 77 = ₹181.

Remaining Sum = 4,445 − 181 = ₹4,264.

Total Ratio = 18 + 3 + 20 = 41 units.

Value of 1 unit = 4,264 ÷ 41 = ₹104.

Difference (B & C) = (20 − 3) units + (77 − 39) = 17 × 104 + 38 = 1,768 + 38 = ₹1,806.

∴ The correct answer is ₹1,806.

Alternate Method

Given: Total sum = ₹4,445. Deductions from A, B, C are ₹65, ₹39, ₹77 respectively. Ratio after deduction = 18 : 3 : 20.

⇒ Sum of remaining shares = 4,445 − (65 + 39 + 77)

⇒ Sum of remaining shares = 4,445 − 181 = ₹4,264

⇒ Total ratio parts = 18 + 3 + 20 = 41 units

⇒ Value of 1 unit = 4,264 ÷ 41 = ₹104

⇒ B’s share after deduction = 3 × 104 = ₹312

⇒ C’s share after deduction = 20 × 104 = ₹2,080

⇒ B’s original share = 312 + 39 = ₹351

⇒ C’s original share = 2,080 + 77 = ₹2,157

⇒ Difference = 2,157 − 351 = ₹1,806

∴ The correct answer is ₹1,806.

Additional Information

Ratio and Proportion Basics

If a total sum S is divided in the ratio a:b:c, then the share of B is [b / (a+b+c)] × S.

Handling Deductions in Ratios

When fixed amounts are deducted from shares to obtain a ratio, always subtract the sum of deductions from the total capital before applying the ratio units.

Difference between Shares

Difference between C and B can be calculated as: (Difference in ratio units × value of 1 unit) + (Difference in deducted amounts).

Shortcut Trick

Ratio of Girls to Boys = 2 : 5

Given, 2 units = 200 (Number of girls)

⇒ 1 unit = 100

∴ 5 units = 5 × 100 = 500 (Number of boys)

∴ The correct answer is 500 boys.

Alternate Method

Given:

Ratio of Girls : Boys = 2 : 5

Total number of girls = 200

Formula Used:

Value of a part = (Total value ÷ Sum of ratios) × Specific ratio part

Calculations:

⇒ Let the number of girls be 2x and boys be 5x.

⇒ 2x = 200

⇒ x = 200 ÷ 2

⇒ x = 100

⇒ Number of boys = 5x = 5 × 100

⇒ Number of boys = 500

∴ The correct answer is 500 boys.

Additional Information

Ratio Definition

A ratio compares two quantities by division. If a : b = c : d, then a × d = b × c.

Proportion Rule

To find an unknown in a ratio, equate the ratio of units to the ratio of actual values: (Part A / Part B) = (Value A / Value B).

Compound Ratio

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

Given:

‘meal visit cat’ is coded as ‘vs mo wm’

‘rain meal mango’ is coded as ‘wm or lo’

Observe the common word:

The common word in both statements is “meal”.

The common code in both code groups is “wm”.

Therefore, “meal” is coded as “wm”.

Hence, the correct answer is "Option 4".

Logic followed here is: In each pair, the letters of the second cluster are obtained by adding 5 to the positional values of the corresponding letters in the first cluster.

Option 1: EK - JP

Option 2: KQ - PV

Option 3: NT - SY

Option 4: OU - TA

Therefore, the letter-cluster pair that DOES NOT belong to that group is OU - TA.

Hence, the correct answer is "Option 4".

Given:

→ Ruman is 20^th from the right end.

→ Daniel is 26^th from the right end.

→ Daniel is exactly between Ruman and Om.

→ Om is 14^th from the left end.

Step 1:

Difference between Daniel and Ruman:

26 - 20 = 6

So, Daniel is 6 places away from Ruman.

Step 2:

Since Daniel is exactly between Ruman and Om, Om will also be 6 places away from Daniel.

Om’s position from the right end:

26 + 6 = 32

Step 3:

Om is 14^th from the left end and 32^nd from the right end.

Total people = Left position + Right position - 1

= 14 + 32 - 1

= 45

Therefore, there are 45 people in the row.

Hence, the correct answer is "Option 1".

Given:

Rekha ranked 18^th from the top and 9^th from the bottom.

Formula used:

Total students = Rank from top + Rank from bottom - 1

Calculation:

= 18 + 9 - 1

= 27 - 1

= 26

Therefore, there are 26 students in the class.

Hence, the correct answer is "Option 4".

Logic followed here is: Mathematical signs are interchanged as per the given instructions, and the equation is solved using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

Signs interchange:

 

Given equation:

11 ÷ 3 + 24 × 4 − 1 = ? + 5

After interchanging the signs:

11 × 3 − 24 ÷ 4 + 1 = ? − 5

⇒ 33 - 6 + 1 = ? - 5

⇒ 34 - 6 = ? - 5

⇒ 28 + 5 = ?

Therefore, the value in place of the question mark is 33.

Hence, the correct answer is "Option 3".

Given: 3, 6, 12, 24, 48, ?

The pattern followed here is:

∴ Following the pattern given above 96 will replace the (?) mark.

Hence, the correct answer is "Option 1".

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)

Logic followed:

We need to find even numbers which are:

→ Immediately preceded by an even number

→ Immediately followed by an even number

Pattern: Even Number → Even Number → Even Number

Checking the series:

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

Only the middle digit 2 in the pattern 2 2 4 satisfies the condition.

Therefore, there is One such even number.

Hence, the correct answer is "Option 1".

Logic followed here is: The letters in the first word of the triad are rearranged in a specific positional pattern to form the second and third words.

Pattern:

If the letters of the first word are numbered as 1, 2, 3, 4:

1. The second word follows the sequence: 2 - 4 - 3 - 1
2. The third word follows the sequence: 4 - 3 - 2 - 1 (Reverse of the first word)

For INFO - NOFI - OFNI

 

For WAKE - AEKW - EKAW

 

The pattern observed is: Word 1 (1234) → Word 2 (2431) → Word 3 (4321).

Similarly, let's check the options:

Option (3) FEAR - ERAF - RAEF:

 

Therefore, the triad 'FEAR - ERAF - RAEF' follows the same logic.

Logic followed here is:

Second number : (1st Number × 2) + 3

Third number : 2nd Number - 9

Set 1 (8, 19, 10) →

2nd Number = (1st Number × 2) + 3 = (8 × 2) + 3 = 19

3rd Number = 2nd Number - 9 = 19 - 9 = 10

Set 2 (13, 29, 20) →

2nd Number = (1st Number × 2) + 3 = (13 × 2) + 3 = 29

3rd Number = 2nd Number - 9 = 29 - 9 = 20

Similarly,

Option 3 (15, 33, 24):

2nd Number = (15 × 2) + 3 = 33

3rd Number = 33 - 9 = 24

Therefore, the set related in the same way is (15, 33, 24).

Hence, the correct answer is "Option 3".

Logic followed here is: The value of the equation is determined by replacing the given symbols with their actual mathematical operators and then solving the expression using the BODMAS rule.

Symbol Replacement Table:

 

Given Equation:

6 − 3 + 32 × 4 ÷ 5 = ?

After replacing the signs as per the table above, the equation becomes:

6 × 3 − 32 ÷ 4 + 5

Applying BODMAS rule:

⇒ (6 × 3) - (32 ÷ 4) + 5 = ?

⇒ 18 - 8 + 5 = ?

⇒ 23 - 8 = ?

⇒ 15 = ?

Therefore, the value that will come in place of the question mark is 15.

Hence, the correct answer is "Option 1".

Given numbers:

741, 753, 789, 761, 745

Step 1:

Arrange the numbers in descending order: 789, 761, 753, 745, 741

Step 2:

The second highest number is 761.

Second digit of 761 = 6

Step 3:

The lowest number is 741.

Third digit of 741 = 1

Step 4:

Required result: 6 + 1 = 7

Therefore, the result is 7.

Hence, the correct answer is "Option 2".