RRB Group D Official Paper (Held On: 15 Dec, 2025 Shift 2) (Previous Year Paper)

The correct answer is 2.2 m/s.

Key Points

• The motion of the ball follows the first equation of motion, which relates final velocity, initial velocity, acceleration, and time.
• Given data: The initial velocity (u) is 12 m/s, the time (t) is 1 second, and the acceleration due to gravity (g) is 9.8 m/s^2.
• When an object is thrown upward, it moves against the force of gravity. Therefore, the acceleration (a) is taken as -g, which is -9.8 m/s^2. Hence, statement 1 is correct in its application of sign convention.
• Applying the formula v = u + at, we substitute the values: v = 12 + (-9.8 × 1).
• The calculation results in v = 12 - 9.8, which equals 2.2 m/s. Hence, the final velocity after 1 second is 2.2 m/s.
• Since the resulting velocity is positive, it indicates that the ball is still moving upward but at a significantly reduced speed due to the constant downward pull of gravity.

Additional Information

• Equations of Motion:
• These equations describe the behavior of a physical system in terms of its motion as a function of time. They are valid only when acceleration is constant.
• First Equation: v = u + at (Velocity-Time relation).
• Second Equation: s = ut + 1/2 at^2 (Position-Time relation), where s is displacement.
• Third Equation: v^2 = u^2 + 2as (Velocity-Position relation).
• Acceleration due to Gravity (g):
• On Earth, the standard value for g is approximately 9.8 m/s^2. It represents the constant acceleration experienced by an object in free fall.
• The value of g varies slightly depending on the location (altitude and latitude), being higher at the poles and lower at the equator.
• Sign Convention in Kinematics:
• In vertical motion, if we consider the upward direction as positive, then initial velocity (u) is positive, and acceleration due to gravity (g) is negative because it acts downward.
• At the maximum height of the trajectory, the final velocity (v) of the object becomes 0 m/s momentarily before it starts descending.
• Scalar vs Vector Quantities:
• Velocity is a vector quantity, meaning it has both magnitude and direction. In this problem, the magnitude is 2.2 m/s and the direction is upward.
• Speed is a scalar quantity and only represents the magnitude of the rate of motion.

Important Points

• If the time given was longer, such as 2 seconds, the velocity would become negative (12 - 19.6 = -7.6 m/s), indicating the ball has already reached its peak and is falling back down.
• The maximum height reached by the ball can be calculated using the formula s = u^2 / 2g. For an initial velocity of 12 m/s, the maximum height would be approximately 7.35 meters.
• The total time of flight for the ball to return to the thrower's hand (ignoring air resistance) is calculated as T = 2u / g, which would be approximately 2.45 seconds.

The correct answer is cytoplasm.

Key Points

• Cytoplasm is the thick, jelly-like fluid content that is enclosed within the plasma membrane and surrounds the nucleus of a cell.
• It serves as the primary arena for most metabolic activities and chemical reactions necessary for life, such as glycolysis (the initial stage of cellular respiration).
• The cytoplasm is not just a simple fluid; it contains many specialised cell organelles like mitochondria, endoplasmic reticulum, ribosomes, and the Golgi apparatus, each performing a specific function for the cell.
• The portion of the cytoplasm that is not occupied by organelles is called the cytosol, which consists mainly of water (about 70% to 90%), salts, and organic molecules like proteins and enzymes.
• It plays a vital role in structural support, maintaining the shape of the cell, and facilitates the movement of materials via cytoplasmic streaming.

Additional Information

• Nucleoplasm:
• Also known as karyoplasm, this is the highly gelatinous fluid found inside the nucleus of eukaryotic cells.
• It provides the medium in which the nucleolus and chromatin (containing DNA) are suspended.
• The nuclear envelope acts as a barrier that separates the nucleoplasm from the cytoplasm, allowing for controlled transport of molecules through nuclear pores.
• Vacuoplasm (Cell Sap):
• This refers to the fluid found inside a vacuole, a storage organelle particularly prominent and large in plant cells.
• It contains various substances such as water, sugars, salts, amino acids, and sometimes metabolic waste products.
• In plants, it helps maintain turgor pressure, which provides the cell with rigidity and structural integrity.
• Plasmin:
• Unlike the other terms, plasmin is not a part of the internal cell structure; it is an important proteolytic enzyme present in the blood.
• Its primary function is to degrade fibrin clots, a process known as fibrinolysis, which prevents blood clots from becoming too large and causing health issues.
• It is derived from its inactive precursor, plasminogen, which is produced in the liver.
• Protoplasm:
• The term protoplasm refers to the entire living substance of the cell, including both the cytoplasm and the nucleus.
• It was first coined by the scientist J.E. Purkinje in 1839 to describe the physical basis of life within a cell.

Important Points

• Plasma Membrane: This is a selectively permeable membrane that regulates the entry and exit of molecules, thereby protecting the internal environment of the cytoplasm.
• Cell Organelles: These are membrane-bound structures within the cytoplasm. For example, mitochondria are the powerhouse of the cell because they generate ATP (Adenosine Triphosphate), the energy currency of the cell.
• Eukaryotic vs. Prokaryotic: In eukaryotic cells (like those in humans and plants), the cytoplasm contains complex organelles, whereas in prokaryotic cells (like bacteria), there are no membrane-bound organelles, and the genetic material is suspended directly in the cytoplasm.

The correct answer is 6 a.

Key Points

• Newton's Second Law of Motion provides the fundamental relationship between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
• The mathematical expression for this law is F = m × a, where F is the force applied, m is the mass of the body, and a is the acceleration produced. From this, the acceleration can be derived as a = F / m.
• In the first scenario provided in the question, the force is F and the mass is m, which results in an acceleration of a = F / m.
• In the second scenario, the force is increased to 3F and the mass is reduced to m/2. To find the new acceleration (a'), we substitute these new values into the formula: a' = (3F) / (m / 2).
• By performing the algebraic calculation, a' = 3F × (2 / m), which simplifies to a' = 6 × (F / m). Since the original acceleration a is F / m, the new value becomes a' = 6a.
• This demonstrates that increasing the force while simultaneously decreasing the mass leads to a significantly higher acceleration. Hence, the statement that the new acceleration is 6a is correct.

Additional Information

• Force (F): It is an external agency capable of changing the state of rest or motion of a particular body. It is a vector quantity, meaning it has both magnitude and direction. The SI unit of force is the Newton (N), which is equivalent to 1 kg·m/s².
• Mass (m): This represents the quantity of matter contained in a body and serves as a measure of the body's inertia. Unlike weight, mass is a scalar quantity and remains constant regardless of the body's location in the universe. Its SI unit is the kilogram (kg).
• Acceleration (a): It is defined as the rate of change of velocity with respect to time. If the velocity of an object changes, it is said to be accelerating. The SI unit for acceleration is meters per second squared (m/s²).
• Newton's Laws of Motion: These three laws were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, published in 1687. These laws form the basis for classical mechanics and describe the physics of moving objects.

Important Points

• Inertia and Mass: The mass of an object is a direct measure of its inertia. A larger mass requires more force to achieve the same acceleration. In this problem, reducing the mass to m/2 effectively halved the resistance to change in motion, contributing to the increase in acceleration.
• Proportionality: The relationship shows that if mass is kept constant, acceleration is doubled when force is doubled. However, if force is kept constant and mass is halved, the acceleration also doubles. When both changes occur simultaneously as per the question (3 times the force and 1/2 the mass), the total effect is 3 × 2 = 6.
• Practical Applications: This principle is used in automotive engineering to design faster cars by either increasing engine force (horsepower) or using lightweight materials like carbon fiber to reduce the mass of the vehicle.

The correct answer is Statement 2 and 3 are correct.

Key Points

• Statement 1 claims that a suspension is a homogeneous mixture. In reality, a suspension is a heterogeneous mixture in which the solute particles do not dissolve but remain suspended throughout the bulk of the medium. In a homogeneous mixture, such as a true solution, the composition is uniform throughout. Hence, statement 1 is incorrect.
• Statement 2 mentions that the particles of a suspension can be seen by the naked eye. This is due to the relatively large size of the particles, which are typically larger than 1000 nanometers (or 1 micrometer) in diameter. Unlike colloids or true solutions, where particles are too small to be seen without magnification, suspension particles are large enough to scatter visible light. Hence, statement 2 is correct.
• Statement 3 states that the solute particles settle down when a suspension is left undisturbed, meaning the suspension is unstable. Because the particles are large and heavy, the force of gravity eventually overcomes the kinetic energy keeping them dispersed, leading to sedimentation. Once the particles settle at the bottom, the mixture no longer scatters light as effectively. Hence, statement 3 is correct.
• The physical state of a suspension can be altered through filtration. Since the particles are larger than the pores of filter paper, they can be easily separated from the solvent or dispersion medium.

Additional Information

• Properties of Suspensions:
• A suspension is heterogeneous in nature, meaning its components are not distributed uniformly.
• The particles scatter a beam of light passing through it and make its path visible; this phenomenon is known as the Tyndall Effect. However, the Tyndall Effect disappears once the particles have settled down.
• Examples of suspensions include muddy water, chalk powder in water, flour in water, and milk of magnesia.
• Comparison with Other Mixtures:
• True Solution: It is a homogeneous mixture with a particle size less than 1 nm. These particles do not scatter light and cannot be seen by the naked eye. They are highly stable and do not settle.
• Colloid: It is a heterogeneous mixture that appears homogeneous. Particle sizes range between 1 nm and 1000 nm. They show the Tyndall Effect but do not settle down, making them stable.
• Applications and Facts:
• In the pharmaceutical industry, many medicines are prepared as suspensions (e.g., antacids) and must be shaken well before use to redistribute the settled solute particles.
• The instability of a suspension is a key characteristic used in water treatment plants, where coagulants are added to help suspended impurities settle faster through sedimentation.

Important Points

• Particle Size: The defining feature of a suspension is a particle size greater than 10^-6 meters.
• Separation: Unlike colloids, which require centrifugation, suspensions can be separated by simple mechanical filtration.
• Optical Properties: An active suspension is opaque or translucent, but the supernatant liquid may become transparent once the particles settle.

The correct answer is Phloem.

Key Points

• Phloem is a complex permanent tissue in vascular plants that acts as a conduction system for organic nutrients, primarily sucrose, produced during photosynthesis.
• Unlike Xylem, which only transports water upwards, the movement of materials in the Phloem is bidirectional. This means it can transport food materials both upwards and downwards depending on the plant's requirements.
• The process of transporting food materials through the phloem is known as translocation. This process is active and requires energy in the form of ATP to move substances against a concentration gradient.
• Phloem consists of four main types of cells: sieve tubes, companion cells, phloem parenchyma, and phloem fibres. Out of these, only phloem fibres are dead cells, while the rest are living tissues.
• The bidirectional flow is essential because it allows the leaves (the source) to send sugar to the roots for storage, and during spring, it allows the storage organs (the sink) to send energy back up to the developing buds.

Additional Information

• Xylem:
• It is responsible for the transport of water and minerals from the roots to the leaves.
• The flow of materials in the xylem is unidirectional (only upwards).
• It provides mechanical strength to the plant and is composed of tracheids, vessels, xylem parenchyma, and xylem fibres.
• Sclerenchyma:
• This is a simple permanent tissue made of dead cells with extremely thickened cell walls containing lignin.
• Its primary function is to provide structural support and rigidity to the plant. It is commonly found in stems, veins of leaves, and the hard coverings of seeds and nuts.
• Fibres:
• Fibres are a type of sclerenchyma cell. They are elongated, tapering, and have thick lignified walls.
• In the commercial world, fibres from plants like jute, flax, and hemp are used to make ropes and textiles due to their high tensile strength.

Important Points

• Mechanism of Transport: The movement in phloem is explained by the Pressure Flow Hypothesis (or Mass Flow Hypothesis), where high osmotic pressure in the leaves forces the phloem sap toward regions of low pressure (the sink).
• Vascular Bundles: Both Xylem and Phloem together constitute the vascular bundles of the plant, which are the main conducting tissues.
• Location: In a typical dicot stem, the Phloem is usually located towards the outer side of the vascular bundle, while the Xylem is located towards the inner side (pith).
• Sieve Tubes and Companion Cells: These are uniquely paired. Sieve tubes lack a nucleus at maturity, so their metabolic activities are controlled by the nucleus of the adjacent companion cells.

The correct answer is -60 cm.

Key Points

• The mirror formula provides a quantitative relationship between the object distance (u), the image distance (v), and the focal length (f) of a spherical mirror. The formula is expressed as 1/f = 1/v + 1/u.
• For a concave mirror, the focal length (f) is always considered negative according to the New Cartesian Sign Convention. In this problem, the given focal length is 20 cm, so we use f = -20 cm.
• The object distance (u) is the distance from the pole of the mirror to the object. Since the object is placed in front of the mirror, the distance is measured against the direction of incident light, making it negative. Here, u = -30 cm.
• To find the image distance (v), we rearrange the mirror formula: 1/v = 1/f - 1/u. Substituting the values: 1/v = 1/(-20) - 1/(-30).
• This simplifies to 1/v = -1/20 + 1/30. By taking the Least Common Multiple (LCM) of 20 and 30, which is 60, the equation becomes 1/v = (-3 + 2) / 60.
• Solving further, 1/v = -1/60, which gives the result v = -60 cm. Hence, the image distance is -60 cm and the correct option is -60 cm.

Additional Information

• Nature of the Image: Since the image distance (v) is negative, the image is formed in front of the mirror. This indicates that the image is real and inverted.
• Magnification (m): Magnification is defined as the ratio of the height of the image to the height of the object, also given by m = -v/u. In this case, m = -(-60) / (-30), which equals -2. A magnification of -2 means the image is magnified (twice the size of the object) and inverted.
• Position relative to Center of Curvature: The Radius of Curvature (R) of this mirror is R = 2f = 40 cm. Since the object is placed at 30 cm (between the Focus at 20 cm and the Center of Curvature at 40 cm), the image must form beyond the Center of Curvature (greater than 40 cm). Our result of 60 cm aligns with this optical principle.
• Concave Mirror Characteristics: Often called a converging mirror, it has the ability to produce real as well as virtual images depending on the object's position relative to the focal point.

Important Points

• Sign Convention Rules: Distances measured in the direction of incident light are taken as positive, while those measured in the opposite direction are negative. All distances are measured from the pole (P) of the mirror.
• Focal Lengths: It is a fact that the focal length of a concave mirror is always negative, whereas the focal length of a convex mirror is always positive.
• Real vs Virtual: A positive image distance (v) for a mirror would imply a virtual image formed behind the mirror, while a negative image distance signifies a real image.
• Practical Uses: Concave mirrors are utilized in torches, search-lights, and vehicle headlights to get powerful parallel beams of light, as well as by dentists to see larger images of teeth.

The correct answer is 2 Ω.

Key Points

• When resistors are connected in parallel, the reciprocal of the equivalent resistance (Req) is the sum of the reciprocals of the individual resistances. The formula is given by: 1/Req = 1/R1 + 1/R2 + 1/R3.
• In this problem, the three resistors are 12 Ω, 6 Ω, and 4 Ω. Substituting these values into the formula: 1/Req = 1/12 + 1/6 + 1/4.
• To solve the fractions, find a common denominator, which is 12. The equation becomes: 1/Req = 1/12 + 2/12 + 3/12.
• Adding the numerators gives: 1/Req = 6/12. Simplifying this fraction, we get 1/Req = 1/2.
• Taking the reciprocal of both sides, the equivalent resistance (Req) is 2 Ω. Hence, the calculated result matches the fourth option.
• A fundamental characteristic of parallel circuits is that the equivalent resistance is always less than the smallest individual resistor in the combination (in this case, 2 Ω is less than 4 Ω).

Additional Information

• Parallel Combination of Resistors:
• In a parallel circuit, the voltage across each resistor remains the same and is equal to the total voltage of the source.
• The total current in the circuit is the sum of the currents passing through each individual branch.
• This type of connection is commonly used in household wiring so that each appliance can operate independently at the same voltage.
• Series Combination of Resistors:
• When resistors are connected in series, the total resistance is the simple sum of individual resistances: R = R1 + R2 + R3.
• In a series circuit, the current remains constant through all components, while the voltage is divided among them.
• If the given resistors (12 Ω, 6 Ω, and 4 Ω) were in series, the equivalent resistance would be 22 Ω.
• Electrical Resistance:
• Resistance is the property of a conductor to oppose the flow of electric current.
• The SI unit of resistance is the Ohm (Ω), named after the German physicist Georg Simon Ohm.
• Factors affecting resistance include the length of the conductor, its cross-sectional area, the nature of the material, and its temperature.

Important Points

• Ohm's Law: This law states that the potential difference (V) across a conductor is directly proportional to the current (I) flowing through it, provided physical conditions remain constant. The relation is V = IR.
• Conductance: The reciprocal of resistance is called conductance (G), and its SI unit is Siemens (S) or mho. In parallel circuits, the total conductance is the sum of individual conductances (G_total = G1 + G2 + G3).
• Power Dissipation: In a parallel circuit, the resistor with the lowest resistance dissipates the highest power because P = V²/R, and V is constant for all.
• Practical Application: If one component in a parallel network fails or is switched off, the other components continue to function, which is not the case in a series circuit.

The correct answer is Propanol.

Key Points

• The name of an organic compound is determined by the number of carbon atoms in its longest chain. A chain with three carbon atoms uses the prefix Prop-.
• The description "having no double bond" indicates that the compound is saturated. In chemical nomenclature, saturated hydrocarbons (alkanes) use the primary suffix -ane. When a functional group is added, the 'e' is typically dropped, making the base name Propan-.
• The presence of an -OH functional group identifies the molecule as an alcohol. According to IUPAC nomenclature, alcohols are designated by the suffix -ol.
• The location of the functional group is specified by numbering the carbon chain. Since the hydroxyl group (-OH) is at the first carbon atom, the systematic name is Propan-1-ol, commonly referred to as Propanol. Hence, the correct answer is Propanol.
• Evaluating other options:
• Propanone: This is a ketone (suffix -one) which requires a carbonyl group (C=O) on a non-terminal carbon. Hence, this option is incorrect.
• Propane: This is a simple alkane consisting only of carbon and hydrogen with no functional groups like -OH. Hence, this option is incorrect.
• Propanal: This is an aldehyde (suffix -al) which contains a -CHO group at the end of the chain rather than an -OH group. Hence, this option is incorrect.

Additional Information

• IUPAC Nomenclature System: This is a globally recognized method of naming organic chemical compounds maintained by the International Union of Pure and Applied Chemistry. It ensures that every compound has a unique and unambiguous name based on its molecular structure.
• Root Word Selection: The prefix is determined by the number of carbons: Meth- (1), Eth- (2), Prop- (3), But- (4), Pent- (5), and Hex- (6). For this question, three carbons lead directly to Prop-.
• Functional Group Suffixes:
• Alcohols (-OH): Use the suffix -ol. Example: Methanol (CH3OH).
• Aldehydes (-CHO): Use the suffix -al. Example: Ethanal (CH3CHO).
• Ketones (>C=O): Use the suffix -one. Example: Butanone (CH3COCH2CH3).
• Carboxylic Acids (-COOH): Use the suffix -oic acid. Example: Ethanoic acid (CH3COOH).
• Saturated vs. Unsaturated: A saturated compound contains only single bonds between carbon atoms, whereas unsaturated compounds contain at least one double bond (alkenes, suffix -ene) or triple bond (alkynes, suffix -yne).

Important Points

• Structure of Propanol: The chemical formula for Propan-1-ol is C3H8O, often written as CH3CH2CH2OH to show the straight-chain structure.
• Isomerism: Propanol has a structural isomer called Propan-2-ol (commonly known as Isopropyl alcohol or Isopropanol), where the -OH group is attached to the second carbon atom instead of the first.
• Physical Properties: Propan-1-ol is a colorless liquid at room temperature and is used extensively as a solvent in the pharmaceutical industry and for making antiseptics.
• Classification: It is classified as a primary alcohol because the carbon atom holding the hydroxyl group is attached to only one other carbon atom.

The correct answer is A - 3; B - 2; C - 1; D - 4.

Key Points

• CuO (Copper Oxide): Most metallic oxides are basic in nature. Copper(II) oxide reacts with dilute hydrochloric acid to produce a salt and water, acting as a base. In the given reaction, it serves as the basic oxide reacting with acid. Hence, A is matched with 3.
• HCl (Hydrochloric Acid): This is a strong mineral acid. In this specific chemical interaction, it acts as the acid reacting with basic oxide (CuO) to neutralize the base. Hence, B is matched with 2.
• CuCl2 (Copper Chloride): When an acid reacts with a metallic oxide, the resulting compound formed by the replacement of the hydrogen of the acid by a metal is a salt. Copper(II) chloride is the salt formed in the reaction, which gives the solution a characteristic blue-green color. Hence, C is matched with 1.
• H2O (Water): In a neutralization reaction between a base (or basic oxide) and an acid, water is always produced alongside the salt. It is considered the neutralization product in this context. Hence, D is matched with 4.
• The overall chemical equation for this process is: CuO + 2HCl → CuCl2 + H2O. This demonstrates the typical behavior of metal oxides reacting with acids.

Additional Information

• Nature of Oxides:
• Basic Oxides: These are oxides of metals (like Na2O, MgO, CaO, and CuO) that react with acids to form salt and water.
• Acidic Oxides: These are typically oxides of non-metals (like CO2, SO2, P2O5) that react with bases to form salt and water.
• Amphoteric Oxides: Some metallic oxides, such as Aluminium oxide (Al2O3) and Zinc oxide (ZnO), show both acidic and basic behaviors.
• Neutralization Reaction: It is a chemical reaction in which an acid and a base react quantitatively with each other. In a reaction in water, neutralization results in there being no excess of hydrogen ions or hydroxide ions present in the solution.
• Copper(II) Oxide (CuO): It is a black solid. When it reacts with HCl, the black color disappears as it dissolves to form the blue-green solution of Copper(II) chloride.
• Indicators: The change in the nature of the solution (from acidic to neutral) during such reactions can often be monitored using pH indicators or litmus paper.

Important Points

• Reaction Observation: During the reaction of CuO with HCl, the observation of the color change is a key identifier of the formation of CuCl2. This is a common laboratory experiment to demonstrate the basic nature of metallic oxides.
• Valency: In Copper(II) oxide, the valency of copper is +2. This is why it requires two molecules of HCl (each providing one Cl- ion) to form one molecule of CuCl2.
• Safety: Handling concentrated HCl requires care due to its corrosive nature. Reactions involving metallic oxides and acids are generally exothermic, meaning they release heat.
• Solubility: While CuO is insoluble in water, it dissolves in acids, which is a characteristic property of basic oxides that are not alkalis (water-soluble bases).

The correct answer is Rohu.

Key Points

• Rohu (Labeo rohita) is a species of fish of the carp family (Cyprinidae) found in freshwater rivers in South Asia.
• It is a large omnivore and is used extensively in aquaculture in countries like India, Bangladesh, Nepal, and Pakistan. Hence, it is not a marine fish.
• It belongs to the category of Indian Major Carps, which also includes other freshwater species like Catla and Mrigal.
• Pomfret is a marine fish found in the Indo-Pacific region and is widely consumed in the coastal regions of India. Hence, statement 1 describes a marine fish.
• Mackerel is a common name applied to a number of different species of pelagic marine fish, mostly from the family Scombridae. Hence, statement 4 describes a marine fish.
• Bhetki, also known as Barramundi, is a euryhaline species (can tolerate various salinities) but is primarily found in estuaries, backwaters, and marine coastal waters. Hence, it is considered a marine/estuarine fish.

Additional Information

• Marine Fisheries:
• These involve the exploitation of fish resources from saltwater bodies such as oceans and seas.
• In India, the marine fishery resources are spread along the 8,118 km long coastline.
• Important marine fish varieties include Hilsa, Sardines, Tuna, Mackerel, and Pomfret.
• The exclusive economic zone (EEZ) of India provides a vast area for deep-sea fishing operations.
• Inland Fisheries:
• This refers to the catching and farming of fish in freshwater resources like canals, ponds, reservoirs, and rivers.
• Rohu is the most important among the three Indian major carp species used in polyculture systems.
• Freshwater aquaculture contributes to a significant portion of the total fish production in India, with states like Andhra Pradesh and West Bengal leading the sector.
• Characteristics of Rohu:
• Age and Growth: It can reach a maximum length of about 2 meters and can live for several years under optimal conditions.
• Features: It has cycloid scales, a depressed snout, and is typically bluish-grey or black on the back with a silvery-white belly.
• Diet: It is mainly a column feeder, consuming phytoplankton and decaying organic matter.

Important Points

• Distinction based on Salinity:
• Marine fish are adapted to high salinity levels and possess specialized osmoregulation mechanisms to prevent dehydration.
• Freshwater fish like Rohu are adapted to low-salinity environments; placing them in seawater would lead to fatal osmotic shock.
• Economic Significance:
• The Blue Revolution in India aimed to increase fish production and productivity from both marine and inland sources.
• Mackerel and Pomfret are high-value export items, contributing significantly to India's foreign exchange earnings.
• Rohu remains the primary source of animal protein for a large section of the population in North and East India.

The correct answer is Only statement 1 is correct.

Key Points

• The Mirror Formula is a mathematical relationship that connects the focal length (f), the distance of the object (u) from the pole, and the distance of the image (v) from the pole. The formula is expressed as 1/v + 1/u = 1/f.
• Statement 1: This formula is universally applicable to all spherical mirrors, which includes both concave mirrors and convex mirrors. To obtain accurate results for different positions of the object, one must strictly follow the New Cartesian Sign Convention, where the pole is treated as the origin. Hence, statement 1 is correct.
• Statement 2: The derivation of the mirror formula assumes that the aperture of the mirror is small compared to its radius of curvature and that the rays of light are paraxial (rays that make small angles with the principal axis). However, the validity of the formula itself does not depend on the size of the object. The object height affects the magnification, but the relationship between the distances (u, v, and f) remains constant regardless of how large or small the object is. Hence, statement 2 is incorrect.
• The Sign Convention is crucial: distances measured in the direction of incident light are taken as positive, while those measured opposite to the direction of incident light are negative. For instance, the object distance (u) is almost always negative in standard problems.
• The focal length (f) for a concave mirror is always negative, whereas for a convex mirror, it is always positive according to the standard convention.

Additional Information

• Concave Mirror:
• A concave mirror is a converging mirror that can form both real and virtual images depending on the position of the object.
• When the object is placed between the focus (F) and the pole (P), it forms a virtual, erect, and magnified image.
• In all other positions beyond the focus, it forms real and inverted images of varying sizes.
• Convex Mirror:
• A convex mirror is a diverging mirror that always forms a virtual, erect, and diminished image, regardless of the object's distance from the mirror.
• Because it provides a wider field of view, it is commonly used as a rear-view mirror in vehicles.
• Linear Magnification (m):
• This is defined as the ratio of the height of the image (h') to the height of the object (h).
• It is also related to distances as m = -v/u. A negative magnification indicates a real image, while a positive magnification indicates a virtual image.

Important Points

• The relationship between the radius of curvature (R) and the focal length (f) is given by R = 2f. This relation holds true for all spherical mirrors with small apertures.
• In a concave mirror, the image is real when v is negative and virtual when v is positive.
• In a convex mirror, the image is always formed behind the mirror, so v is always positive, and since u is negative, the magnification m is always positive and less than 1.
• The Mirror Formula was first derived using the principles of Euclidean geometry and the laws of reflection (Angle of Incidence = Angle of Reflection).

The correct answer is particles have moderate gaps.

Key Points

• Compressibility is the measure of the relative volume change of a fluid or solid as a response to a pressure change. In the case of liquids, the particles are packed relatively close to each other, meaning the intermolecular spaces are much smaller than those in gases.
• Liquids have particles that have moderate gaps compared to gases. While these particles can slide past one another (giving liquids their fluid property), they are close enough that there is very little "empty" space to be squeezed out when pressure is applied. Hence, statement 4 is correct.
• The presence of dissolved gases in a liquid might slightly increase its overall compressibility or lead to bubble formation under pressure changes, but it is not the primary reason why the liquid state itself is less compressible. Hence, statement 1 is incorrect.
• While the density of most liquids is indeed less than that of their solid counterparts (with water and ice being a notable exception), density is a measure of mass per unit volume, not the mechanism of compression. Compressibility is determined by the availability of space between particles to be reduced. Hence, statement 2 is incorrect.
• All matter in a closed system has a fixed mass according to the Law of Conservation of Mass. Having a fixed mass is a universal property of matter and does not explain why one state of matter (liquid) resists volume changes more effectively than another (gas). Hence, statement 3 is incorrect.

Additional Information

• States of Matter and Spacing:
• Solids: Particles are tightly packed in a regular pattern with negligible intermolecular spaces. They are considered incompressible for most practical purposes.
• Liquids: Particles are close together with moderate gaps and no regular arrangement. They take the shape of the container but maintain a fixed volume.
• Gases: Particles have very large gaps between them and move randomly at high speeds. Because of these vast spaces, gases are highly compressible.
• Kinetic Molecular Theory:
• This theory states that the distance between liquid particles is small enough that intermolecular attractive forces (like Van der Waals forces or Hydrogen bonding) hold them together, preventing the significant volume reduction seen in gases.
• Bulk Modulus:
• The resistance of a substance to compression is quantified by the Bulk Modulus. A high Bulk Modulus indicates that a substance is difficult to compress. Liquids like water have a very high Bulk Modulus (approximately 2.2 GPa), making them nearly incompressible in hydraulic systems.
• Applications:
• The low compressibility of liquids is the fundamental principle behind Hydraulic Machinery (like Pascal's Law). Since liquids do not compress easily, pressure applied at one point is transmitted efficiently throughout the fluid to perform work, such as in car brakes or industrial lifts.

The correct answer is Broken glass pieces.

Key Points

• Broken glass pieces are categorized as non-biodegradable waste. This means they cannot be broken down or decomposed by the action of microorganisms like bacteria, fungi, or earthworms. Hence, option 2 is correct.
• Mango seeds are organic matter derived from living organisms. They are biodegradable and will eventually rot and decompose into the soil, though they take slightly longer than soft tissues due to their hard shell. Hence, statement 1 is incorrect.
• Pieces of bread consist of organic compounds like carbohydrates. They are highly biodegradable and are quickly decomposed by molds and bacteria when exposed to moisture in the soil. Hence, statement 3 is incorrect.
• Apple peels are typical kitchen waste and are biodegradable. They decompose rapidly (usually within few weeks) and turn into compost, adding nutrients back to the environment. Hence, statement 4 is incorrect.
• Persistence in the environment refers to the duration a substance remains in the soil without changing its chemical structure. Glass is made primarily of silica (Silicon Dioxide), which is extremely stable and can persist for 1 million years or more.

Additional Information

• Biodegradable Substances:
• These are substances that are degraded by natural processes into simpler, non-toxic substances.
• Examples include agricultural residues, paper, wood, cloth, and animal dung.
• The decomposition process is carried out by decomposers, which recycle nutrients back into the ecosystem.
• Non-biodegradable Substances:
• These substances do not degrade or take a very long time (hundreds to thousands of years) to break down.
• Common examples include plastics, polythene bags, synthetic fibers, metal cans, and radioactive waste.
• They often act as pollutants and can enter the food chain, leading to biological magnification.
• Environmental Impact of Glass:
• While glass is chemically inert and does not leach toxic chemicals like some plastics, its physical presence causes land pollution.
• In soil, broken glass poses a physical hazard to macro-fauna such as earthworms and can hinder the penetration of plant roots.
• Because it does not oxidize or corrode easily, it remains unchanged in the geological strata for eras.

Important Points

• Microbial Enzymes: Microorganisms produce specific enzymes to break down organic bonds (like C-C or C-H bonds in food). However, they do not possess enzymes to break the silicon-oxygen bonds found in glass.
• Waste Management: To prevent environmental accumulation, non-biodegradable items like glass and metals should be recycled. Glass is 100% recyclable and can be melted and reformed infinitely without loss in quality.
• Decomposition Timeframes: While an apple peel takes about 2 to 4 weeks to disappear, a plastic bottle may take 450 years, and glass effectively never disappears under normal environmental conditions.

The correct answer is 2 ΩΩΩΩΩΩΩ

      .

Key Points

• The resistance of a conductor is defined by the formula R = ρ (l / A), where ρ is the resistivity of the material, l is the length, and A is the cross-sectional area.

• In the first case, the wire has a resistance of 12 ΩΩΩΩΩΩΩ

        with **length l** and **area A**. This establishes the base equation: **12 = ρ (l / A)**. ***Hence, the initial relationship is defined.***
  

• In the second case, the wire is made of the same material, which means the resistivity (ρ) remains constant because it is an intrinsic property of the material.

• The new length is given as l / 3 and the new cross-sectional area is 2A. Substituting these values into the resistance formula gives: R’ = ρ ( (l/3) / (2A) ). Hence, the new parameters are applied.

• Simplifying the expression, we get R’ = ρ (l / (3 × 2A)) = ρ (l / 6A). This can be rewritten as R’ = (1 / 6) × (ρ l / A).

• Since we know from the first case that ρ (l / A) = 12 ΩΩΩΩΩΩΩ

       , we substitute this value: **R’ = (1 / 6) × 12 = 2 ΩΩΩΩΩΩΩ**
  
       . ***Hence, the calculated resistance is correct.***
  

• ***Therefore, the correct answer is 2 ΩΩΩΩΩΩΩ

       .***
  

Additional Information

• Factors Affecting Resistance:

• Length (l): The resistance of a conductor is directly proportional to its length. If the length increases, the electrons encounter more collisions, increasing the resistance.

• Cross-sectional Area (A): The resistance is inversely proportional to the area of cross-section. A thicker wire (larger area) provides a wider path for electron flow, reducing resistance.

• Nature of Material: Different materials offer different levels of opposition to current flow, characterized by their resistivity (ρ). Conductors like Copper and Silver have low resistivity.

• Temperature: For most metallic conductors, resistance increases with an increase in temperature due to more frequent atomic vibrations obstructing electron flow.

• Resistivity (ρ):

• It is the resistance offered by a wire of unit length and unit cross-sectional area.

• The SI unit of resistivity is **Ohm-meter (ΩΩΩΩΩΩΩ

       ⋅⋅⋅⋅⋅⋅⋅
  
       m)**.
  

• Unlike resistance, resistivity does not change with the dimensions (length or area) of the wire; it only changes with the material and temperature.

Important Points

• Ohm’s Law: This fundamental law states that the current (I) flowing through a conductor is directly proportional to the potential difference (V) across its ends, provided physical conditions like temperature remain constant (V = IR).

• SI Unit of Resistance: The standard unit is the **Ohm (ΩΩΩΩΩΩΩ

       )**, named after the German physicist **Georg Simon Ohm**.
  

• Practical Application: In household wiring, thicker wires are used for high-power appliances (like air conditioners) because their larger cross-sectional area results in lower resistance, preventing the wires from overheating.

• Calculation Summary: When the length is divided by 3 and the area is multiplied by 2, the total resistance becomes 1/6^th of the original value (1/3 × 1/2 = 1/6).

The correct answer is 50%.

Key Points

• In humans, the sex of an individual is determined by a specific pair of chromosomes known as sex chromosomes or allosomes. Humans have a total of 23 pairs of chromosomes, out of which 22 pairs are autosomes and the 23^rd pair determines the sex.
• Females possess two identical sex chromosomes, denoted as XX. Because they only carry X chromosomes, all the eggs or ova produced by a female will contain one X chromosome. This condition is referred to as being homogametic.
• Males possess two different sex chromosomes, denoted as XY. During the process of meiosis (gamete formation), males produce two types of sperm: approximately 50% carry the X chromosome and the other 50% carry the Y chromosome. This makes males heterogametic.
• The sex of the child is determined at the moment of fertilization by the type of sperm that fuses with the egg. If a sperm carrying an X chromosome fertilizes the egg (X), the zygote becomes XX, resulting in a female child.
• If a sperm carrying a Y chromosome fertilizes the egg (X), the zygote becomes XY, resulting in a male child. Since the ratio of X-carrying sperm to Y-carrying sperm is 1:1, the mathematical probability or percentage of having a male child is exactly 50%.

Additional Information

• The SRY Gene: The Y chromosome is significantly smaller than the X chromosome and carries a crucial gene called SRY (Sex-determining Region Y). This gene triggers the development of male characteristics in the embryo. In its absence, the embryo develops as a female.
• Heterogamy vs. Homogamy: In the XY sex-determination system, the male is the heterogametic sex (XY) and the female is the homogametic sex (XX). This system is prevalent in most mammals and some insects like Drosophila.
• Punnett Square Analysis: When crossing a mother (XX) and a father (XY), the possible combinations are XX, XX, XY, and XY. This clearly illustrates that two out of four possibilities result in a male, which equates to a 50% frequency.
• Other Systems: It is important to note that not all species follow the XY system. For example, in birds and some reptiles, the ZW system is used where the female is the heterogametic sex (ZW) and the male is homogametic (ZZ).

Important Points

• Scientific Responsibility: In many societies, women are traditionally blamed for the sex of the child. However, biological evidence confirms that the sex of the baby is determined solely by the genetic contribution of the father, as only he can provide the Y chromosome.
• Historical Context: The role of chromosomes in sex determination was first identified in the early 20^th century (around 1905) by scientists like Nettie Stevens and Edmund Beecher Wilson, who independently discovered the XY and XO systems.
• Statistical Reality: While the theoretical probability is 50%, global demographic data often shows a slight natural skew (approximately 105 males to 100 females) due to various biological factors influencing sperm motility or embryo survival, though the genetic chance remains 1 in 2.

The correct answer is By the inheritance of X or Y chromosome from the father.

Key Points

• In human beings, every cell contains 23 pairs of chromosomes, making a total of 46 chromosomes. Out of these, 22 pairs are called autosomes and are identical in both males and females.
• The 23^rd pair of chromosomes is known as the sex chromosome, which differs between the two sexes. Females possess two X chromosomes (XX), while males possess one X and one Y chromosome (XY).
• During the process of gametogenesis, the mother produces eggs (ova) that always carry a single X chromosome. Therefore, the maternal contribution to the child's sex is always the same.
• The father produces two types of sperm: approximately 50% carry an X chromosome and the other 50% carry a Y chromosome.
• The sex of the child is determined at the moment of fertilization. If a sperm carrying an X chromosome fertilizes the egg, the resulting zygote will be XX, developing into a female. If a sperm carrying a Y chromosome fertilizes the egg, the zygote will be XY, developing into a male. Hence, statement 2 is the correct mechanism.

Additional Information

• Heterogametic Sex: In humans, the male is the heterogametic sex because he produces two different types of gametes (X and Y), whereas the female is homogametic as she produces only one type (X).
• The SRY Gene: The Sex-determining Region Y (SRY) gene, located on the Y chromosome, is the master switch that triggers the development of male physical characteristics. In the absence of this gene, the embryo develops into a female.
• Probability of Sex: Statistically, there is a 50:50 chance of having a boy or a girl in each pregnancy, as the segregation of X and Y chromosomes into sperm occurs randomly during meiosis.
• Genetic vs. Environmental Determination: Unlike some reptiles (such as crocodiles and turtles) where the incubation temperature of the eggs determines the sex, human sex determination is strictly genetic and is not influenced by environmental conditions during the development of the foetus.
• Chromosomal Abnormalities: Sometimes, errors in cell division can lead to variations in the 23^rd pair, such as Turner syndrome (XO) or Klinefelter syndrome (XXY), which affect physical and reproductive development.

Important Points

• Mendelian Inheritance: The inheritance of sex follows Mendel’s laws of segregation, where alleles (in this case, sex chromosomes) separate during gamete formation.
• Historical Context: The discovery of the XY sex-determination system was made by Nettie Stevens and Edmund Beecher Wilson independently in 1905.
• Misconceptions: It is a common biological fact that the mother does not determine the sex of the child; however, in many cultures, women have been historically blamed for not giving birth to a male child despite the biological determination coming from the paternal sperm.

The correct answer is Both force and displacement, and the angle between them.

Key Points

• Work done by a constant force is defined as the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between the force and the displacement vectors.
• The mathematical expression for work is W = Fs cos θ, where W is the work done, F is the constant force, s is the displacement, and θ is the angle between the force vector and the displacement vector.
• It depends on the magnitude of force: If the force applied is doubled while keeping other factors constant, the work done will also double. Hence, option 3 is only a partial factor.
• It depends on the displacement: If an object does not move (s = 0), no work is done, regardless of how much force is applied. An example is pushing against a stationary brick wall. Hence, option 1 is also only a partial factor.
• It depends on the angle (θ): The direction of the force relative to the motion is crucial. If the force is perpendicular to the displacement (θ = 90°), the work done is zero because cos 90° = 0.
• Mass is not a direct variable in the basic formula for work done by a constant force, although it may influence the force required to move an object (via F = ma). Hence, statement 4 is incorrect.
• Hence, the correct answer is Both force and displacement, and the angle between them.

Additional Information

• Nature of Work:
• Positive Work: Occurs when the angle θ is between 0° and 90°. In this case, the force has a component in the direction of the displacement. An example is lifting an object upwards where the applied force is in the direction of movement.
• Negative Work: Occurs when the angle θ is between 90° and 180°. The force acts in a direction opposite to the displacement. Frictional force doing work on a sliding block is a classic example of negative work.
• Zero Work: Occurs when the force is perpendicular to displacement or either force or displacement is zero. A coolie carrying a load on his head and walking on a horizontal platform does zero work against gravity because the force of gravity is vertical while the displacement is horizontal.
• Units and Dimensions:
• The SI unit of work is the Joule (J). 1 Joule is defined as the work done by a force of 1 Newton causing a displacement of 1 meter.
• In the CGS system, the unit of work is the Erg. The relationship between them is 1 Joule = 10⁷ Ergs.
• The dimensional formula for work is [ML²T⁻²], which is the same as that for energy and torque.

Important Points

• Scalar Quantity: Even though force and displacement are vector quantities, their dot product, Work, is a scalar quantity. This means it only has magnitude and no specific direction in space.
• Work-Energy Theorem: This theorem states that the net work done on an object is equal to the change in its Kinetic Energy (KE). This connects the concept of work directly to the velocity and mass of the object.
• External vs Internal Force: The calculation of work depends on the specific force being considered. For example, while gravity does work on a falling apple, air resistance simultaneously does negative work on it.

The correct answer is Vibration.

Key Points

• Vibration is defined as the rapid to-and-fro or back-and-forth motion of an object about its mean position. This mechanical motion is the fundamental requirement for the production of sound. Hence, the correct answer is Vibration.
• Every source of sound, whether it is a musical instrument like a guitar string, a tuning fork, or the human vocal cords, must undergo vibration to generate pressure waves in the surrounding medium.
• When an object vibrates, it transfers its energy to the particles of the medium (solid, liquid, or gas) surrounding it. These particles then start vibrating, which leads to the propagation of sound waves.
• The frequency of these vibrations determines the pitch of the sound produced. A higher frequency of vibration results in a high-pitched sound, while a lower frequency results in a low-pitched sound.
• The amplitude of the vibration determines the loudness or intensity of the sound. Larger vibrations (greater displacement from the mean position) produce louder sounds.

Additional Information

• Compression:
• In a longitudinal wave (like sound), compression is the region where the particles of the medium are crowded together.
• This region is characterized by high pressure and high density. Hence, statement 4 is incorrect as it describes a state of the medium, not the motion of the source.
• Rarefaction:
• Rarefaction is the region in a longitudinal wave where the particles are spread apart.
• This area exhibits low pressure and low density. Hence, statement 1 is incorrect because it refers to the wave's structure rather than the initiating motion.
• Refraction:
• Refraction is a phenomenon where a wave changes direction (bends) as it passes from one medium to another with a different optical or acoustic density.
• It occurs due to a change in the speed of the wave. Hence, statement 2 is incorrect as it relates to wave behavior at boundaries, not sound production.
• Characteristics of Sound Waves:
• Sound waves are mechanical waves and require a material medium for travel. They cannot propagate through a vacuum.
• The speed of sound in dry air at 20 degrees Celsius (68 degrees Fahrenheit) is approximately 343 meters per second.
• Sound travels fastest in solids (e.g., steel at approx 5960 m/s), followed by liquids (e.g., water), and slowest in gases.

Important Points

• Human Audibility: The range of audible frequencies for the human ear is typically between 20 Hz and 20,000 Hz (20 kHz).
• Infrasound: Frequencies below 20 Hz are known as infrasonic sounds, often produced by earthquakes or whales.
• Ultrasound: Frequencies above 20,000 Hz are called ultrasonic sounds, used in medical imaging (sonography) and by bats for navigation.
• Oscillation: While often used interchangeably with vibration, oscillation usually refers to slower, more rhythmic back-and-forth movements (like a pendulum), whereas vibration implies a more rapid movement.
• Echo: The reflection of sound arriving at the listener with a delay after the direct sound is known as an echo. To hear a distinct echo, the minimum distance from the reflecting surface should be approximately 17.2 meters.

The correct answer is Weight of the book and normal reaction by the table.

Key Points

• When a book is lying stationary on a table, it is in a state of static equilibrium, meaning the net force acting on the object is zero.
• The Weight of the book is the gravitational force exerted by the Earth on the book, acting vertically downwards toward the center of the Earth.
• The Normal reaction force is the support force exerted by the surface of the table onto the book, acting vertically upwards, perpendicular to the surface.
• According to Newton's First Law of Motion, an object at rest stays at rest unless acted upon by an unbalanced force; since the book is not moving, the upward normal force must exactly magnitude-wise equal the downward gravitational force.
• These two forces are balanced forces because they act on the same object (the book), are equal in magnitude, and opposite in direction, resulting in zero acceleration. Hence, the correct answer is Weight of the book and normal reaction by the table.

Additional Information

• Newton's First Law (Law of Inertia): It states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. In this scenario, the book's velocity is 0 and remains 0 because the forces are balanced.
• Weight vs. Mass: Mass is the amount of matter in the book (measured in kilograms), while Weight is a force (measured in Newtons) calculated as W = mg, where g is the acceleration due to gravity (approximately 9.8 m/s²).
• Frictional Force: Friction acts parallel to the surface and opposes relative motion. If the book is just lying on a horizontal table without any horizontal external force trying to push it, the static friction is zero. Hence, options involving friction are incorrect for a book simply lying at rest.
• Balanced vs. Unbalanced Forces: Balanced forces do not change the state of motion of an object. Unbalanced forces cause acceleration (change in speed or direction) as per Newton's Second Law (F = ma).

Important Points

• Newton's Third Law Confusion: A common misconception is that the Weight and the Normal Reaction are an action-reaction pair. This is incorrect.
• Action-Reaction pairs act on different bodies (e.g., Book pushes Table down, Table pushes Book up).
• Balanced forces (Weight and Normal force here) act on the same body (the Book).
• Free Body Diagram (FBD): In a Free Body Diagram of the book, you would draw two arrows originating from the center of the book: one pointing down labeled W (mg) and one pointing up labeled N (Normal Force). Since there is no movement in the y-axis, N - mg = 0, or N = mg.
• Nature of Normal Force: The Normal Force is an electromagnetic force resulting from the atoms of the table resisting being compressed by the atoms of the book.

The correct answer is a and c.

Key Points

• Indicators are substances that change their color or odor when added to an acidic or basic solution. They are essential in acid-base titration to determine the endpoint of a reaction.
• Turmeric is a well-known natural indicator. It contains curcumin, which remains yellow in acidic and neutral environments but changes its color to reddish-brown when it comes into contact with a base (such as soap or sodium hydroxide). Hence, statement a is correct.
• Phenolphthalein is a synthetic indicator often used in laboratories. It remains colourless in acidic and neutral solutions. However, it turns pink or magenta in a basic medium once the pH level exceeds approximately 8.2. The statement provided reverses these characteristics. Hence, statement b is incorrect.
• Olfactory indicators are specific substances whose smell changes in acidic or basic media. This property allows individuals, including those with visual impairments, to identify the nature of a solution. Clove oil is a classic example of an olfactory indicator, along with onion and vanilla essence. Hence, statement c is correct.
• Hence, the correct answer is a and c.

Additional Information

• Classification of Indicators:
• Natural Indicators: These are obtained from natural sources like plants. Examples include Litmus (extracted from Lichens), Red Cabbage extract, and Turmeric. Litmus turns red in acid and blue in base.
• Synthetic Indicators: These are man-made chemicals. Methyl orange and Phenolphthalein are the most common. Methyl orange gives a red color in acidic solutions and a yellow color in basic solutions.
• Olfactory Indicators: These work based on odor. In a basic medium, the characteristic smell of onion or clove oil is generally destroyed, while it persists in an acidic medium.
• Uses and Applications:
• Titration: Indicators are used to find the equivalence point where the acid and base have neutralized each other.
• Soil Testing: Farmers use indicators to check the pH of the soil to ensure optimal growth for specific crops.
• Water Treatment: Testing the alkalinity or acidity of water in swimming pools or industrial effluents.

Important Points

• Universal Indicator: Unlike simple indicators, a Universal Indicator is a mixture of several indicators. It shows different colors at different concentrations of hydrogen ions (pH scale from 0 to 14).
• Clove Oil Specifics: In the presence of hydrochloric acid (acid), the smell of clove oil is retained, but when mixed with sodium hydroxide (base), the smell cannot be detected.
• Phenolphthalein Range: It is widely used because its color transition is very sharp and occurs near the neutralization point for strong acid-strong base titrations.

The correct answer is Phloem.

Key Points

• Complex permanent tissue is defined as a collection of different types of cells that work together as a single unit to perform a common specialized function.
• Phloem is a prime example of complex permanent tissue because it consists of four distinct types of cells: sieve tubes, companion cells, phloem parenchyma, and phloem fibers. Hence, statement 3 is correct.
• The primary function of Phloem is the translocation of soluble organic compounds, specifically sucrose, from the leaves (the source) to the roots and other storage organs (the sink).
• Unlike simple tissues, complex tissues like Phloem and Xylem are vital for the survival of vascular plants as they facilitate the long-distance transport of nutrients and water.
• Lateral meristem is not a permanent tissue; it is a type of meristematic tissue responsible for the secondary growth (increase in diameter) of the plant stem and roots. Hence, statement 1 is incorrect.
• Aerenchyma is a specialized type of simple permanent tissue (a variation of parenchyma) that contains large air cavities, providing buoyancy to aquatic plants. Hence, statement 2 is incorrect.
• Collenchyma is a simple permanent tissue composed of living cells with unevenly thickened walls (due to cellulose and pectin) that provide mechanical support and flexibility. Hence, statement 4 is incorrect.

Additional Information

• Xylem:
• Along with phloem, Xylem is the second type of complex permanent tissue in plants.
• It is responsible for the conduction of water and minerals from the roots to the aerial parts of the plant.
• It consists of four elements: tracheids, vessels, xylem parenchyma, and xylem fibers.
• Except for xylem parenchyma, all other xylem cells are dead at maturity.
• Simple Permanent Tissue:
• These tissues are made up of only one type of cell which are similar in structure and function.
• Parenchyma: The most common tissue, involved in photosynthesis, storage, and secretion.
• Sclerenchyma: Consists of long, narrow cells with thick lignified walls; it provides structural strength and is usually dead.
• Vascular Bundles:
• In plants, Xylem and Phloem are organized into vascular bundles, which act as the plumbing system of the plant.
• The arrangement of these bundles varies between monocots and dicots.

Important Points

• Direction of Flow: In Xylem, the transport is strictly unidirectional (upwards from roots), whereas in Phloem, the transport is bidirectional (multi-directional) to ensure nutrients reach all parts of the plant.
• Companion Cells: These are specialized parenchyma cells in the Phloem that help regulate the metabolic activities of sieve tube elements, which lack a nucleus at maturity.
• Sieve Plates: The end walls of sieve tube cells have pores and are called sieve plates, allowing the flow of cytoplasmic strands between adjacent cells.
• Secondary Growth: The Lateral meristems, specifically the vascular cambium, produce new layers of secondary Xylem and secondary Phloem every year in woody plants.

The correct answer is They are brittle and poor conductors of electricity.

Key Points

• Non-metals are elements that do not possess the properties of metals, such as being able to conduct electricity or heat, or being malleable or ductile.
• One of the most defining physical properties of solid non-metals is brittleness. Unlike metals, which can be hammered into thin sheets, solid non-metals like Sulfur and Phosphorus will shatter or break into pieces when struck. Hence, statement 1 is correct.
• Non-metals are generally poor conductors of heat and electricity. This is because, unlike metals, they do not have a "sea of free electrons" available to carry thermal energy or electric charge. Hence, statement 3 is incorrect.
• Properties like being malleable (capable of being beaten into sheets) and sonorous (producing a ringing sound when hit) are characteristic of metals, not non-metals. Hence, statement 2 is incorrect.
• Non-metals lack metallic lustre; they are usually dull in appearance and cannot be drawn into wires (ductility). Hence, statement 4 is incorrect.
• In the Periodic Table, non-metals are primarily located on the right-hand side, specifically within groups 14, 15, 16, 17, and 18, with the exception of Hydrogen, which is placed in Group 1.

Additional Information

• Physical States: Non-metals can exist in all three states of matter at room temperature. For example, Carbon and Sulfur are solids, Bromine is a liquid, and Oxygen and Nitrogen are gases.
• Bromine: It is the only non-metal that remains in a liquid state at standard room temperature and pressure. It is a reddish-brown liquid with a suffocating odor.
• Exceptions to Conductivity: While most non-metals are insulators, Graphite (an allotrope of Carbon) is an excellent conductor of electricity due to its unique layered structure that allows electron mobility.
• Exceptions to Hardness and Lustre: Diamond, another allotrope of Carbon, is the hardest natural substance known on Earth. Furthermore, while non-metals are usually dull, Iodine crystals exhibit a distinct metallic-like lustre (shininess).
• Density and Melting Points: Generally, non-metals have low densities and low melting and boiling points compared to metals, making them less dense and more easily vaporized.

Important Points

• Chemical Properties: Non-metals are electronegative in nature. They tend to gain electrons during chemical reactions to form anions (negatively charged ions).
• Nature of Oxides: When non-metals react with Oxygen, they typically form acidic oxides (like CO2 or SO2) or neutral oxides (like N2O or CO). This contrasts with metals, which usually form basic oxides.
• Common Examples and Uses:
• Hydrogen: Used in the manufacture of ammonia and as rocket fuel.
• Oxygen: Essential for respiration and combustion processes.
• Nitrogen: Used to provide an inert atmosphere and in the production of fertilizers.
• Chlorine: Widely used for water purification and as a bleaching agent.
• Silicon: A metalloid (sharing some non-metal properties) used extensively in the semiconductor industry for making microchips.
• Valence Electrons: Most non-metals have 4 to 8 electrons in their outermost shell, which determines their high ionization energy and chemical reactivity.

The correct answer is Prostate and seminal vesicles.

Key Points

• The male reproductive system consists of various accessory glands, primarily the seminal vesicles, the prostate gland, and the bulbourethral glands.
• These glands play a crucial role in reproduction by secreting fluids that combine with sperms to form semen. These secretions are collectively known as seminal plasma.
• The seminal vesicles secrete a significant portion of the fluid that contains fructose, which serves as the primary nutrition and energy source for sperm motility. Hence, the role of seminal vesicles in nutrition is vital.
• The prostate gland contributes a milky, slightly alkaline fluid that contains enzymes and citric acid. This fluid helps in the transport of sperm and enhances their motility while neutralizing the acidic environment of the female reproductive tract. Hence, the prostate gland is essential for easier transport.
• Since both the prostate and seminal vesicles contribute to these specific functions of nutrition and transport, the combined option is the most accurate. Hence, the correct answer is Prostate and seminal vesicles.

Additional Information

• Seminal Vesicles:
• These are a pair of simple tubular glands located behind the urinary bladder in males.
• They produce approximately 60% to 70% of the total volume of semen.
• The fluid is rich in proteins, enzymes, fructose, vitamin C, and prostaglandins. The prostaglandins help in the movement of sperm by inducing contractions in the female reproductive tract.
• Prostate Gland:
• It is a single, walnut-sized gland located just below the bladder and surrounds the urethra.
• It secretes a thin, milky fluid that makes up about 20% to 30% of the semen.
• Its alkaline nature protects the sperms from the acidic nature of the male urethra and the female vagina, ensuring their survival.
• Bulbourethral Glands (Cowper's Glands):
• These are small, pea-sized glands located inferior to the prostate gland.
• They secrete an alkaline mucus during sexual arousal that lubricates the penis and neutralizes any acidic urine remaining in the urethra.
• Pineal Gland:
• Unlike the reproductive glands, the Pineal gland is an endocrine gland located in the epithalamus of the brain.
• It is responsible for secreting the hormone melatonin, which regulates the body's circadian rhythm and sleep-wake cycles. It has no direct involvement in sperm transport or nutrition.

Important Points

• Semen Composition: Semen is a mixture of sperms (produced in the testes) and seminal plasma (the collective secretions of the accessory glands).
• Fructose: It is an important forensic marker because it is produced only by the seminal vesicles and is not found elsewhere in the body's secretions.
• Sperm Motility: Without the secretions from the prostate and seminal vesicles, sperms would lack the energy and the fluid medium necessary to reach the egg for fertilization.

The correct answer is CaSO4 . 2H2O + heat -> CaSO4 . 1/2H2O + 1 1/2H2O.

Key Points

• Plaster of Paris (POP) is chemically known as Calcium Sulphate Hemihydrate. Its chemical formula is CaSO4 . 1/2H2O.
• It is prepared from Gypsum, which is Calcium Sulphate Dihydrate (CaSO4 . 2H2O).
• The preparation involves heating Gypsum at a carefully controlled temperature of 373 K (which equals 100 degrees Celsius).
• At this temperature, Gypsum loses one and a half molecules (or three-fourths) of its water of crystallization to become the hemihydrate form. Hence, option 3 is correct.
• In the chemical reaction CaSO4 . 2H2O + heat -> CaSO4 . 1/2H2O + 1 1/2H2O, the reactant is Gypsum and the product is Plaster of Paris.
• Option 1 represents the setting process where Plaster of Paris reacts with water to turn back into hard Gypsum. Hence, option 1 is incorrect.
• Option 2 shows the total dehydration of Gypsum which occurs when it is heated strongly above 373 K, resulting in dead burnt plaster. Hence, option 2 is incorrect.
• Option 4 shows the thermal decomposition of Calcium Carbonate (Limestone) into Calcium Oxide (Quicklime) and Carbon Dioxide. Hence, option 4 is incorrect.

Additional Information

• Origin of Name: It is called Plaster of Paris because Gypsum was found in abundance in the Montmartre district of Paris, where it was first used for architectural purposes on a large scale.
• Medical Uses: One of the most common applications of POP is in orthopedics. Doctors use it as a bandage or cast to support and immobilize fractured bones, ensuring they heal in the correct location and alignment.
• Dead Burnt Plaster: If Gypsum is heated to a temperature higher than 373 K, all the water of crystallization is removed. The resulting Anhydrous Calcium Sulphate (CaSO4) does not set like POP when mixed with water and is termed dead burnt plaster.
• Physical Properties: Plaster of Paris is a white powder. A unique feature is its ability to expand slightly upon setting, which makes it excellent for casting molds as it picks up the fine details of the cast.
• Artistic and Industrial Uses: Apart from construction, it is used for making statues, toys, cheap ornaments, and blackboard chalk. In dentistry, it is used to make impressions of teeth.
• Chemical Detail: In the formula CaSO4 . 1/2H2O, it is understood that two formula units of CaSO4 share one molecule of water (2CaSO4 . H2O).

Important Points

• Storage: Plaster of Paris should be stored in moisture-proof containers because it absorbs moisture from the air and undergoes hydration to form a hard mass of Gypsum, rendering it useless.
• Reaction Temperature: Temperature control is vital. If the temperature is not maintained at exactly 373 K, the quality and setting properties of the plaster are compromised.
• Water Ratio: When mixing POP for use, the water-to-powder ratio determines the strength and setting time of the final product.

The correct answer is PO4-n.

Key Points

• A polyatomic ion, also known as a molecular ion, is a charged chemical species composed of two or more atoms covalently bonded that act as a single unit. The "n" value in the context of these ions refers to the magnitude of the ionic charge.
• PO4-n (Phosphate Ion): The phosphate ion consists of one phosphorus atom surrounded by four oxygen atoms in a tetrahedral arrangement. Its chemical formula is PO4 3-. Therefore, the value of n is 3. Hence, this statement is correct.
• SO4-n (Sulfate Ion): The sulfate ion is a polyatomic anion consisting of one sulfur atom and four oxygen atoms. The standard charge for a sulfate ion is -2, which means the n value is 2. Hence, this statement is incorrect.
• CO3-n (Carbonate Ion): The carbonate ion is composed of one carbon atom and three oxygen atoms. It carries a net charge of -2, making the n value equal to 2. Hence, this statement is incorrect.
• HCO3-n (Bicarbonate Ion): Also known as the hydrogen carbonate ion, it consists of one hydrogen, one carbon, and three oxygen atoms. It carries a charge of -1, meaning the n value is 1. Hence, this statement is incorrect.

Additional Information

• Phosphate Ion (PO4 3-):
• This ion is trivalent, meaning it has a valency of 3.
• It is a vital component in biological systems, forming the backbone of DNA and RNA.
• It is also a key part of ATP (Adenosine Triphosphate), which is the primary energy currency of the cell.
• In agriculture, phosphates are used extensively in fertilizers to promote plant growth.
• Sulfate Ion (SO4 2-):
• Sulfates are widely used in the chemical industry and are found in common substances like Epsom salts (Magnesium sulfate) and Gypsum (Calcium sulfate).
• In environmental science, atmospheric sulfates are major contributors to acid rain.
• Carbonate (CO3 2-) and Bicarbonate (HCO3 -) Ions:
• Carbonates are found in minerals like Calcite and are the primary component of limestone and shells of marine organisms.
• Bicarbonate plays a crucial role in the human body as a buffer system, helping to maintain the pH balance of blood (keeping it around 7.4).

Important Points

• The charge of a polyatomic ion is the result of the total number of electrons being different from the total number of protons in the combined group of atoms.
• Most polyatomic ions are oxyanions, which are polyatomic ions that contain oxygen. Examples include Nitrate (NO3 -), Nitrite (NO2 -), and Sulfite (SO3 2-).
• Valency vs. Charge: While valency refers to the combining capacity of an element or ion, the charge (n) indicates the actual electrical state. For the ions mentioned, the magnitude of the charge matches the common valency.
• Ammonium (NH4 +) is one of the few common positively charged (cationic) polyatomic ions, with an n value of 1.

Shortcut Trick

If all linear dimensions (length, breadth, and height) of a 3D figure are multiplied by a factor k, then its surface area increases by a factor of k².

Here, every dimension is doubled, so k = 2.

New Area = Original Area × (2)²

⇒ New Area = 4 × Original Area

∴ The correct answer is 4 times.

Alternate Method

Given:

Original dimensions: Length = L, Breadth = B, Height = H

New dimensions: Length (L₁) = 2L, Breadth (B₁) = 2B, Height (H₁) = 2H

Formula Used:

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

Calculations:

⇒ Original TSA = 2(LB + BH + HL)

⇒ New TSA = 2(L₁B₁ + B₁H₁ + H₁L₁)

⇒ New TSA = 2[(2L × 2B) + (2B × 2H) + (2H × 2L)]

⇒ New TSA = 2[4LB + 4BH + 4HL]

⇒ New TSA = 4 × [2(LB + BH + HL)]

⇒ New TSA = 4 × Original TSA

∴ The correct answer is 4 times.

Additional Information

Volume Scaling

If each dimension of a cuboid is multiplied by k, its volume becomes k³ times the original volume. For doubling (k=2), volume becomes 8 times.

Lateral Surface Area

The lateral surface area of a cuboid is 2h(l + b). If dimensions are doubled, the lateral surface area also becomes 4 times.

Surface Area of a Cube

For a cube with side a, the total surface area is 6a². If the side is doubled, the area becomes 6(2a)² = 24a², which is 4 times the original.

Shortcut Trick

Use the Algebraic Identity: (x − y)² = (x + y)² − 4xy

Given Sum (x + y) = 10 and Product (xy) = 24.96

Substitute the values: (x − y)² = 10² − 4 × 24.96

(x − y)² = 100 − 99.84 = 0.16

∴ The correct answer is 0.16.

Alternate Method

Given:

Sum of variables (x + y) = 10

Product of variables (xy) = 24.96

Formula Used:

The standard expansion of the square of a binomial leads to the relationship:

(x − y)² = (x + y)² − 4xy

Calculations:

⇒ (x + y)² = 10² = 100

⇒ 4xy = 4 × 24.96

⇒ 4xy = 99.84

⇒ (x − y)² = 100 − 99.84

⇒ (x − y)² = 0.16

∴ The correct answer is 0.16.

Additional Information

Square of a Sum

The formula is (a + b)² = a² + b² + 2ab.

Square of a Difference

The formula is (a − b)² = a² + b² − 2ab.

Difference of Squares

The relation (a + b)² − (a − b)² = 4ab is useful for finding the difference when the sum and product are known.

Sum of Squares

The value of a² + b² can be found using (a + b)² − 2ab.

Shortcut Trick

Convert mixed fractions: 8 1/4 = 33/4, 4 1/7 = 29/7, 5 1/2 = 11/2, 1 1/7 = 8/7, 5 1/3 = 16/3

First solve “of”: 12/11 of 33/4 = 9

Then 2/3 ÷ 9 = 2/27

Now simplify: 2/27 + 29/7 − 44/7 + 16/3

⇒ 2/27 − 15/7 + 16/3 = 617/189

∴ Answer = 617/189

Alternate Method

Given:

2/3 ÷ (12/11 of 8 1/4) + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

Concept Used:

BODMAS rule

Calculation:

⇒ 2/3 ÷ (12/11 × 33/4) + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

⇒ 2/3 ÷ (396/44) + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

⇒ 2/3 ÷ 9 + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

⇒ 2/3 × 1/9 + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

⇒ 2/27 + 4 1/7 − 5 1/2 × 1 1/7 + 5 1/3

⇒ 2/27 + 29/7 − 11/2 × 8/7 + 16/3

⇒ 2/27 + 29/7 − 88/14 + 16/3

⇒ 2/27 + 29/7 − 44/7 + 16/3

⇒ 2/27 − 15/7 + 16/3

⇒ LCM = 189

⇒ 2/27 = 14/189

⇒ −15/7 = −405/189

⇒ 16/3 = 1008/189

⇒ 14 − 405 + 1008 = 617

⇒ Result = 617/189

∴ The correct answer is 617/189.

Additional Information

BODMAS Full Form

Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Order of Operations

Solve ‘of’ first, then division and multiplication from left to right.

Mixed Fraction Conversion

a b/c = (ac + b)/c before applying operations.

LCM Technique

Use common denominator to combine multiple fractions efficiently.

Shortcut Trick

Apply MDW formula: (M × D) ÷ W = Constant.

Case 1: (5M + 2W) × 3 ÷ 48 ⇒ (5M + 2W) ÷ 16

Case 2: (3M + 1W) × 6 ÷ 54 ⇒ (3M + 1W) ÷ 9

Equating: 9(5M + 2W) = 16(3M + 1W) ⇒ 45M + 18W = 48M + 16W ⇒ 2W = 3M ⇒ 1M = (2/3)W.

Substitute in Case 3: (6M + xW) × 14 ÷ 378 ⇒ (4W + xW) ÷ 27. Equating with Case 2: (3W) ÷ 9 = (4+x)W ÷ 27 ⇒ 9 = 4 + x ⇒ x = 5.

∴ The correct answer is 5 women.

Alternate Method

Given: 5 men and 2 women reap 48 hectares in 3 days; 3 men and 1 woman reap 54 hectares in 6 days.

Formula Used: (M₁ × D₁) ÷ W₁ = (M₂ × D₂) ÷ W₂

⇒ (5M + 2W) × 3 / 48 = (3M + 1W) × 6 / 54

⇒ (5M + 2W) / 16 = (3M + 1W) / 9

⇒ 45M + 18W = 48M + 16W

⇒ 2W = 3M ⇒ Efficiency ratio M:W = 2:3

⇒ Total work rate from Case 2: (3 × 2 + 1 × 3) × 6 / 54 = 9 × 6 / 54 = 1 unit

⇒ For Case 3: (6 × 2 + x × 3) × 14 / 378 = 1

⇒ (12 + 3x) × 14 = 378

⇒ 12 + 3x = 378 / 14 = 27

⇒ 3x = 15 ⇒ x = 5

∴ The correct answer is 5 women.

Additional Information

Man-Days-Work Relation

The amount of work done is directly proportional to the number of men and days: M₁D₁H₁/W₁ = M₂D₂H₂/W₂.

Efficiency Ratio

If M men can do as much work as W women, their efficiency ratio is inversely proportional to their numbers for the same work.

Work and Wages

Wages are always distributed in the ratio of the actual work done by each person, which is Efficiency × Time.

Shortcut Trick

If 20% is added to 72 to get 100%, then 72 must represent the remaining 80%.

80% of Number = 72

10% of Number = 72 ÷ 8 = 9

70% of Number = 9 × 7 = 63

∴ The correct answer is 63.

Alternate Method

Let the number be x.

According to the question:

⇒ 20% of x + 72 = x

⇒ 0.2x + 72 = x

⇒ x − 0.2x = 72

⇒ 0.8x = 72

⇒ x = 72 ÷ 0.8 = 90

Now, find 70% of the number:

⇒ 70% of 90 = (70/100) × 90

⇒ 0.7 × 90 = 63

∴ The correct answer is 63.

Additional Information

Percentage Change

Percentage change is calculated as (Change / Original Value) × 100.

Fraction Conversion

To convert a percentage to a fraction, divide by 100 (e.g., 20% = 1/5, 80% = 4/5).

Base Value

In problems like this, the 'number' is always treated as 100% (the base).

Shortcut Trick

Rate of typing = 2 pages / 40 minutes = 1 page per 20 minutes.

Pages in 60 minutes = 60 ÷ 20 = 3 pages.

∴ The correct answer is 3 pages.

Alternate Method

Given:

Time 1 (T₁) = 40 minutes

Work 1 (W₁) = 2 pages

Time 2 (T₂) = 60 minutes

Formula Used:

Work is directly proportional to time: W₁ / T₁ = W₂ / T₂

Calculations:

⇒ 2 / 40 = W₂ / 60

⇒ 1 / 20 = W₂ / 60

⇒ W₂ = 60 / 20

⇒ W₂ = 3 pages

∴ The correct answer is 3 pages.

Additional Information

Direct Proportion

When one quantity increases as the other increases at a constant rate, it is Direct Proportion (Work ∝ Time).

Unitary Method

First find the value of a single unit (e.g., pages per minute) and then multiply by the required quantity.

Work, Men, and Time Relation

The general formula is (M₁ × D₁ × T₁) / W₁ = (M₂ × D₂ × T₂) / W₂.

Shortcut Trick

Ratio of Income : Expenditure = 2 : 1 ⇒ Saving = 2 − 1 = 1 unit.

Initial Expenditure (1 unit) = ₹35,000 ⇒ Initial Income (2 units) = ₹70,000.

New Income = 70,000 × 130% = ₹91,000.

New Expenditure = 35,000 × 90% = ₹31,500.

Final Saving = 91,000 − 31,500 = ₹59,500.

∴ The correct answer is ₹59,500.

Alternate Method

Given Ratio = 2 : 1 (Income : Expenditure)

Given Initial Expenditure = ₹35,000

⇒ 1 unit = ₹35,000

⇒ Initial Income = 2 × 35,000 = ₹70,000

⇒ Increase in Income = 30% of 70,000 = ₹21,000

⇒ New Income = 70,000 + 21,000 = ₹91,000

⇒ Decrease in Expenditure = 10% of 35,000 = ₹3,500

⇒ New Expenditure = 35,000 − 3,500 = ₹31,500

⇒ Final Saving = New Income − New Expenditure

⇒ Final Saving = 91,000 − 31,500 = ₹59,500

∴ The correct answer is ₹59,500.

Additional Information

Basic Income Formula

The relationship between income, expenditure, and savings is always Income = Expenditure + Savings.

Percentage Change

New Value = Original Value × (100 ± Change%)/100. Use (+) for increase and (−) for decrease.

Savings Percentage

The percentage of savings is calculated as (Savings ÷ Income) × 100.

Shortcut Trick

Put a specific value for θ, let θ = 0°.

√((1 + sin 0°)/(1 − sin 0°)) = √((1 + 0)/(1 − 0)) = √1 = 1.

Check options at θ = 0°: (1) sec 0° + tan 0° = 1 + 0 = 1.

Since option (1) matches the value, it is the correct expression.

∴ The correct answer is secθ + tanθ.

Alternate Method

Given: √((1 + sinθ)/(1 − sinθ))

Formula Used: Rationalization, sin²θ + cos²θ = 1, secθ = 1/cosθ, tanθ = sinθ/cosθ

⇒ Multiply numerator and denominator by (1 + sinθ) under the square root.

⇒ √([(1 + sinθ)(1 + sinθ)] / [(1 − sinθ)(1 + sinθ)])

⇒ √((1 + sinθ)² / (1 − sin²θ))

⇒ √((1 + sinθ)² / cos²θ)

⇒ (1 + sinθ) / cosθ

⇒ (1 / cosθ) + (sinθ / cosθ)

⇒ secθ + tanθ

∴ The correct answer is secθ + tanθ.

Additional Information

Trigonometric Identities

The three fundamental identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Conjugate Multiplication

In square root expressions involving (1 ± sinθ) or (1 ± cosθ), rationalizing the denominator using its conjugate often simplifies the expression to a perfect square.

Shortcut Trick

Let the Assumed Mean be 60.

Calculate deviations from 60: (+1, −27, +34, −12, −24, +33, +13, +34, −7).

Sum of deviations = (1 + 34 + 33 + 13 + 34) − (27 + 12 + 24 + 7) = 115 − 70 = 45.

Average deviation = 45 ÷ 9 = 5.

Arithmetic Mean = Assumed Mean + Average Deviation = 60 + 5 = 65.

∴ The correct answer is 65.

Alternate Method

Given: Observations = 61, 33, 94, 48, 36, 93, 73, 94, 53

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

⇒ Total number of observations (n) = 9

⇒ Sum of observations = 61 + 33 + 94 + 48 + 36 + 93 + 73 + 94 + 53

⇒ Sum of observations = 585

⇒ Arithmetic Mean = 585 ÷ 9

⇒ Arithmetic Mean = 65

∴ The correct answer is 65.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of a set of observations from their arithmetic mean is always zero: Σ(x_i − Mean) = 0.

Weighted Arithmetic Mean

When different observations have different levels of importance, the mean is calculated as Σ(w_i × x_i) ÷ Σw_i, where w_i is the weight of each observation.

Change in Origin and Scale

If a constant 'k' is added to or multiplied with every observation, the new mean will also be increased by 'k' or multiplied by 'k' respectively.

Shortcut Trick

The Buy X, Get Y Free scheme means you pay for X items but receive (X + Y) items in total.

Discount% = (Free Items ÷ Total Items) × 100.

Here, Free Items = 8 and Total Items = 2 + 8 = 10.

Discount% = (8 ÷ 10) × 100 = 80%.

∴ The correct answer is 80%.

Alternate Method

Given → Number of Purchased Items = 2, Number of Free Items = 8.

Formulas: Discount% = (Discount ÷ MP) × 100 | MP = Price of total items received.

⇒ Let the Marked Price (MP) of 1 item be Re. 1.

⇒ Total items given to the customer = 2 + 8 = 10 items.

⇒ Total MP of 10 items = 10 × 1 = Rs. 10.

⇒ The customer pays for only 2 items, so Selling Price (SP) = Rs. 2.

⇒ Discount amount = MP − SP = 10 − 2 = Rs. 8.

⇒ Discount% = (8 ÷ 10) × 100 = 80%.

∴ The correct answer is 80%.

Additional Information

Marked Price and Discount

In "Buy X Get Y Free" schemes, the Marked Price is always calculated on the total number of items (X+Y) handed over to the customer.

Effective Discount Formula

Effective Discount% = [Free Items ÷ (Paid Items + Free Items)] × 100.

Shortcut Trick

Excess runs scored above the old average = 219 − 88 = 131 runs.

These extra runs are distributed equally among all 11 innings to find the average increase.

Average increase = 131 ÷ 11 ≈ 11.909 runs.

Rounding off to the nearest whole number, we get 12 runs.

∴ The correct answer is 12 runs.

Alternate Method

Given:

Average of 10 innings = 88 runs

Score in 11^th inning = 219 runs

Calculations:

⇒ Total runs in 10 innings = 88 × 10 = 880 runs

⇒ Total runs in 11 innings = 880 + 219 = 1099 runs

⇒ New average = Total runs ÷ Total innings = 1099 ÷ 11

⇒ New average = 99.9090... runs

⇒ Increase in average = New average − Old average

⇒ Increase = 99.909 − 88 = 11.909 runs

⇒ Rounded increase ≈ 12 runs

∴ The correct answer is 12 runs.

Additional Information

Basic Average Formula

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

Net Deviation Method

If a new value is added, the change in average = (New Value − Old Average) ÷ New Total Number of items.

Weighted Average

When combining two groups, New Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is count and x is average.

Shortcut Trick

To convert Speed from km/hr to m/sec, multiply the speed by the factor 5/18.

Speed = 126 × (5/18) m/sec

Speed = 7 × 5 = 35 m/sec

∴ The correct answer is 35 m/sec.

Alternate Method

Given:

Speed in km/hr = 126 km/hr

Formula Used:

Speed (m/sec) = Speed (km/hr) × (1000 meters / 3600 seconds)

Speed (m/sec) = Speed (km/hr) × (5/18)

Calculations:

⇒ Speed = 126 × (5/18)

⇒ Speed = (126 ÷ 18) × 5

⇒ Speed = 7 × 5

⇒ Speed = 35 m/sec

∴ The correct answer is 35 m/sec.

Additional Information

Conversion from m/sec to km/hr

To convert speed from meters per second to kilometers per hour, multiply the value by 18/5.

Basic Speed Formula

Speed is defined as the Distance traveled per unit Time. Formula: Speed = Distance ÷ Time.

Average Speed

For a journey with two equal distances at speeds x and y, the Average Speed is given by (2xy) ÷ (x + y).

Shortcut Trick

Total ratio units = 6 + 5 = 11 units

Given 11 units = 44 years

1 unit = 4 years

Amit age (6 units) = 6 × 4 = 24 years

∴ The correct answer is 24 years.

Alternate Method

Given:

Ratio of ages of Amit and Geeta = 6 : 5

Sum of their ages = 44 years

Let the ages be 6x and 5x.

⇒ 6x + 5x = 44

⇒ 11x = 44

⇒ x = 4

⇒ Amit's age = 6x = 6 × 4 = 24 years

∴ The correct answer is 24 years.

Additional Information

Ratio and Proportion

To divide a quantity S in the ratio a : b, the parts are [a/(a+b)] × S and [b/(a+b)] × S.

Age Problems

If the current age is x, then age after n years is (x + n) and age n years ago was (x − n).

Shortcut Trick

Since the Principal and Time are same for both banks, find the Rate difference.

Rate difference = 7.5% − 3.5% = 4% per annum.

Total interest difference for 4 years = 4% × 4 = 16%.

Required Difference = 16% of ₹2,00,000 = 16 × 2,000 = ₹32,000.

∴ The correct answer is ₹32,000.

Alternate Method

Given:

Principal (P) = ₹2,00,000 | Time (T) = 4 years

Rate A (R₁) = 3.5% | Rate B (R₂) = 7.5%

Formula Used:

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

Calculations:

⇒ SI from Bank A = (2,00,000 × 3.5 × 4) ÷ 100

⇒ SI from Bank A = 2,000 × 14 = ₹28,000

⇒ SI from Bank B = (2,00,000 × 7.5 × 4) ÷ 100

⇒ SI from Bank B = 2,000 × 30 = ₹60,000

⇒ Difference = ₹60,000 − ₹28,000

⇒ Difference = ₹32,000

∴ The correct answer is ₹32,000.

Additional Information

Simple Interest Concept

In Simple Interest, the interest is always calculated on the original Principal amount throughout the entire loan period.

Total Amount Formula

The final amount (A) to be paid back is the sum of Principal and Simple Interest: A = P + (P × R × T) ÷ 100.

Direct Difference Rule

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

Shortcut Trick

First, calculate the semi-perimeter (s): s = (34 + 88 + 78) ÷ 2 = 100 cm.

Find the Area using Heron's formula: √(100 × 66 × 12 × 22) = 1320 cm².

The altitude (h) on side 88 cm is: h = (2 × Area) ÷ Base = (2 × 1320) ÷ 88 = 30 cm.

∴ The correct answer is 30 cm.

Alternate Method

Given:

Sides of the triangle: a = 34 cm, b = 88 cm, c = 78 cm.

We need to find the altitude (h) corresponding to side b = 88 cm.

Formula Used:

1. Semi-perimeter (s) = (a + b + c) ÷ 2
2. Area of triangle = √[s(s − a)(s − b)(s − c)]
3. Area of triangle = (1 ÷ 2) × base × altitude

Calculations:

⇒ s = (34 + 88 + 78) ÷ 2 = 200 ÷ 2 = 100 cm

⇒ Area = √[100 × (100 − 34) × (100 − 88) × (100 − 78)]

⇒ Area = √(100 × 66 × 12 × 22)

⇒ Area = √(100 × 11 × 6 × 6 × 2 × 11 × 2)

⇒ Area = 10 × 11 × 6 × 2 = 1320 cm²

⇒ 1320 = (1 ÷ 2) × 88 × h

⇒ 1320 = 44 × h

⇒ h = 1320 ÷ 44 = 30 cm

∴ The correct answer is 30 cm.

Additional Information

Heron's Formula

Used to find the area of a triangle when all three sides are known: Area = √[s(s−a)(s−b)(s−c)].

Altitude of a Triangle

The altitude is the perpendicular distance from a vertex to the opposite side (base). Height = (2 × Area) ÷ Base.

Pythagorean Triplets in Scalene Triangles

An altitude divides a triangle into two right-angled triangles. Here, the altitude 30 cm creates triplets (30, 72, 78) and (16, 30, 34).

Shortcut Trick

The total area of the triangular garden with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

Area = 0.5 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Area = 0.5 × |2(11 − 7) + 8(7 − 3) + 14(3 − 11)| = 0.5 × |8 + 32 − 112| = 36 square units.

Since the landscape designer wants to cover exactly half of this triangular area with flowers:

Required Area = 36 ÷ 2 = 18 square units.

∴ The correct answer is 18 square units.

Alternate Method

Given:

⇒ Vertices of the triangular garden: A(2, 3), B(8, 11), and C(14, 7)

⇒ Fraction of area to be covered with flowers: Exactly half (1/2)

Formula Used:

⇒ Area of a triangle = 0.5 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Calculations:

Let the coordinates of the vertices be:

⇒ (x₁, y₁) = (2, 3)

⇒ (x₂, y₂) = (8, 11)

⇒ (x₃, y₃) = (14, 7)

Substitute these coordinates into the area formula:

⇒ Total Area = 0.5 × |2 × (11 − 7) + 8 × (7 − 3) + 14 × (3 − 11)|

⇒ Total Area = 0.5 × |2 × 4 + 8 × 4 + 14 × (−8)|

⇒ Total Area = 0.5 × |8 + 32 − 112|

⇒ Total Area = 0.5 × |40 − 112|

⇒ Total Area = 0.5 × |−72|

⇒ Total Area = 36 square units

The area to be covered with flowers is exactly half of the total area:

⇒ Required Area = Total Area ÷ 2

⇒ Required Area = 36 ÷ 2 = 18 square units

Justification of Options:

⇒ Option (1) is correct because the total area of the triangle is 36 square units, and exactly half of this area is 18 square units.

⇒ Option (2) is incorrect because 12 square units represents only one-third of the total area, not half.

⇒ Option (3) is incorrect because 36 square units is the total area of the triangular garden, not the half area to be covered with flowers.

⇒ Option (4) is incorrect because 24 square units represents two-thirds of the total area, not half.

∴ The correct answer is 18 square units.

Additional Information

Area of a Triangle using Coordinates

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated as 0.5 × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If the result is negative, we take the absolute value because area is always positive.

Median of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. A median divides the triangle into two smaller triangles of equal area.

Collinearity of Three Points

If the calculated area of a triangle formed by three points is exactly 0, then the three points are collinear (they lie on the same straight line).

Shortcut Trick

Equate the common entity Blue tokens by multiplying the ratios.

Orange : Blue = 11 : 18 and Blue : Yellow = 5 : 2

To find Orange : Yellow, multiply the terms: (11 × 5) : (18 × 2)

Orange : Yellow = 55 : 36

∴ The correct answer is 55 : 36.

Alternate Method

Given:

Ratio of Blue to Yellow (B : Y) = 5 : 2

Ratio of Orange to Blue (O : B) = 11 : 18

Calculations:

⇒ To combine ratios, make Blue (B) equal in both.

⇒ Multiply (B : Y) by 18 → B : Y = (5 × 18) : (2 × 18) = 90 : 36

⇒ Multiply (O : B) by 5 → O : B = (11 × 5) : (18 × 5) = 55 : 90

⇒ Combined Ratio Orange : Blue : Yellow = 55 : 90 : 36

⇒ Therefore, Orange : Yellow = 55 : 36

∴ The correct answer is 55 : 36.

Additional Information

Compound Ratio

The compound ratio of a : b and c : d is calculated by multiplying corresponding terms: (a × c) : (b × d).

Ratio Balancing

When two ratios have a common element, find the LCM of the common element's values to merge the ratios into a single continuous form.

Duplicate Ratio

The duplicate ratio of a : b is a² : b², often used in area-related ratio problems.

Shortcut Trick

Ratio of Expenditure to Savings = 80 : 20 = 4 : 1

By Alligation/Weighted Average: Net Change in Income = (4 × 35% + 1 × x%) ÷ 5 = 19%

⇒ 140 + x = 95 ⇒ x = −45% (Savings decrease by 45%)

Initial Savings = 20% of 91,700 = ₹18,340

Change in Savings = 45% of 18,340 = 0.45 × 18,340 = ₹8,253

∴ The correct answer is decrease by ₹8,253.

Alternate Method

Given: Initial Income = ₹91,700 | Savings = 20% | Expenditure = 80%

⇒ Initial Savings = 20 ÷ 100 × 91,700 = ₹18,340

⇒ Initial Expenditure = 91,700 − 18,340 = ₹73,360

⇒ New Income = 91,700 + (19 ÷ 100 × 91,700) = 91,700 + 17,423 = ₹109,123

⇒ New Expenditure = 73,360 + (35 ÷ 100 × 73,360) = 73,360 + 25,676 = ₹99,036

⇒ New Savings = 109,123 − 99,036 = ₹10,087

⇒ Decrease in Savings = 18,340 − 10,087 = ₹8,253

∴ The correct answer is decrease by ₹8,253.

Additional Information

Income Expenditure Relation

The basic identity is Income = Expenditure + Savings. Any percentage change must balance this equation.

Successive Percentage

For two successive changes a% and b%, the effective change is (a + b + ab/100)%, though for income-expenditure, weighted averages are more efficient.

Base Change Property

When income increases, if expenditure increases by a higher percentage than income, the savings percentage will always decrease relative to the original savings.

Shortcut Trick

Combined efficiency of A, B, and C = (1/2 + 1/2 + 1/2) = 3/2 work per day.

Total work to be done = 3 × (Original Work) = 3 × 1 = 3 units.

Time taken = (Total Work) ÷ (Combined Efficiency) = 3 ÷ (3/2) = 2 days.

∴ The correct answer is 2 days.

Alternate Method

Given: Individual time for A, B, and C = 2 days each.

⇒ Total Efficiency of (A + B + C) = 1 + 1 + 1 = 3 units/day.

⇒ Original Total Work = 2 units.

⇒ Thrice the same work = 3 × 2 = 6 units.

⇒ Time taken = (New Total Work) ÷ (Total Efficiency) = 6 ÷ 3 = 2 days.

∴ The correct answer is 2 days.

Additional Information

Work and Time Relationship

Time is directly proportional to the amount of work to be done. If work becomes thrice, time also becomes thrice for the same efficiency.

Combined Work Formula

If n people with same efficiency do a work in 'D' days, then together they do it in D/n days.

Pipes & Cisterns

Similar logic applies to filling tanks; Efficiency is the rate of filling/emptying per unit time.

Shortcut Trick

Successive multiplier: MP × (100 − d₁)% × (100 − d₂)% = SP.

88,000 × 75/100 × (100 − Z)/100 = 33,000.

66,000 × (100 − Z)/100 = 33,000.

(100 − Z)/100 = 33,000 ÷ 66,000 = 1/2.

100 − Z = 50 ⇒ Z = 50.

∴ The correct answer is 50.

Alternate Method

Given → Marked Price (MP) = ₹88,000, Selling Price (SP) = ₹33,000, First Discount (d₁) = 25%, Second Discount (d₂) = Z%. Formulas: Price after first discount = MP × (100 − d₁)% | Discount% = (Price Difference ÷ Intermediate Price) × 100.

⇒ Price after first discount = 88,000 − (25% of 88,000) = 88,000 − 22,000 = ₹66,000. ⇒ Second discount amount = Price after first discount − Final Selling Price. ⇒ Discount amount = 66,000 − 33,000 = ₹33,000. ⇒ Z% (Second Discount) = (33,000 ÷ 66,000) × 100. ⇒ Z = (1/2) × 100 = 50.

∴ The correct answer is 50.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The total discount is not the sum of individual percentages but is calculated on the reducing balance. Net Discount = a + b − (a × b)/100.

Marked Price & Selling Price

Marked Price is the price printed on the article. Selling Price = MP × (100 − d₁)% × (100 − d₂)%...

Shortcut Trick

Relative speed (P & Q) = 72 − 42 = 30 km/hr = 30 × (5/18) m/s.

Total distance (L_P + L_Q) = Speed × Time = (30 × 5/18) × 90 = 750 m.

Length of Train P (L_P) = 750 − 393 = 357 m.

Relative speed (P & R) = 72 + 63 = 135 km/hr = 135 × (5/18) = 37.5 m/s.

Time to cross R = (L_P + L_R) ÷ Relative speed = (357 + 468) ÷ 37.5 = 825 ÷ 37.5 = 22 seconds.

∴ The correct answer is 22 seconds.

Alternate Method

Given:

Speed of Train P (S_P) = 72 km/hr, Length of Train Q (L_Q) = 393 m, Speed of Q (S_Q) = 42 km/hr, Time₁ = 1.5 min = 90 s.

Length of Train R (L_R) = 468 m, Speed of R (S_R) = 63 km/hr.

Formula Used:

Relative Speed (Same) = S₁ − S₂ | Relative Speed (Opposite) = S₁ + S₂

Distance = Speed × Time | 1 km/hr = 5/18 m/s

Calculations:

⇒ Relative speed P and Q = 72 − 42 = 30 km/hr = 30 × 5/18 = 25/3 m/s

⇒ L_P + L_Q = Relative Speed × Time = (25/3) × 90 = 750 m

⇒ L_P = 750 − 393 = 357 m

⇒ Relative speed P and R (opposite) = 72 + 63 = 135 km/hr

⇒ 135 km/hr = 135 × 5/18 = 37.5 m/s

⇒ Time to cross R = (L_P + L_R) ÷ Relative Speed = (357 + 468) ÷ 37.5

⇒ Time = 825 ÷ 37.5 = 22 seconds

∴ The correct answer is 22 seconds.

Additional Information

Relative Speed Concept

When two objects move in the same direction, subtract speeds (u − v). When they move in opposite directions, add speeds (u + v).

Crossing Distance

When a train crosses another train or a bridge, the total distance covered is always the sum of their lengths (L₁ + L₂).

Unit 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

Ratio of quantities = 45 : 15 = 3 : 1.

Total units = 3 + 1 = 4.

Weighted average profit/loss = [(3 × 20%) + (1 × −40%)] ÷ 4.

Net result = (60% − 40%) ÷ 4 = 20% ÷ 4 = 5%.

Since the result is positive, it is a 5% profit.

∴ The correct answer is 5% profit.

Alternate Method

Given: Total CP of 60 pens = ₹900, Pens sold at profit = 45, Pens sold at loss = 15. Formulas: CP of one pen = Total CP ÷ Total Quantity | Profit = CP × Profit% | Loss = CP × Loss% | Net% = (Net Profit/Loss ÷ Total CP) × 100.

⇒ CP of 1 pen = 900 ÷ 60 = ₹15. ⇒ CP of 45 pens = 45 × 15 = ₹675. ⇒ Profit on 45 pens = 675 × 20% = ₹135. ⇒ CP of remaining 15 pens = 15 × 15 = ₹225. ⇒ Loss on 15 pens = 225 × 40% = ₹90. ⇒ Net Profit = 135 − 90 = ₹45. ⇒ Net Profit% = (45 ÷ 900) × 100 = 5%.

∴ The correct answer is 5% profit.

Additional Information

Weighted Average Method

When parts of a total quantity are sold at different profit/loss percentages, the overall profit/loss percentage can be calculated using (n₁p₁ + n₂p₂) ÷ (n₁ + n₂).

Profit and Loss Basics

Profit occurs when Selling Price (SP) is greater than Cost Price (CP). Loss occurs when CP is greater than SP. Both are always calculated on the CP unless stated otherwise.

Shortcut Trick

For two numbers a and b, the third proportional is given by b²/a.

Here, a = 75x² and b = 30xy.

Third Proportional = (30xy × 30xy) ÷ 75x²

Third Proportional = 900x²y² ÷ 75x² = 12y².

∴ The correct answer is 12y².

Alternate Method

Given: First term (a) = 75x², Second term (b) = 30xy.

Formula Used: If c is the third proportional to a and b, then a : b :: b : c, which means c = b²/a.

⇒ Let the third proportional be c.

⇒ 75x² / 30xy = 30xy / c

⇒ c = (30xy × 30xy) / 75x²

⇒ c = 900x²y² / 75x²

⇒ c = 12y²

∴ The correct answer is 12y².

Additional Information

Mean Proportional

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

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is such that a/b = c/d, so d = (b × c) / a.

Continued Proportion

Three numbers a, b, c are in continued proportion if a/b = b/c, which implies b² = ac.

Shortcut Trick

Assume all 45 notes are of the lower denomination (₹5).

Total value would be = 45 × 5 = ₹225.

Difference from the actual total value = 350 − 225 = ₹125.

Difference in the value of one ₹10 note and one ₹5 note = 10 − 5 = ₹5.

Number of ₹10 notes = 125 ÷ 5 = 25.

∴ The correct answer is 25.

Alternate Method

Given:

Total number of notes = 45

Total value of notes = ₹350

Denominations available = ₹10 and ₹5

Calculations:

⇒ Let the number of ₹10 notes be x.

⇒ Then the number of ₹5 notes is (45 − x).

⇒ According to the question: 10 × x + 5 × (45 − x) = 350

⇒ 10x + 225 − 5x = 350

⇒ 5x + 225 = 350

⇒ 5x = 350 − 225

⇒ 5x = 125

⇒ x = 125 ÷ 5

⇒ x = 25

∴ The correct answer is 25.

Additional Information

Rule of Alligation

This problem can be solved by finding the average value per note (350 ÷ 45) and applying the alligation cross-method to find the ratio of the quantities.

Linear Equations in Two Variables

For problems involving two unknowns, we use two independent equations: x + y = Total Quantity and (Value₁ × x) + (Value₂ × y) = Total Value.

Denomination Logic

The total amount is always the sum of (Value of each unit × Number of units) for every different denomination present.

Shortcut Trick

Time taken per page = 40 ÷ 4 = 10 minutes.

For 111 pages = 111 × 10 = 1110 minutes.

∴ The correct answer is 1110 minutes.

Alternate Method

Given:

Pages (W₁) = 4, Time (T₁) = 40 minutes

Pages (W₂) = 111, Time (T₂) = ?

Formula Used:

Work is directly proportional to Time (W₁/T₁ = W₂/T₂)

Calculations:

⇒ 4 / 40 = 111 / T₂

⇒ 1 / 10 = 111 / T₂

⇒ T₂ = 111 × 10

⇒ T₂ = 1110 minutes

∴ The correct answer is 1110 minutes.

Additional Information

Unitary Method

Finding the value of a single unit first (1 page) and then multiplying by the required quantity (111 pages).

Time and Work Relation

When efficiency is constant, the time taken is directly proportional to the amount of work done (T ∝ W).

BODMAS Rule:

Decoding the given information:

Given equation: 7 B 6 C 3 D 24 A 4 = ?

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

⇒ 7 × 6 + 3 − 24 ÷ 4 = ?

⇒ 7 × 6 + 3 − 6 = ?

⇒ 42 + 3 − 6 = ?

⇒ 45 − 6 = ?

⇒ 39 = ?

Hence, the correct answer is "Option 1".

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

Now,

Even numbers, each of which is immediately preceded by an odd number and also immediately followed by an odd number.

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

Thus, there are 2 even numbers, each of which is immediately preceded by an odd number and also immediately followed by an odd number.

Hence, the correct answer is "Option 3".