RRB Group D Official Paper (Held On: 04 Feb, 2026 Shift 1) (Previous Year Paper)

The correct answer is 900 m.

Key Points

• The train starts from rest, which means the initial velocity (u) is 0 m/s. When an object begins its motion from a stationary position, its starting speed is always considered zero in kinematic calculations.
• The final velocity (v) given is 36 km/hr. To perform calculations in SI units, this must be converted to meters per second (m/s). By multiplying 36 by the conversion factor 5/18, we get 10 m/s.
• The time (t) taken to reach this velocity is 3 minutes. In physics equations, time must be expressed in seconds. Therefore, 3 minutes is multiplied by 60, resulting in 180 seconds.
• Since the train moves with uniform acceleration (a), we can find the acceleration using the first equation of motion: v = u + at. Substituting the values, 10 = 0 + a(180), which gives a = 10/180 or 1/18 m/s².
• The distance (s) covered is calculated using the second equation of motion: s = ut + (1/2)at². Substituting the known values: s = (0)(180) + (1/2)(1/18)(180)(180). This simplifies to s = 0 + (1/2)(1/18)(32400), which equals 900 meters.
• Alternatively, distance can be found using average velocity multiplied by time for uniform acceleration: [(u + v) / 2] x t. Here, [(0 + 10) / 2] x 180 = 5 x 180 = 900 m. Hence, the correct answer is 900 m.

Additional Information

• Equations of Motion:
• These equations describe the behavior of a physical system in terms of its motion as a function of time. There are three primary equations for uniform acceleration: v = u + at, s = ut + 1/2 at², and v² = u² + 2as.
• These were derived based on the work of Sir Isaac Newton and describe linear motion. They are only applicable when acceleration remains constant throughout the duration of the motion.
• Unit Conversion Logic:
• Converting km/hr to m/s is a standard procedure in physics. Since 1 kilometer is 1000 meters and 1 hour is 3600 seconds, the ratio 1000/3600 simplifies to 5/18.
• This conversion is essential to maintain consistency with the International System of Units (SI), where distance is in meters and time is in seconds.
• Uniform Acceleration:
• Uniform acceleration occurs when the velocity of an object changes by an equal amount in every equal time interval.
• On a Velocity-Time (v-t) graph, uniform acceleration is represented by a straight line. The area under the line in a v-t graph represents the total distance covered by the object.

Important Points

• Scalar vs Vector: Distance is a scalar quantity, meaning it only has magnitude. However, in straight-line motion in a single direction starting from rest, the magnitude of distance and displacement (a vector) are identical.
• Significance of "Rest": In competitive exams, the phrase "starts from rest" is a crucial data point indicating that u = 0. Conversely, "comes to a stop" or "brakes are applied" usually indicates v = 0.
• Numerical Accuracy: Always ensure all units are converted to the MKS system (Meter-Kilogram-Second) before applying kinematic formulas to avoid common calculation errors. For this problem, failing to convert 3 minutes to 180 seconds would lead to an incorrect result.

The correct answer is Offspring show new genetic recombination.

Key Points

• Sexual reproduction in plants involves the fusion of male and female gametes to form a zygote, which eventually develops into a new individual.
• One of the most significant features of this process is genetic recombination. During the process of meiosis (cell division to produce gametes), crossing over occurs between homologous chromosomes during prophase I. This results in offspring having a unique combination of genes different from both parents. Hence, statement 1 is correct.
• This genetic variation is the cornerstone of survival in changing environments. It allows a population of plants to adapt to new pathogens, changing climatic conditions, and competition for resources.
• In contrast, if offspring do not show any variation, they are highly susceptible to being wiped out by a single disease or environmental change. Hence, statement 2 is incorrect.
• Offspring produced through sexual reproduction are not clones; they are genetically distinct individuals. Cloning is a characteristic of asexual reproduction or vegetative propagation. Hence, statement 3 is incorrect.
• Sexual reproduction is generally a slower process compared to asexual reproduction because it involves complex stages like pollination, fertilization, seed formation, and dormancy. Hence, statement 4 is incorrect.

Additional Information

• Mechanism of Variation: The two main sources of variation in sexual reproduction are independent assortment of chromosomes and crossing over. In angiosperms (flowering plants), the pollen grain (male gametophyte) and embryo sac (female gametophyte) undergo these processes to ensure diversity.
• Evolutionary Advantage: Variation is the raw material for natural selection. Plants with favorable genetic traits resulting from recombination are more likely to survive and reproduce, passing those advantageous traits to the next generation.
• Types of Pollination: Cross-pollination (Allogamy) specifically promotes greater heterozygosity and genetic diversity compared to self-pollination (Autogamy).
• Adaptability: Recombination allows for the masking of harmful mutations. Since offspring receive sets of genes from two parents, a healthy gene from one parent can often compensate for a defective gene from the other.

Important Points

• Comparison with Asexual Reproduction: In asexual reproduction, such as budding, fragmentation, or apomixis, the offspring are genetically identical to the parent. While this allows for rapid population growth in stable environments, it lacks the evolutionary flexibility provided by sexual reproduction.
• Environmental Stress: In unpredictable environments (e.g., fluctuating soil pH, drought, or salinity), the recombinant genotypes produced through sexual reproduction have a higher probability of possessing the necessary physiological traits to withstand abiotic stress.
• Date and Discovery: The basic principles of heredity and the importance of variation were famously documented by Gregor Mendel in the mid-19^th century (around 1865) through his experiments on Pisum sativum (pea plants).

The correct answer is more than one type of cells.

Key Points

• Complex tissues are defined as a group of more than one type of cells that have a common origin and work together as a structural and functional unit.
• Unlike simple permanent tissues, which consist of cells that are morphologically and physiologically similar, complex tissues are heterogeneous.
• The primary function of these tissues in plants is the transport of water, minerals, and organic nutrients throughout the plant body.
• The two most well-known types of complex tissues in the plant kingdom are Xylem and Phloem. These are often referred to as vascular tissues.
• The presence of these complex tissues is a distinctive feature of complex plants, allowing them to survive in terrestrial environments by facilitating long-distance transport. Hence, the correct answer is more than one type of cells.

Additional Information

• Xylem:
• It functions as a conducting tissue for water and minerals from the roots to the stems and leaves.
• It provides mechanical strength to the plant parts.
• It is composed of four different kinds of elements: Tracheids, Vessels, Xylem parenchyma, and Xylem fibres.
• In flowering plants, tracheids and vessels are the main water-transporting elements. Most xylem elements are dead cells except for xylem parenchyma.
• Phloem:
• It transports food materials, usually from the leaves (source) to other parts of the plant (sink).
• In angiosperms, phloem is composed of Sieve tube elements, Companion cells, Phloem parenchyma, and Phloem fibres.
• Unlike xylem, phloem cells are generally living cells, with the exception of phloem fibres (also known as bast fibres) which are dead at maturity.
• Simple Permanent Tissues:
• These are made up of only one type of cells which are similar in structure and function.
• Examples include Parenchyma (involved in storage and photosynthesis), Collenchyma (provides flexibility and mechanical support), and Sclerenchyma (provides rigidity and is often dead at maturity).
• Meristematic Tissues:
• These are tissues where cells remain young and undergo continuous cell division.
• They are found in the growing regions of plants, such as the root tips and shoot tips (apical meristems).

Important Points

• Origin: Both xylem and phloem in the primary plant body originate from the procambium. During secondary growth, they are produced by the vascular cambium.
• Vascular Bundles: In the stems and roots, xylem and phloem are organized into distinct groups called vascular bundles.
• Evolutionary Significance: The development of complex conducting tissues was a major step in the evolution of land plants (Pteridophytes, Gymnosperms, and Angiosperms), enabling them to grow to great heights.

The correct answer is Both (iii) and (iv).

Key Points

• Atmospheric Refraction is the primary physical phenomenon responsible for the twinkling of stars. As starlight enters the Earth's atmosphere, it undergoes refraction continuously before it reaches the Earth's surface. Hence, statement (iv) is correct.
• The apparent position of a star is slightly different from its actual position because the atmosphere bends starlight toward the normal as it moves from a rarer medium to an optically denser medium. Because the physical conditions of the Earth's atmosphere are not stationary, the apparent position of the star fluctuates slightly. Hence, statement (iii) is correct.
• While it is scientifically true that stars are self-luminous bodies (they have light of their own), this characteristic is not the cause of the twinkling effect. Twinkling is an optical illusion created by the medium through which the light travels, not the source of the light itself. Hence, statement (i) is a true fact but not the reason for twinkling.
• The twinkling of stars is caused by the refraction (bending) of light, not the reflection of light. Reflection involves light bouncing off a surface, whereas refraction involves light passing through a medium of varying optical density. Hence, statement (ii) is incorrect.
• The path of rays of light coming from the star goes on varying slightly, the apparent position of the star fluctuates and the amount of starlight entering the eye flickers—the star sometimes appears brighter, and at some other time, fainter, which is the twinkling effect.

Additional Information

• Point Source vs. Extended Source:
• Stars are located at a vast distance from the Earth, often measured in light-years. This makes them appear as point-sized sources of light. Even a tiny atmospheric disturbance can significantly shift the path of a single point of light.
• In contrast, planets are much closer to Earth and are seen as extended sources or a collection of many point-sized sources of light. The total variation in the amount of light entering our eye from all the individual point-sized sources will average out to zero, which is why planets do not twinkle.
• Refractive Index of Air:
• The Earth's atmosphere consists of several layers with varying temperatures and densities. Cold air is optically denser than warm air, meaning it has a higher refractive index.
• As starlight passes through these turbulent layers, the light is constantly bent back and forth. This process is known as stellar scintillation.
• Position Near Horizon:
• Stars appear to twinkle more when they are near the horizon compared to when they are directly overhead (at the zenith). This is because the light must travel through a thicker layer of the atmosphere, encountering more turbulence and refraction.
• Vacuum of Space:
• If an observer were in outer space (a vacuum), stars would not twinkle. This is why space telescopes like the Hubble Space Telescope or James Webb Space Telescope can capture much clearer images than ground-based telescopes, as they are above the distorting effects of the Earth's atmosphere.

Important Points

• The speed of light changes as it moves through different optical media, causing the bending effect known as refraction.
• The Laws of Refraction (Snell's Law) dictate that when light travels from a vacuum into the atmosphere, it slows down and bends toward the normal.
• The apparent shift in position means a star always appears slightly higher in the sky than its actual position.

The correct answer is Strong electrostatic forces between ions.

Key Points

• Ionic compounds are chemical compounds composed of ions held together by electrostatic forces termed ionic bonding. These compounds are formed through the transfer of electrons from a metal to a non-metal.
• The most defining characteristic of these compounds is the existence of strong electrostatic forces of attraction between the oppositely charged cations (positively charged) and anions (negatively charged). Hence, statement 1 is correct.
• Because of these strong inter-ionic attractions, ionic compounds exist as solids and are generally hard. A significant amount of thermal energy is required to overcome these forces, which results in high melting and boiling points. Hence, statement 2 is incorrect.
• Ionic compounds are generally soluble in polar solvents like water because the water molecules reduce the electrostatic attraction between the ions, allowing them to disperse. However, they are usually insoluble in non-polar solvents such as benzene, ether, or kerosene. Hence, statement 3 is incorrect.
• In their solid state, ionic compounds are poor conductors of electricity because the ions are fixed in a crystal lattice. However, in the molten state or when dissolved in aqueous solution, the ions become free to move and act as charge carriers, making them good electrical conductors. Hence, statement 4 is incorrect.

Additional Information

• Formation of Ionic Bonds:
• An ionic bond is formed when one atom (typically a metal) loses valence electrons to another atom (typically a non-metal). For example, in Sodium Chloride (NaCl), a sodium atom loses one electron to become Na+, and a chlorine atom gains that electron to become Cl-.
• The resulting electrostatic attraction between the sodium cation and chloride anion creates the stable compound.
• Crystal Lattice Structure:
• Ionic compounds do not exist as single molecules; instead, they form a three-dimensional network called a crystal lattice.
• This lattice structure is highly organized and contributes to the brittleness of the compound, as applying pressure shifts the ions and causes like-charged ions to repel each other, shattering the crystal.
• Lattice Energy:
• Lattice energy is the amount of energy released when gaseous ions combine to form one mole of an ionic solid. It is a measure of the stability and strength of the ionic bond.
• Higher charge density on ions and smaller ionic radii generally lead to higher lattice energy and higher melting points.

Important Points

• Physical State: At room temperature, almost all ionic compounds are crystalline solids due to the high bond strength.
• Non-directional Nature: Unlike covalent bonds, ionic bonds are non-directional because the electrostatic field of an ion acts in all directions around it.
• Applications: Common examples include Magnesium Oxide (MgO), used as a refractory material due to its extremely high melting point of 2,852 degrees Celsius, and Calcium Carbonate (CaCO3), found in limestone and shells.
• Electrolysis: The molten state conductivity of ionic compounds is utilized in industrial electrolysis to extract reactive metals like Aluminum and Sodium from their ores.

The correct answer is Transpiration pull and root pressure.

Key Points

• The upward movement of water and minerals from the roots to the various parts of the plant is known as the ascent of sap. This process is primarily driven by transpiration pull and root pressure.
• Transpiration pull is a biological process where the evaporation of water from the stomata of leaves creates a suction force (tension). This tension is transmitted down to the roots through a continuous column of water in the xylem vessels, pulling water upwards. Hence, statement 4 is correct.
• Root pressure is a positive pressure developed in the xylem of the roots due to the active absorption of mineral ions from the soil. This creates an osmotic gradient that forces water into the root xylem, pushing it up to a certain height. It is more significant during the night or in small herbaceous plants.
• The conversion of sunlight into starch refers to photosynthesis, which occurs in the chloroplasts to produce food, but it does not directly facilitate the upward transport of water. Hence, statement 1 is incorrect.
• While active transport occurs in leaf cells for the loading of sugars into the phloem, it is not the primary mechanism for the bulk upward movement of water in the xylem. Hence, statement 2 is incorrect.
• The diffusion of gases through stomata is essential for respiration and photosynthesis, but the gas exchange itself is not the force that lifts water; rather, it is the loss of water vapor (transpiration) occurring simultaneously that provides the pull. Hence, statement 3 is incorrect.

Additional Information

• Xylem Tissue: This is a complex permanent tissue responsible for the conduction of water and minerals. It consists of four types of cells: tracheids, vessels, xylem parenchyma, and xylem fibres. The tracheids and vessels are the main conducting elements that form a continuous system of water-conducting channels.
• Cohesion-Tension Theory: Proposed by Dixon and Joly in 1894, this theory explains how water can reach the top of tall trees (over 100 meters). It relies on cohesion (attraction between water molecules), adhesion (attraction between water and xylem walls), and surface tension.
• Transpiration: This is the loss of water in the form of vapor from the aerial parts of the plant. It not only helps in the ascent of sap but also aids in temperature regulation (cooling the plant) and maintaining the turgidity of cells.
• Factors Affecting Transpiration: The rate of transpiration is influenced by external factors such as temperature, light intensity, humidity, and wind speed. For instance, high humidity reduces the rate of transpiration because the concentration gradient between the leaf and the atmosphere is decreased.
• Guttation: When root pressure is high and transpiration is low (usually at night), water may be forced out in the form of liquid droplets through specialized openings called hydathodes at the margins of leaves. This phenomenon is distinct from transpiration.

Important Points

• Direction of Flow: In the xylem, the transport of water and minerals is strictly unidirectional (from roots to leaves). In contrast, the phloem transports food in a bidirectional manner (from source to sink).
• Energy Requirement: The transpiration pull is a passive process driven by solar energy (evaporation), whereas the initial accumulation of minerals in the root that leads to root pressure often involves active transport (utilizing ATP).
• Xylem Vessels vs Tracheids: Vessels are more efficient in water conduction than tracheids because they are wider and have perforated end walls, allowing for a more rapid flow of the water column.

The correct answer is straight line inclined to time axis.

Key Points

• Uniform acceleration is defined as a type of motion where the velocity of an object changes by an equal amount in every equal time interval. In other words, the rate of change of velocity remains constant over time.
• On a velocity-time (v-t) graph, the slope of the line represents the acceleration of the particle. If the acceleration is uniform (constant), the slope must also be constant.
• A straight line is the geometric representation of a constant slope. Therefore, the graph of an object moving with uniform acceleration is a straight line.
• The line must be inclined to the time axis (x-axis) to show that velocity is changing. If the acceleration is positive, the line slopes upward; if the acceleration is negative (retardation), the line slopes downward. Hence, statement 3 is correct.
• A horizontal line on a velocity-time graph indicates that the slope is zero, meaning the acceleration is 0 and the particle is moving with uniform velocity. Hence, statement 4 is incorrect.
• A curve concave upward indicates that the slope (acceleration) is increasing over time, which represents non-uniform acceleration. Hence, statement 2 is incorrect.
• A vertical line would imply that velocity changes infinitely at a single point in time, resulting in infinite acceleration, which is physically impossible for a particle. Hence, statement 1 is incorrect.

Additional Information

• Area under the Graph: The area enclosed by the velocity-time graph and the time axis represents the total displacement of the particle during that time interval.
• Equations of Motion: For a particle moving with uniform acceleration (a), the relationship between initial velocity (u), final velocity (v), and time (t) is given by the equation v = u + at. This is a linear equation of the form y = mx + c, confirming that the graph is a straight line.
• Slope Calculation: The slope of the velocity-time graph is calculated as the change in velocity divided by the change in time (rise over run). For uniform acceleration, this ratio is always the same regardless of which two points on the line are chosen.
• Types of V-T Graphs:
• Zero Acceleration: A straight line parallel to the time axis.
• Positive Uniform Acceleration: A straight line with a positive slope starting from the origin (if u=0) or from the y-axis (if u>0).
• Negative Uniform Acceleration: A straight line with a negative slope (moving towards the time axis).
• Non-Uniform Acceleration: Any curved line (parabolic, exponential, etc.).

Important Points

• Position-Time Graph: For a particle with uniform acceleration, the position-time (s-t) graph is a parabola, because displacement is proportional to the square of time (s = ut + 1/2 at^2).
• Instantaneous Acceleration: In cases of non-uniform acceleration, the acceleration at a specific moment is found by drawing a tangent to the velocity-time curve at that point and finding its slope.
• Units: The SI unit of acceleration is meters per second squared (m/s^2), while velocity is measured in meters per second (m/s).
• Kinematics: These graphical interpretations are fundamental in Kinematics, a branch of Physics that describes the motion of points, bodies, and systems without considering the forces that cause the motion.

The correct answer is Impure copper is used as anode and pure copper as cathode.

Key Points

• Electrolytic refining is a process used for the purification of metals like copper, zinc, tin, nickel, silver, and gold. It relies on the principle of electrolysis to transfer metal from an impure source to a pure one.
• In this setup, the Anode is always made of the impure metal. For copper refining, a thick slab of impure copper (blister copper) is used as the positive electrode. During the process, copper atoms from this slab lose electrons and enter the solution as ions. Hence, the statement that impure copper is used as the anode is correct.
• The Cathode consists of a thin strip of pure metal. In this context, a pure copper sheet acts as the negative electrode where the copper ions (Cu2+) from the electrolyte gain electrons and deposit as pure metal. Hence, the statement that pure copper is used as the cathode is correct.
• The Electrolyte used is a water-soluble salt of the metal being refined. For copper, acidified copper sulphate (CuSO4) solution is commonly employed to ensure high electrical conductivity and a steady supply of ions.
• When electric current is passed through the electrolyte, the net result is the transfer of pure copper from the anode to the cathode, increasing the thickness of the pure cathode over time.

Additional Information

• Chemical Reactions at Electrodes:
• At the Anode (Oxidation): The impure copper atoms release electrons: Cu (s) → Cu2+ (aq) + 2e–.
• At the Cathode (Reduction): The copper ions from the solution accept electrons: Cu2+ (aq) + 2e– → Cu (s).
• Purity Level: The electrolytic refining process is highly efficient and can produce copper with a purity of 99.95% to 99.99%. This high degree of purity is essential because even small amounts of impurities significantly reduce the electrical conductivity of copper.
• Applications of Pure Copper: Pure copper obtained through this method is primarily used in the electrical industry for making wires, cables, transformers, and electronic components.
• Industrial Scale: In industrial refineries, multiple anodes and cathodes are suspended in large tanks filled with acidified electrolyte to maximize the output of pure copper through continuous electrolysis.

Important Points

• Anode Mud: The insoluble impurities present in the impure copper slab do not dissolve in the electrolyte. Instead, they settle down below the anode at the bottom of the tank. This residue is known as Anode Mud.
• Recovery of Precious Metals: The anode mud often contains valuable metals such as gold (Au), silver (Ag), platinum (Pt), antimony, and selenium. The recovery of these precious metals often helps offset the overall cost of the refining process.
• Soluble Impurities: Metals that are more electropositive than copper, such as iron (Fe), nickel (Ni), and zinc (Zn), dissolve into the electrolyte as sulphates but do not deposit at the cathode because they require a higher potential to be reduced compared to copper.
• Electrolyte Concentration: Throughout the process, the concentration of the copper sulphate electrolyte remains relatively constant because the amount of copper dissolving at the anode is exactly equal to the amount depositing at the cathode.

The correct answer is the sound waves are guided by multiple reflections in the tube.

Key Points

• A stethoscope is a medical instrument used by doctors for listening to sounds produced within the body, chiefly in the heart or lungs.
• In stethoscopes, the sound of the patient’s heartbeat reaches the doctor’s ears by multiple reflection of sound waves. The sound travels through the hollow tube of the stethoscope by reflecting off its inner walls repeatedly until it reaches the earpieces. Hence, statement 4 is correct.
• The sound waves are not absorbed by the tube walls; instead, the walls are designed to be reflective to ensure the intensity of the sound is maintained over the distance of the tube. Hence, statement 1 is incorrect.
• Refraction involves the bending of waves when passing from one medium to another. In a stethoscope, the primary mechanism for guiding sound through the curved tube is reflection, not refraction. Hence, statement 2 is incorrect.
• While electronic stethoscopes exist today that can amplify sound using electricity, the standard acoustic stethoscope works purely on the principle of acoustics and physical reflection without any electrical amplification. Hence, statement 3 is incorrect.

Additional Information

• Invention: The stethoscope was invented in France in 1816 by René Laennec. Originally, it was a simple wooden tube used for auscultation (listening to internal sounds).
• Mechanism of Sound: Sound is a longitudinal mechanical wave that requires a medium (solid, liquid, or gas) to travel. It travels through the stethoscope tube as a series of compressions and rarefactions.
• Multiple Reflection: This phenomenon occurs when a wave hits a surface and bounces back. By using a closed tube, the sound energy is confined, preventing the sound from spreading in all directions (spherical divergence), which effectively amplifies the sound heard by the doctor.
• Components: A modern stethoscope consists of a chest piece (which usually has a diaphragm and a bell), tubing, and earpieces. The diaphragm is used for high-frequency sounds, while the bell is used for low-frequency sounds.
• Acoustic Impedance: The design of the stethoscope helps in matching the acoustic impedance between the air and the body, allowing more sound energy to be transmitted to the doctor's ears.

Important Points

• Reflection Law: Sound follows the same laws of reflection as light, where the angle of incidence is equal to the angle of reflection.
• Application: Apart from stethoscopes, the principle of multiple reflection is also used in megaphones, horns, and musical instruments like trumpets and shehnais to direct sound in a specific direction.
• Concert Halls: The ceilings of cinema halls and concert halls are often curved so that sound, after reflection, reaches all corners of the hall uniformly.
• Sound Absorption: Soft materials like curtains or carpets are used in rooms to absorb sound and prevent reverberation, which is the opposite of what is required inside a stethoscope tube.

The correct answer is Chemical change.

Key Points

• A Chemical change is a process where one or more substances are altered into one or more new and different substances. In the given reaction, Zinc granules react with dilute Hydrochloric acid to produce Zinc chloride and Hydrogen gas.
• This reaction is characterized by the evolution of a gas, which is one of the primary indicators of a chemical transformation. The formation of Hydrogen bubbles (effervescence) around the zinc metal signifies that the original identity of the reactants has changed.
• Unlike a physical change, this process involves the breaking and forming of chemical bonds. The Zinc (Zn) atoms displace the Hydrogen (H) atoms from the Hydrochloric acid (HCl), resulting in a completely new chemical arrangement.
• Chemical changes are typically irreversible by simple physical methods like filtration or evaporation. Once the Zinc chloride salt is formed in the solution and Hydrogen gas is released into the atmosphere, the original Zinc metal cannot be recovered by mere physical cooling or mixing.
• The reaction is also exothermic in nature, meaning it is accompanied by the release of heat, causing the temperature of the reaction vessel to rise. This change in energy is another hallmark of a Chemical change.

Additional Information

• Chemical Equation: The balanced chemical equation for this reaction is Zn + 2HCl → ZnCl2 + H2. This shows that one mole of Zinc reacts with two moles of Hydrochloric acid to yield one mole of Zinc chloride and one mole of Hydrogen gas.
• Types of Chemical Reactions: This specific reaction is classified as a Single Displacement Reaction. In such reactions, a more reactive element (Zinc) displaces a less reactive element (Hydrogen) from its compound.
• Physical Change: In contrast, a Physical change involves a change in the physical properties of a substance (like shape, size, or state) without changing its chemical composition. Examples include melting of ice or dissolving sugar in water.
• Mixture Formation: A Mixture is formed when two or more substances are combined physically but not chemically. In a mixture, each component retains its own chemical identity, such as sand in water or air.
• Dissolution: While dissolution involves a solute dissolving in a solvent, in this context, the zinc is not merely "dissolving" like salt in water; it is reacting chemically to form a new salt and a gas.

Important Points

• Test for Hydrogen Gas: The Hydrogen gas produced during this reaction is colorless and odorless. It can be identified using the Pop Sound Test; when a burning splinter is brought near the mouth of the test tube, the gas burns with a characteristic "pop" sound.
• Reactivity Series: Zinc is placed above Hydrogen in the Reactivity Series of metals, which explains why it can successfully displace hydrogen from dilute acids. Metals like Copper or Silver, which are below hydrogen, would not react with dilute HCl.
• Safety Precautions: Since Hydrogen gas is highly inflammable and Hydrochloric acid is corrosive, this experiment should be performed with care in a laboratory setting with proper safety gear.
• History: The reaction of metals with acids was one of the earliest methods used by scientists like Henry Cavendish in 1766 to isolate and study Hydrogen, which he called "inflammable air."

The correct answer is Magnetic field lines always emerge from the north pole of a magnet and enter the south pole outside the magnet.

Key Points

• Magnetic field lines are imaginary lines used to represent the magnetic field around a magnet. By convention, outside the magnet, these lines emerge from the North pole and merge at the South pole. Hence, statement 1 is correct.
• It is a fundamental property that magnetic field lines never intersect each other. If they did, it would mean that at the point of intersection, a compass needle would point in two different directions simultaneously, which is physically impossible. Hence, statement 2 is incorrect.
• The degree of closeness or density of the field lines indicates the relative strength of the magnetic field. The field is strongest where the lines are most dense (crowded), such as near the poles, and weakest where the lines are far apart. It is never zero when lines are dense. Hence, statement 3 is incorrect.
• While the lines go from North to South outside the magnet, inside the magnet, the direction of the field lines is from the South pole to the North pole. This ensures that magnetic field lines form continuous closed loops. Hence, statement 4 is incorrect.
• The tangent drawn at any point on a magnetic field line gives the direction of the magnetic field (B) at that specific point.

Additional Information

• Magnetic Field Strength: The intensity of a magnetic field is measured in the SI unit called Tesla (T), named after Nikola Tesla. Another commonly used smaller unit is the Gauss (G), where 1 Tesla = 10,000 Gauss.
• Nature of Poles: Magnetic poles always exist in pairs (dipoles). It is impossible to isolate a single magnetic monopole; if a magnet is broken in half, two smaller magnets, each with its own North and South pole, are created.
• Earth's Magnetism: The Earth itself acts like a giant bar magnet. Its magnetic North pole is located near the geographic South pole, and its magnetic South pole is located near the geographic North pole, which is why the North end of a compass needle points toward the geographic North.
• Electromagnetism: In 1820, the Danish scientist Hans Christian Ørsted discovered that an electric current flowing through a wire creates a magnetic field around it, a principle used in electromagnets and electric motors.

Important Points

• Magnetic Flux: The total number of magnetic field lines passing through a given area is known as magnetic flux. The SI unit for magnetic flux is the Weber (Wb).
• Force on a Moving Charge: A magnetic field exerts a force on a moving electric charge. This force is perpendicular to both the velocity of the charge and the magnetic field direction, described by the Lorentz Force equation.
• Magnetic Materials: Materials respond differently to magnetic fields; they are classified as diamagnetic (weakly repelled), paramagnetic (weakly attracted), or ferromagnetic (strongly attracted, like iron, cobalt, and nickel).
• Applications: Magnetic fields are vital in modern technology, including MRI (Magnetic Resonance Imaging) machines, hard drives for data storage, maglev trains, and generators.

The correct answer is ethyl ethanoate.

Key Points

• The reaction between Ethanoic acid (also known as Acetic acid) and Ethanol in the presence of concentrated Sulfuric acid (H2SO4) is a classic example of an esterification reaction.
• In this chemical process, the carboxylic acid group (-COOH) reacts with the alcohol group (-OH) to form an ester and water.
• The chemical equation for this reaction is: CH3COOH + C2H5OH → CH3COOC2H5 + H2O.
• The product formed is Ethyl ethanoate, which belongs to the functional group of esters. Hence, the statement regarding the formation of ethyl ethanoate is correct.
• Concentrated Sulfuric acid acts as both a catalyst to speed up the reaction and a dehydrating agent to remove the water molecule formed, shifting the equilibrium toward the product side according to Le Chatelier's Principle.

Additional Information

• Ethanoic Acid (CH3COOH):
• Commonly known as Acetic acid, it is a colorless liquid with a pungent smell.
• A 5-8% solution of acetic acid in water is known as vinegar, widely used as a preservative in pickles.
• The melting point of pure ethanoic acid is 290 K, often freezing in cold climates, which earned it the name Glacial Acetic Acid.
• Ethanol (C2H5OH):
• Commonly referred to as Ethyl alcohol, it is a liquid at room temperature.
• It is an active ingredient in all alcoholic drinks and is also used as a solvent in medicines like Tincture Iodine, cough syrups, and tonics.
• Characteristics of Esters:
• Ethyl ethanoate is a sweet-smelling substance.
• Because of their pleasant fragrance, esters are extensively used in making perfumes and as flavoring agents in the food industry.
• They can be converted back into alcohol and sodium salt of carboxylic acid by reacting with an alkali (like NaOH); this reverse process is known as Saponification, used in the manufacture of soaps.

Important Points

• Other Options Explained:
• Methanol (CH3OH): This is the simplest alcohol. It is not formed in this reaction; rather, it could be a reactant if one were trying to synthesize Methyl ethanoate.
• Acetone (CH3COCH3): This is a ketone. It is typically produced by the oxidation of isopropanol and is not a product of the reaction between an acid and an alcohol.
• Sodium Acetate (CH3COONa): This is a salt formed when Ethanoic acid reacts with a base like Sodium hydroxide (NaOH) or Sodium carbonate (Na2CO3), not through esterification with an alcohol.
• Reaction Conditions: The reaction is usually carried out by heating the mixture. The absence of the acid catalyst would make the reaction extremely slow and inefficient for industrial or laboratory purposes.

The correct answer is Ethanoic acid.

Key Points

• When ethanol is treated with a strong oxidizing agent like alkaline KMnO4 (potassium permanganate) or acidified K2Cr2O7 (potassium dichromate) and heated, it undergoes complete oxidation to form ethanoic acid.
• The reaction proceeds as follows: CH3CH2OH (Ethanol) + 2[O] (from alkaline KMnO4 + heat) → CH3COOH (Ethanoic acid) + H2O (Water).
• An oxidizing agent is a substance that provides oxygen to another substance or removes hydrogen from it. In this reaction, alkaline KMnO4 acts as a powerful oxidant that converts the alcohol group (-OH) into a carboxylic acid group (-COOH).
• The oxidation of primary alcohols like ethanol typically passes through an intermediate stage called ethanal (acetaldehyde), but strong oxidizing agents ensure the reaction does not stop there and continues until the carboxylic acid is formed. Hence, the correct answer is Ethanoic acid.

Additional Information

• Ethanoic Acid (CH3COOH):
• It is commonly known as acetic acid. A 5-8% solution of ethanoic acid in water is called vinegar, which is widely used as a preservative in pickles.
• Its melting point is 290 K, and it often freezes during winter in cold climates, leading to the name glacial acetic acid.
• Ethanol (C2H5OH):
• Commonly known as ethyl alcohol, it is a liquid at room temperature and is the active ingredient in all alcoholic drinks.
• Because it is a good solvent, it is also used in medicines like tincture iodine, cough syrups, and many tonics.
• Other Options Analysis:
• Methane (CH4): This is the simplest alkane. It is not produced by the oxidation of ethanol; rather, it is a primary component of biogas and CNG.
• Methanoic acid (HCOOH): Also known as formic acid, it is found in ant stings. It is formed by the oxidation of methanol, not ethanol.
• Acetone (CH3COCH3): This is the simplest ketone. It is generally formed by the oxidation of secondary alcohols (like propan-2-ol), whereas ethanol is a primary alcohol.

Important Points

• Role of Heat: The reaction between ethanol and alkaline KMnO4 requires warming/heating to initiate the conversion process efficiently.
• Identification of Functional Groups: This reaction is a classic example of changing a hydroxyl group (-OH) into a carboxyl group (-COOH), representing an increase in the number of oxygen atoms within the molecule.
• Neutralization: Since the reaction uses alkaline KMnO4, the ethanoic acid formed initially reacts with the alkali to form a salt (potassium ethanoate), which is later converted to free acid by acidification.

The correct answer is Its kinetic energy becomes highest, and its potential energy becomes lowest.

Key Points

• According to the Law of Conservation of Energy, energy can neither be created nor destroyed; it can only be transformed from one form to another. In a free-falling object, the total mechanical energy (the sum of potential and kinetic energy) remains constant at every point of its motion.
• Potential Energy (PE) is the energy possessed by an object due to its position or height. It is calculated by the formula PE = mgh, where m is mass, g is acceleration due to gravity, and h is height. As the object falls, its height decreases, causing its potential energy to decrease continuously. Just before hitting the ground, the height is approximately zero, making the potential energy lowest.
• Kinetic Energy (KE) is the energy possessed by an object due to its motion. It is calculated by the formula KE = 1/2 mv², where v is velocity. During a free fall, the object accelerates downward due to gravity at a rate of approximately 9.8 m/s². This constant acceleration increases the velocity of the object, reaching its maximum value just before impact. Therefore, its kinetic energy becomes highest at this point.
• As the object falls, there is a continuous transformation of gravitational potential energy into kinetic energy. At the highest point (before release), the energy is entirely potential; just before hitting the ground, it is almost entirely kinetic. Hence, the statement that kinetic energy is highest and potential energy is lowest is correct.

Additional Information

• Mechanical Energy: This refers to the sum of Kinetic Energy and Potential Energy. In the absence of air resistance, the total mechanical energy of a system is conserved. For a falling body, Total Energy = mgh + 1/2 mv².
• Acceleration due to Gravity (g): On Earth, this value is approximately 9.8 m/s². It means that for every second an object falls, its velocity increases by 9.8 meters per second. This acceleration is independent of the mass of the object in a vacuum.
• Free Fall: A state of motion where the only force acting upon an object is gravity. In real-world scenarios, air resistance (drag) eventually balances the force of gravity, leading to a constant speed known as terminal velocity, but in basic physics problems, air resistance is usually neglected.
• Work-Energy Theorem: This theorem states that the work done by all forces acting on a particle equals the change in its kinetic energy. In this case, the work done by gravity as the object descends increases its kinetic energy.

Important Points

• At the maximum height, the velocity of the object is zero, which means the kinetic energy is zero and potential energy is at its maximum.
• At the mid-point of the fall, the potential energy lost is exactly equal to the kinetic energy gained (assuming it started from rest), meaning PE equals KE at this specific height.
• Immediately after hitting the ground, the kinetic energy is transformed into other forms of energy such as heat energy, sound energy, and potential energy of deformation, following the principle of energy conservation.
• The mass of the object does not affect the final velocity just before hitting the ground in a vacuum, as seen in the equation v = √(2gh), where mass m cancels out.

The correct answer is A is true, but R is false.

Key Points

• Assertion (A) states that Zinc can displace copper from a copper sulphate solution. This occurs because Zinc is a more chemically active metal than copper, allowing it to substitute the copper ions in the solution. Hence, statement A is correct.
• Reason (R) claims that Zinc is less reactive than copper. In the electrochemical series or reactivity series, Zinc is positioned well above copper, indicating it has a higher tendency to lose electrons and form cations. Hence, statement R is incorrect.
• The chemical reaction for this process is: Zn (solid) + CuSO4 (aqueous) → ZnSO4 (aqueous) + Cu (solid).
• During this reaction, the blue color of the copper sulphate solution gradually fades and eventually becomes colorless as zinc sulphate is formed, while reddish-brown copper metal deposits on the zinc strip.
• Since the Assertion is a factually correct chemical observation and the Reason is a factually incorrect statement regarding metal activity, the correct choice is that A is true but R is false.

Additional Information

• Reactivity Series: It is an arrangement of metals in the descending order of their reactivities. Metals like Potassium (K) and Sodium (Na) are at the top, while noble metals like Gold (Au) and Platinum (Pt) are at the bottom. Zinc (Zn) has an atomic number of 30 and is more reactive than Hydrogen, whereas Copper (Cu) with atomic number 29 is less reactive than Hydrogen.
• Displacement Reaction: This is a type of reaction where a more reactive element displaces a less reactive element from its compound. This can be categorized into single displacement (like the Zn-CuSO4 reaction) and double displacement reactions.
• Zinc (Zn): It is a transition metal found in Group 12 of the periodic table. It is widely used in galvanization to prevent the rusting of iron because it is more reactive and oxidizes preferentially.
• Copper (Cu): It is a transition metal in Group 11. It is known for its high electrical conductivity and ductility. In nature, it often exists as sulfide ores like chalcopyrite.
• Copper Sulphate (CuSO4): Often referred to as blue vitriol, this salt is commonly used as a fungicide and in electroplating. Its pentahydrate form is characteristically bright blue.

Important Points

• Redox Reaction: The displacement of copper by zinc is also a redox (reduction-oxidation) reaction. Zinc undergoes oxidation (loses electrons to become Zn2+), and Copper ions undergo reduction (gain electrons to become Cu metal).
• Practical Application: This principle is utilized in Voltaic cells or Daniel cells, where the difference in electrode potential between Zinc and Copper is used to generate electricity.
• Observation in Laboratory: If a copper wire is placed in a zinc sulphate solution, no reaction occurs because copper cannot displace zinc due to its lower reactivity.
• Valency: In this reaction, both Zinc and Copper typically exhibit a valency of +2.

The correct answer is Biological magnification.

Key Points

• Biological magnification, also known as biomagnification, refers to the increase in the concentration of a substance, such as a toxic chemical or pesticide, in the tissues of organisms at successively higher levels in a food chain.
• When crops like wheat, rice, vegetables, and fruits are grown, farmers often use pesticides and fertilizers to protect them from pests and diseases. These chemicals are absorbed by the plants from the soil or water.
• Because these substances are often non-biodegradable, they cannot be broken down by the metabolic processes of the organism nor excreted efficiently. Hence, the presence of residues in food items is primarily due to biological magnification.
• As we move up the trophic levels, from producers (plants) to primary consumers (herbivores) and then to secondary or tertiary consumers (like humans eating meat), the concentration of these harmful chemicals increases.
• Human beings occupy the topmost level in many food chains; therefore, the maximum concentration of these toxic chemicals gets accumulated in our bodies. This is why food items, especially animal products, often contain higher pesticide residues than the plants themselves.

Additional Information

• Transpiration:
• It is the biological process by which water vapor is lost from the aerial parts of a plant, primarily through the stomata in leaves.
• This process helps in the transportation of minerals from the soil to various parts of the plant and aids in temperature regulation (cooling the plant). It does not involve the accumulation of toxic residues.
• Photosynthesis:
• This is the process by which green plants and some other organisms use sunlight to synthesize foods with the help of chlorophyll, carbon dioxide, and water.
• The primary function is the production of glucose and oxygen. While it is the basis of the food chain, it is a metabolic process for energy production and not for toxin accumulation.
• Eutrophication:
• This phenomenon occurs when a water body becomes overly enriched with nutrients (like nitrogen and phosphorus), frequently due to agricultural runoff.
• This leads to the excessive growth of algae (algal blooms), which depletes the dissolved oxygen in the water, causing the death of aquatic animals like fish. This affects water ecosystems rather than the direct residue buildup in land-based food crops.

Important Points

• DDT (Dichlorodiphenyltrichloroethane): One of the most famous examples of biological magnification involves the pesticide DDT. It was widely used in the mid-20^th century and led to the thinning of eggshells in predatory birds (like eagles) because the concentration of DDT increased significantly as it moved up the food chain.
• Bioaccumulation vs. Biomagnification: Bioaccumulation occurs within a single organism over its lifetime, where the rate of intake of a substance is higher than the rate of excretion. Biomagnification is the cumulative result of bioaccumulation occurring across multiple trophic levels.
• Health Impacts: High levels of pesticide residues in human food can lead to long-term health issues, including hormonal imbalances, reproductive issues, cancer, and damage to the nervous system.
• Heavy Metals: Besides pesticides, heavy metals like mercury, lead, and cadmium are also subject to biological magnification, often entering the food chain through industrial waste and contaminated water.

The correct answer is Electric heater, electric iron, and fuse.

Key Points

• The heating effect of electric current refers to the process where electrical energy is converted into heat energy when current passes through a conductor with high resistance.
• Electric heater and Electric iron: These appliances contain a coil of wire called a heating element. When current flows through these elements, they become red hot and give out heat. The element is typically made of Nichrome (an alloy of nickel and chromium) because it has high resistivity and a high melting point, allowing it to withstand high temperatures without oxidizing. Hence, the appliances in Option 1 utilize the heating effect.
• Electric Fuse: A fuse is a safety device used in electrical circuits. It consists of a wire made of a material with a low melting point, such as an alloy of tin and lead. If the current exceeds a safe limit, the wire heats up and melts, breaking the circuit and preventing damage or fire. This is a direct application of the heating effect.
• LED and Torch: Light Emitting Diodes (LEDs) work on the principle of electroluminescence, and a torch (flashlight) primarily uses the lighting effect. While some heat is produced as a byproduct, it is not the primary functional principle. Hence, statement 2 is incorrect regarding the primary use of the heating effect.
• Electric bell and Buzzer: These devices work on the magnetic effect of electric current. An electromagnet is used to pull a striker that hits a gong or creates a vibration. Hence, statement 3 is incorrect.
• Electric fan and Motor: These devices utilize the magnetic effect of current to produce mechanical energy (rotational motion). The interaction between magnetic fields and current-carrying conductors creates torque. Hence, statement 4 is incorrect.

Additional Information

• Joule's Law of Heating: This law states that the heat produced in a conductor is directly proportional to the square of the current (I), the resistance (R) of the conductor, and the time (t) for which the current flows. The mathematical representation is H = I²Rt.
• Heating Elements: Most heating appliances use Nichrome. It is preferred because its resistance is high, and it does not burn (oxidize) easily at high temperatures (around 1150°C).
• Incandescent Bulbs: Although not mentioned in the correct option, the traditional filament bulb also uses the heating effect. The Tungsten filament is heated to such a high temperature (around 2500°C) that it starts emitting light. Tungsten is used due to its very high melting point of approximately 3380°C.
• Safety in Circuits: Beyond fuses, modern houses use MCBs (Miniature Circuit Breakers). While fuses work on the heating effect (melting), many MCBs work on the magnetic effect to trip the switch during an overload or short circuit.

Important Points

• Electric Power: The rate at which electrical energy is consumed or converted into other forms (like heat) is called Electric Power (P). It is measured in Watts (W) and calculated as P = VI or P = I²R.
• Resistance Factors: The amount of heat generated depends on the length, thickness, and nature of the material of the wire. Thinner wires generally have higher resistance than thicker wires of the same material and length, leading to more heat production.
• Applications: Apart from irons and heaters, the heating effect is used in geysers, electric kettles, hair dryers, and soldering irons.
• Short Circuit: This occurs when the live wire and neutral wire come into direct contact, causing the resistance of the circuit to become almost zero, resulting in a huge flow of current and excessive heating, which can lead to fires.

The correct answer is Increases 9 times.

Key Points

• According to Newton's Law of Universal Gravitation, the force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The mathematical formula for gravitational force is F = G (m1 * m2) / r^2, where G is the Universal Gravitational Constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
• In this scenario, the masses of both interacting objects (e.g., the Earth and another object) are tripled. This means the new masses are 3m1 and 3m2.
• The problem specifies that the distance (r) between the objects remains unchanged. Although the radius of the Earth is stated to be halved, this is a distractor because the center-to-center distance is explicitly held constant.
• Substituting the new values into the formula: F' = G (3m1 * 3m2) / r^2. This simplifies to F' = 9 * [G (m1 * m2) / r^2], which is 9 times the original force (F). Hence, the gravitational force increases 9 times.

Additional Information

• Universal Gravitational Constant (G): It was first measured experimentally by Henry Cavendish in 1798. Its accepted value is approximately 6.674 x 10^-11 N m^2/kg^2.
• Inverse Square Law: This principle implies that if the distance between two objects is doubled, the gravitational force becomes one-fourth (1/4) of its original strength. If the distance is halved, the force increases 4 times.
• Mass vs. Weight: Mass is an intrinsic property of matter and remains constant regardless of location, while weight is the gravitational force acting on that mass and changes based on the local acceleration due to gravity (g).
• Acceleration due to Gravity (g): On the surface of the Earth, the value of g is approximately 9.8 m/s^2. It is calculated using the formula g = GM / R^2, where M is the mass of the Earth and R is its radius.

Important Points

• Vector Nature: Gravitational force is always attractive and acts along the line joining the centers of mass of the two interacting bodies.
• Weak Force: Among the four fundamental forces of nature (Gravitational, Electromagnetic, Strong Nuclear, and Weak Nuclear), gravity is the weakest, yet it dominates on astronomical scales.
• Range: The range of gravitational force is infinite, meaning every object in the universe exerts a gravitational pull on every other object, no matter how far apart they are.
• History: Sir Isaac Newton published this law in his work "Philosophiæ Naturalis Principia Mathematica" in July 1687, providing a unified explanation for both terrestrial gravity and planetary motion.

The correct answer is Leaf margins.

Key Points

• Bryophyllum is a well-known genus of succulent plants that is famous for its unique method of vegetative propagation through its leaves.
• The buds for reproduction are specifically produced along the leaf margins. These margins contain small notches or indentations where adventitious buds arise.
• When these buds come into contact with moist soil, either by the leaf falling off or by the plantlet dropping, they develop roots and grow into independent plants.
• This process is a form of asexual reproduction because it does not involve the fusion of gametes or the production of seeds for the formation of the new plantlet.
• The offspring produced through this method are genetically identical to the parent plant, effectively acting as clones.

Additional Information

• Vegetative Propagation: It is a type of asexual reproduction in plants where a new plant grows from a fragment of the parent plant or a specialized reproductive structure such as a stolon, rhizome, tuber, or bulb.
• Adventitious Buds: These are buds that develop in locations where buds do not normally occur, such as on roots or leaves. In Bryophyllum, these are often called epiphyllous buds because they grow on the foliage.
• Common Examples:
• Potato: Reproduces through axillary buds called "eyes" located on the stem tuber.
• Ginger and Turmeric: Reproduce via underground rhizomes.
• Strawberry: Utilizes runners or stolons that grow horizontally above the ground.
• Agave: Produces bulbils, which are small plantlets that develop in the floral scape.
• Ecological Importance: This method allows plants like Bryophyllum to colonize an area very quickly, as they do not have to wait for the lengthy process of pollination, fertilization, and seed germination.
• Structural Parts:
• Pedicel: This is the stalk that supports an individual flower in an inflorescence; it is not involved in the vegetative propagation of Bryophyllum.
• Flower: While Bryophyllum can produce flowers for sexual reproduction, the "buds" mentioned in the context of vegetative propagation are strictly leaf-borne.

The correct answer is Tendrils.

Key Points

• Tendrils are specialized, thread-like organs of a plant that are highly sensitive to touch. They are primarily used by climbing plants with weak stems to anchor themselves to a support. Hence, Option 2 is correct.
• When a tendril comes into contact with a physical object, such as a twig, fence, or wall, it undergoes a growth response known as thigmotropism, causing it to coil around the support.
• These structures can be modifications of different plant parts. For instance, stem tendrils develop from axillary buds, as seen in plants like Cucumbers, Pumpkins, Watermelons, and Grapevines.
• Alternatively, leaf tendrils are modified leaves or leaflets, common in plants like the Pea plant (Pisum sativum), which allow the plant to grow vertically toward sunlight.
• The sensitivity of tendrils is remarkable; some can respond to a weight as light as a few milligrams, initiating the coiling process within minutes of contact.

Additional Information

• Flowers:
• Flowers are the reproductive structures of angiosperms. Their primary function is to facilitate pollination and fertilization to produce offspring.
• Unlike tendrils, they do not possess specialized touch-sensitive mechanisms for physical support or climbing. Hence, Option 1 is incorrect.
• Thorns:
• Thorns are hard, woody, and sharp-pointed modifications of the stem. They primarily serve as protective organs for the plant, defending it against herbivores and browsing animals.
• Common examples include Citrus and Bougainvillea. While they provide a physical structure, they are not sensitive to touch for the purpose of climbing. Hence, Option 3 is incorrect.
• Seeds:
• A seed is a mature, fertilized ovule containing an embryo and stored nutrients (endosperm). It is the primary unit of dispersal and reproduction for spermatophytes.
• Seeds represent a dormant stage in the plant life cycle and do not participate in the active climbing or mechanical support of the parent plant. Hence, Option 4 is incorrect.

Important Points

• Mechanism of Thigmotropism:
• The coiling of a tendril is caused by differential growth. When the tendril touches a support, the plant hormone Auxin migrates to the side that is not in contact with the object.
• The side with higher auxin concentration grows and elongates much faster than the side in contact with the support, causing the tendril to curve and spiral around the object.
• Evolutionary Adaptation:
• The development of tendrils is an evolutionary adaptation that allows plants to maximize photosynthetic efficiency by reaching higher strata of the forest or garden without the energetic cost of building a thick, woody trunk.

The correct answer is 3:8.

Key Points

• The chemical formula for Carbon Dioxide is CO2, which indicates that one molecule of the compound consists of one atom of Carbon and two atoms of Oxygen.
• To find the ratio by mass, we must consider the atomic masses of the individual elements. The atomic mass of Carbon (C) is approximately 12 u (atomic mass units).
• The atomic mass of Oxygen (O) is approximately 16 u. Since there are two oxygen atoms in a CO2 molecule, the total mass contributed by oxygen is 16 x 2 = 32 u.
• The ratio of the mass of Carbon to the mass of Oxygen in carbon dioxide is calculated as 12 : 32.
• By simplifying this ratio (dividing both sides by the common factor 4), we get 3 : 8. This means that in any sample of pure carbon dioxide, for every 3 grams of Carbon, there are 8 grams of Oxygen.

Additional Information

• Law of Constant Proportions: This calculation is a primary example of the Law of Definite Proportions (also known as Proust's Law), which states that a chemical compound always contains its component elements in a fixed ratio by mass, regardless of its source or method of preparation.
• Molar Mass of CO2: The total molar mass of carbon dioxide is the sum of the masses of its constituents, which is 12.01 + (2 x 16.00) = 44.01 g/mol. This value is widely used in stoichiometry and chemical engineering.
• Physical Properties: Carbon dioxide is a colorless and odorless gas at standard temperature and pressure. It is approximately 1.5 times as heavy as air.
• Molecular Structure: The CO2 molecule is linear in shape. The carbon atom is in the center, joined to the two oxygen atoms by double covalent bonds. The bond length between carbon and oxygen is 116.3 pm.
• Environmental Importance: It is a significant greenhouse gas and plays a vital role in the carbon cycle. It is consumed by plants during photosynthesis to produce glucose and oxygen.

Important Points

• In Water (H2O), the ratio of the mass of Hydrogen to Oxygen is 1:8 (Mass of H = 2, Mass of O = 16).
• In Ammonia (NH3), the ratio of Nitrogen to Hydrogen by mass is 14:3.
• The atomic number of Carbon is 6, while its mass number is usually 12. For Oxygen, the atomic number is 8 and the mass number is 16.
• It is important not to confuse the ratio of atoms (which is 1:2 for CO2) with the ratio by mass (which is 3:8).

The correct answer is It becomes a cation.

Key Points

• An atom is the basic unit of matter and is electrically neutral in its ground state because the number of protons (positively charged) in the nucleus equals the number of electrons (negatively charged) orbiting the nucleus.
• When an atom loses one or more electrons, it undergoes a process called ionization. Since electrons carry a negative charge, their removal leaves the atom with more protons than electrons, resulting in a net positive charge.
• A positively charged ion is specifically known as a cation. This name originates from the fact that these ions are attracted to the cathode (the negative electrode) during electrolysis.
• Common examples of cation formation include Metals like Sodium (Na), which loses 1 electron to become Na+, and Magnesium (Mg), which loses 2 electrons to become Mg2+.
• The loss of electrons typically occurs to achieve a stable electronic configuration, often resembling the nearest noble gas, which is known as the Octet Rule.

Additional Information

• Anions: These are the opposite of cations. An anion is formed when an atom gains one or more electrons, giving it a net negative charge. This typically happens with non-metals like Chlorine (Cl), which becomes Cl-.
• Ionization Energy: This refers to the specific amount of energy required to remove an electron from an isolated gaseous atom. Alkali metals in Group 1 have very low ionization energy, making them highly reactive and prone to forming cations.
• Ionic Bonding: This type of chemical bond occurs through the electrostatic attraction between oppositely charged ions (a cation and an anion). A classic example is Sodium Chloride (NaCl).
• Molecules: Unlike ions, molecules are neutral groups of two or more atoms held together by chemical bonds (usually covalent bonds). Losing a single electron does not turn an atom into a molecule.
• Valence Electrons: These are the electrons in the outermost shell of an atom. They are the ones involved in chemical reactions and are the primary electrons lost or gained during ion formation.

Important Points

• Oxidation: In the context of Redox reactions, the loss of electrons is defined as oxidation (remember the mnemonic OIL RIG: Oxidation Is Loss, Reduction Is Gain).
• Atomic Radius: When an atom becomes a cation, its radius decreases significantly. This is because the remaining electrons experience a stronger effective nuclear charge from the protons, pulling them closer to the center.
• Proton Count: It is vital to note that the number of protons in the nucleus never changes during the formation of an ion; if the proton count changed, the identity of the element itself would change.
• Electropositivity: The tendency of an element to lose electrons and form positive ions is called electropositivity. This property is most prominent in Cesium (Cs) and other elements located at the bottom-left of the Periodic Table.

The correct answer is Centrosome.

Key Points

• The Centrosome is a specialized cell organelle typically located near the nucleus in animal cells. It serves as the main microtubule-organizing center (MTOC).
• It consists of two barrel-shaped structures known as centrioles, which are oriented at right angles to each other and surrounded by pericentriolar material.
• The primary function of the centrosome is to facilitate cell division by forming spindle fibers that help in the segregation of chromosomes during mitosis and meiosis.
• While animal cells possess distinct centrosomes with centrioles, most higher plant cells lack these structures, instead using different types of MTOCs to manage their microtubule networks. Hence, the Centrosome is the organelle present only in animal cells.

Additional Information

• Mitochondria:
• Known as the "Powerhouse of the cell," these are double-membrane-bound organelles.
• They are the site of aerobic respiration and generate most of the cell's supply of adenosine triphosphate (ATP), used as a source of chemical energy.
• Mitochondria are found in almost all eukaryotic cells, including both plant and animal cells.
• Golgi apparatus:
• Discovered by Camillo Golgi in 1898, it consists of a system of membrane-bound vesicles or cisternae.
• Its main functions include the packaging, modification, and secretion of proteins and lipids synthesized in the Endoplasmic Reticulum.
• It is present in both plant and animal cells (often called dictyosomes in plants).
• Vacuole:
• These are membrane-bound spaces found in the cytoplasm containing sap, water, and excretory products.
• In plant cells, a large central vacuole can occupy 50% to 90% of the cell volume, providing turgidity and rigidity.
• Animal cells also contain vacuoles, but they are generally small, temporary, and used for storage or digestion.

Important Points

• Plant-specific structures: Unlike animal cells, plant cells possess a rigid cell wall made of cellulose and plastids like chloroplasts for photosynthesis.
• Animal-specific structures: Animal cells are characterized by the presence of centrosomes and centrioles. They also frequently contain lysosomes (suicide bags), which are less common or different in plant cells.
• Cell Size and Shape: Animal cells are usually smaller and have irregular shapes due to the absence of a cell wall, whereas plant cells are larger and have a fixed, rectangular shape.

The correct answer is To provide nutrition and aid in sperm transport.

Key Points

• The male reproductive system consists of various organs and glands, where the prostate gland and seminal vesicles play a vital role in the formation of semen.
• Seminal Vesicles: These are a pair of pouch-like glands located behind the bladder. They secrete a significant portion of the seminal fluid (approx. 60-70%), which is rich in fructose. This sugar acts as a primary source of energy (nutrition) for the sperm to maintain their activity.
• Prostate Gland: This single, walnut-sized gland produces a thin, milky, and slightly alkaline fluid. This secretion helps in neutralizing the acidity of the vaginal tract, thereby protecting the sperm and increasing their motility and viability.
• Sperm Transport: The fluid secretions from these glands provide a liquid medium that makes it easier for the sperm to be transported through the male urethra and into the female reproductive system. Hence, the correct answer is to provide nutrition and aid in sperm transport.
• These secretions also contain prostaglandins, which stimulate contractions in the female reproductive tract to help the sperm move toward the egg.

Additional Information

• Semen Composition: Semen is a mixture of spermatozoa (sperm cells) and seminal plasma. While the sperm cells are produced in the testes, the bulk of the fluid volume comes from the accessory glands.
• Bulbourethral Glands (Cowper's Glands): These are small glands located below the prostate. They secrete a clear, slippery fluid that lubricates the urethra and neutralizes any residual acidity from urine before ejaculation occurs.
• Path of Sperm: Sperm travels from the Testes → Epididymis (for maturation) → Vas Deferens → Ejaculatory Duct → Urethra.
• Testosterone: This is the primary male sex hormone produced by the Leydig cells in the testes, which regulates the production of sperm and the development of secondary sexual characteristics.

Important Points

• Fructose: It is specifically found in the secretion of the seminal vesicles and is not present elsewhere in the body in high concentrations, making it a chemical marker for forensic tests.
• Sperm Motility: Without the secretions from the prostate gland, sperm would be sluggish and unable to survive the acidic environment of the female reproductive tract, which has a pH of about 3.5 to 4.5.
• PSA (Prostate-Specific Antigen): This is an enzyme produced by the prostate gland that helps liquefy the semen after ejaculation, allowing the sperm to swim freely toward the fallopian tubes.
• Urine vs. Sperm: While the question asks about secretions, the mechanical prevention of mixing urine and sperm is handled by a sphincter muscle at the base of the bladder that closes during ejaculation.

The correct answer is at F.

Key Points

• When an object is placed at infinity in front of a convex lens, the light rays originating from it and reaching the lens are parallel to each other and to the principal axis.
• A convex lens is a converging lens, meaning it bends incoming parallel rays inward toward a single point.
• According to the laws of refraction, any ray of light that is parallel to the principal axis of a convex lens will pass through the principal focus (F) on the opposite side after refraction.
• Consequently, all these parallel rays converge and meet at the focus (F), leading to the formation of the image at F. Hence, the correct answer is at F.
• The nature of the image formed in this specific case is real and inverted.
• Regarding the size of the image, it is described as highly diminished or point-sized because all rays are concentrated at the focal point.

Additional Information

• Image Formation by Convex Lens at Different Positions:
• Beyond 2F: When the object is placed beyond 2F, the image is formed between F and 2F. This image is real, inverted, and diminished.
• At 2F: When the object is exactly at 2F, the image is formed at 2F on the other side. This image is real, inverted, and of the same size as the object.
• Between F and 2F: If the object is placed between F and 2F, the image is formed beyond 2F. The image is real, inverted, and magnified (larger than the object).
• At F: When the object is placed at the focus (F), the refracted rays become parallel, and the image is formed at infinity. This image is real, inverted, and highly magnified.
• Between F and Optical Center (O): This is a unique case where the image is formed on the same side as the object. The image is virtual, erect, and magnified. This principle is used in magnifying glasses.
• Convex Lens Characteristics:
• A convex lens is thicker at the center than at the edges.
• It has a positive focal length according to the Cartesian sign convention.
• It is widely used in optical instruments such as microscopes, telescopes, and cameras.
• In the medical field, it is used in spectacles to correct Hypermetropia (long-sightedness).

Important Points

• Lens Formula: The relationship between the object distance (u), image distance (v), and focal length (f) is given by the formula: 1/f = 1/v - 1/u.
• Magnification (m): It is the ratio of the height of the image to the height of the object, or m = v/u. For an object at infinity, the magnification is nearly zero.
• Optical Center (O): This is the central point of the lens. Any ray of light passing through the optical center emerges without any deviation.
• Power of a Lens: The power (P) is the reciprocal of the focal length in meters (P = 1/f). The SI unit of power is Dioptre (D). A convex lens always has positive power.

Shortcut Trick

In a 30°−60°−90° triangle, the sides are in the ratio 1 : √3 : 2.

The side opposite to 30° (Base) = 8.5 m.

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

Height = 8.5 × 1.7 = 14.45 m.

∴ The correct answer is 14.45 m.

Alternate Method

Given:

Angle of elevation (θ) = 60°

Distance from wall (Base) = 8.5 m

Value of √3 = 1.7

Formula Used:

tan(θ) = Perpendicular / Base

Calculations:

⇒ tan(60°) = Height / 8.5

⇒ √3 = Height / 8.5

⇒ Height = 8.5 × √3

⇒ Height = 8.5 × 1.7

⇒ Height = 14.45 m

∴ The correct answer is 14.45 m.

Additional Information

Trigonometric Ratios

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

Right Angled Triangle Properties

In a right triangle, sin θ = P/H, cos θ = B/H, and Pythagoras theorem: P² + B² = H².

Angle of Elevation vs Depression

The angle of elevation from an observer to an object is always equal to the angle of depression from the object to the observer.

Shortcut Trick

Ratio of Curved Surface Area to Volume is 2πrh ÷ πr²h = 2/r.

Given 2/r = 1/7 ⇒ r = 14 units.

Diameter (d) = 2 × 14 = 28 units. Given d : h = 7 : 6.

Since 7 units = 28, then 6 units (height) = 24 units.

Total Surface Area = 2 × (22/7) × 14 × (14 + 24) = 88 × 38 = 3344 unit².

∴ The correct answer is 3344 unit².

Alternate Method

Given:

Ratio of CSA to Volume = 1 : 7

Ratio of Diameter to Height = 7 : 6

Formula Used:

Curved Surface Area (CSA) = 2πrh

Volume (V) = πr²h

Total Surface Area (TSA) = 2πr(r + h)

Calculations:

⇒ CSA / Volume = 2πrh / πr²h = 2/r

⇒ 2/r = 1/7

⇒ r = 14 units

⇒ Diameter (d) = 2 × r = 2 × 14 = 28 units

⇒ d / h = 7 / 6

⇒ 28 / h = 7 / 6

⇒ h = (28 × 6) ÷ 7 = 4 × 6 = 24 units

⇒ TSA = 2 × (22/7) × 14 × (14 + 24)

⇒ TSA = 2 × 22 × 2 × 38

⇒ TSA = 88 × 38

⇒ TSA = 3344 unit²

∴ The correct answer is 3344 unit².

Additional Information

Cylinder Properties

For a cylinder with base radius r and height h, the volume is V = πr²h and the curved surface area is CSA = 2πrh.

Total Surface Area

TSA includes both the top and bottom circular faces plus the curved side: TSA = 2πrh + 2πr² = 2πr(r + h).

Hollow Cylinder

Volume of a hollow cylinder with outer radius R and inner radius r is V = π(R² − r²)h.

Shortcut Trick

Age ratio (8 years ago) = 13 : 1

Age ratio (after 12 years) = 3 : 1 (or 6 : 2 to match the age difference)

Equalizing the gap: 13 − 1 = 12 units and 3 − 1 = 2 units. Multiply second ratio by 6 → 18 : 6.

Ratio change: 13 → 18 (5 units) over 20 years (8 + 12). So, 1 unit = 4 years.

Age difference = (13 − 1) × 4 = 12 × 4 = 48 years.

∴ The correct answer is 48 years.

Alternate Method

Let the granddaughter's age 8 years ago be x and the grandmother's age be 13x.

Present ages: Granddaughter = x + 8, Grandmother = 13x + 8.

Ages after 12 years: Granddaughter = x + 20, Grandmother = 13x + 20.

According to the question:

⇒ 13x + 20 = 3 × (x + 20)

⇒ 13x + 20 = 3x + 60

⇒ 10x = 40

⇒ x = 4

Age difference (remains constant):

⇒ Difference = 13x − x = 12x

⇒ Difference = 12 × 4 = 48 years.

∴ The correct answer is 48 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two people always remains the same, regardless of how many years pass.

Ratio Balancing Method

To solve age problems quickly, make the difference between ratio parts equal by multiplying the ratios with the difference of the other ratio.

Linear Equation in Ages

For any "n" years in the future, add n to the present age; for "n" years in the past, subtract n from the present age.

Shortcut Trick

Check the unit digit: 6³ ends in 6 and 0³ ends in 0.

The unit digit of the result must be 6 − 0 = 6, which narrows it to 3096 or 3064.

Since 16³ is 4096 and 10³ is 1000, the difference is 3096.

∴ The correct answer is 3096.

Alternate Method

Given:

Expression = 16³ − 10³

Method Used: Direct expansion of cubes

Calculations:

⇒ 16³ = 16 × 16 × 16

⇒ 16³ = 256 × 16 = 4096

⇒ 10³ = 10 × 10 × 10 = 1000

⇒ Value = 4096 − 1000

⇒ Value = 3096

∴ The correct answer is 3096.

Additional Information

Difference of Cubes Identity

The expression can also be solved using the identity: a³ − b³ = (a − b)(a² + ab + b²).

Sum of Cubes Identity

Similarly, for addition, the formula used is: a³ + b³ = (a + b)(a² − ab + b²).

BODMAS Rule in Powers

According to the BODMAS rule, exponents or orders (cubes, squares) are calculated before performing addition or subtraction.

Shortcut Trick

Total decimal places in the numerator = 1 + 4 + 1 = 6.

Total decimal places in the denominator = 3 + 4 + 1 = 8.

Adjust the decimal difference by multiplying the numerator by 10^{8 − 6} = 100.

Simplify the integer fraction: (95 × 85 × 189) ÷ (21 × 17 × 19) = 5 × 5 × 9 = 225.

Evaluate the full expression: √16 + √900 + √(225 × 100) = 4 + 30 + 150 = 184.

∴ The correct answer is 184.

Alternate Method

Given:

Expression: $\sqrt{16} + \sqrt{900} + \sqrt{\frac{9.5 × 0.0085 × 18.9}{0.021 × 0.0017 × 1.9}}$

Calculation:

⇒ $\sqrt{16} + \sqrt{900} + \sqrt{\frac{9.5 × 0.0085 × 18.9}{0.021 × 0.0017 × 1.9}}$

⇒ $4 + 30 + \sqrt{\frac{95 × 85 × 189}{0.21 × 17 × 19}}$

⇒ $34 + \sqrt{\frac{5 × 5 × 189 × 100}{21 }}$

⇒ $34 + \sqrt{{5 × 5 × 9 × 100}}$

⇒ 34 + 5 × 3 × 10

⇒ 4 + 30 + 150

⇒ 184

∴ The correct answer is 184.

Additional Information

BODMAS Rule

When solving simplifications, follow the order of operations: Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction.

Properties of Square Roots

Multiplication and division inside roots can be separated: √(a × b) = √a × √b and √(a ÷ b) = √a ÷ √b. This allows for simplifying complex terms easily before evaluating the root.

Exponent Division Rule

When dividing algebraic terms or numbers with the same base, subtract the exponent of the denominator from the exponent of the numerator: x^a ÷ x^b = x^{a − b}. In this problem, 10⁻⁶ ÷ 10⁻⁸ = 10².

Shortcut Trick

In decimal multiplication, the total number of decimal places in the product is the sum of the decimal places in the multipliers.

Here, the number 4.3 has 1 decimal place.

So, 4.3 × 4.3 will have 1 + 1 = 2 decimal places in total.

We are given that 43 × 43 = 1849.

Placing the decimal point 2 places from the right gives 18.49.

∴ The correct answer is 18.49.

Alternate Method

Given:

43 × 43 = 1849

Calculation:

⇒ We need to find the exact value of 4.3 × 4.3

⇒ First, express the decimal numbers as fractions:

⇒ 4.3 = 43 ÷ 10

⇒ 4.3 × 4.3 = (43 ÷ 10) × (43 ÷ 10)

⇒ 4.3 × 4.3 = (43 × 43) ÷ (10 × 10)

⇒ Substitute the given value of 43 × 43 = 1849

⇒ 4.3 × 4.3 = 1849 ÷ 100

⇒ 4.3 × 4.3 = 18.49

∴ The correct answer is 18.49.

Additional Information

Multiplying Decimals Rule

Multiply the numbers as if they were whole numbers ignoring the decimals. Then, count the total number of decimal digits in the question and place the decimal point so the product has the exact same total number of decimal places.

Dividing by Powers of 10

To divide any number by 10, 100, 1000, etc., shift the decimal point to the left by the exact number of zeros present in the denominator.

Squares Ending in 3

The square of any number that ends with the digit 3 will always have 9 as its unit digit. For example, 3² = 9, 13² = 169, 23² = 529, and 43² = 1849.

Shortcut Trick

When the distance traveled is the same for both parts of the journey, the Average Speed is the harmonic mean of the speeds.

Average Speed = (2 × s₁ × s₂) ÷ (s₁ + s₂)

Average Speed = (2 × 52 × 118) ÷ (52 + 118) = 12272 ÷ 170

Average Speed = 72.188... ≈ 72.19 km/hr

∴ The correct answer is 72.19 km/hr.

Alternate Method

Given: Distance (D) = 966 km, Forward Speed (s₁) = 52 km/hr, Return Speed (s₂) = 118 km/hr.

Formula Used: Average Speed = Total Distance ÷ Total Time

⇒ Total Distance = 966 + 966 = 1932 km

⇒ Time to go (t₁) = 966 ÷ 52 = 18.5769 hours

⇒ Time to return (t₂) = 966 ÷ 118 = 8.1864 hours

⇒ Total Time = 18.5769 + 8.1864 = 26.7633 hours

⇒ Average Speed = 1932 ÷ 26.7633 = 72.188...

∴ The correct answer is 72.19 km/hr.

Additional Information

Average Speed (Equal Distances)

If a person covers equal distances at speeds x and y, the average speed is 2xy ÷ (x + y).

Average Speed (Equal Time)

If a person travels for equal time intervals at speeds x and y, the average speed is (x + y) ÷ 2.

General Average Speed Formula

Average Speed = (Total Distance) ÷ (Total Time taken for the journey).

Shortcut Trick

Total Capacity (LCM of 12, 18) = 36 units

Efficiency (P + Q) = 36 ÷ 12 = 3 units/hr

Efficiency (P) = 36 ÷ 18 = 2 units/hr

Efficiency (Q) = 3 − 2 = 1 unit/hr

Time for Q to fill 1/4^th = (36 × 1/4) ÷ 1 = 9 hours

∴ The correct answer is 9 hours.

Alternate Method

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

⇒ Efficiency of Q = 3 − 2 = 1 unit/hr

⇒ Total capacity of cistern = 36 units

⇒ One-fourth of the cistern = 36 ÷ 4 = 9 units

⇒ Time taken by Q = 9 units ÷ 1 unit/hr = 9 hours

∴ The correct answer is 9 hours.

Additional Information

Pipes & Cisterns Efficiency

The rate of work done by a pipe is its Efficiency. Total Work = Time taken × Efficiency.

Inlet and Outlet Pipes

Inlet pipes (filling) have positive efficiency, while outlet pipes (emptying/leakage) have negative efficiency.

Time Calculation

If a pipe fills a tank in 'x' hours, its 1-hour work is 1/x of the total capacity.

Shortcut Trick The best scheme for the shopkeeper is the one offering the lowest effective discount. Scheme A: 19 + 25 − (19 × 25) ÷ 100 = 44 − 4.75 = 39.25% Scheme B: [6 ÷ (7 + 6)] × 100 = 6 ÷ 13 × 100 ≈ 46.15% Scheme C: 50% (Fixed single discount) Scheme D: 3 + 39 − (3 × 39) ÷ 100 = 42 − 1.17 = 40.83% Comparing discounts: 39.25% < 40.83% < 46.15% < 50%. ∴ The correct answer is A.

Alternate Method Given → Four different discount schemes A, B, C, and D. Formulas: Successive Discount = a + b − (ab ÷ 100) | Buy X Get Y Free Discount% = [Y ÷ (X + Y)] × 100.

⇒ Calculate effective discount for Scheme A: 19 + 25 − 4.75 = 39.25%. ⇒ Calculate effective discount for Scheme B (Buy 7 Get 6 Free): (6/13) × 100 = 46.15%. ⇒ Calculate effective discount for Scheme D: 3 + 39 − 1.17 = 40.83%. ⇒ Scheme C is given as a flat 50% discount. ⇒ For a shopkeeper, the minimum discount results in the maximum profit. ⇒ Minimum value among {39.25, 46.15, 50, 40.83} is 39.25%. ∴ The correct answer is A.

Additional Information Perspective of Best Scheme The Best Scheme for Customer is the one with the highest discount, whereas the Best Scheme for Shopkeeper is the one with the lowest discount. Buy X Get Y Free Discount% = [Number of free items ÷ Total number of items (Paid + Free)] × 100.

Shortcut Trick

Initial votes share = 63%

Decrease in support = 24%

Remaining support = 100% − 24% = 76%

Final percentage = 63 × 0.76 = 47.88%

∴ The correct answer is 47.88%.

Alternate Method

Given:

Votes in first round = 63% of total

Decrease in second round = 24% of his current share

Calculations:

⇒ Amount of decrease = 24% of 63%

⇒ Decrease = (24 ÷ 100) × 63 = 0.24 × 63 = 15.12%

⇒ Final votes = 63% − 15.12%

⇒ Final votes = 47.88%

∴ The correct answer is 47.88%.

Additional Information

Percentage Decrease

To calculate a percentage decrease on a base value, multiply the base by (100 − decrease %)/100.

Successive Percentage

When a value changes successively, the final value is A × (1 ± x/100) × (1 ± y/100).

Base Value Concept

Always identify if the percentage change is applied to the original total or the intermediate value.

Shortcut Trick

For Mean Proportion b between a and c: b² = a × c

Given b = 42 and c = 294, find x (where x is a).

x = (42 × 42) ÷ 294

x = 1764 ÷ 294 = 6

∴ The correct answer is 6.

Alternate Method

Given:

Mean Proportion (b) = 42

First term (a) = x

Third term (c) = 294

Formula Used:

If b is the mean proportion between a and c, then a : b = b : c ⇒ b² = ac

Calculations:

⇒ 42² = x × 294

⇒ 1764 = x × 294

⇒ x = 1764 ÷ 294

⇒ x = 6

∴ The correct answer is 6.

Additional Information

Mean Proportion

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

Third Proportion

If a : b = b : c, then c is called the third proportion to a and b, calculated as c = b² ÷ a.

Fourth Proportion

If a : b = c : d, then d is the fourth proportion, calculated as d = (b × c) ÷ a.

Shortcut Trick

Check options for numerator = denominator − 6.

(1) 5 − 4 = 1 | (2) 11 − 6 = 5 | (3) 8 − 5 = 3 | (4) 13 − 7 = 6.

Only 7/13 has a difference of 6. Let us verify the condition: (7+2)/(13+2) = 9/15 = 3/5.

Wait, if the condition becomes 2/3, let us check (x − 6 + 2) / (x + 2) = 2/3 ⇒ (x − 4) / (x + 2) = 2/3 ⇒ 3x − 12 = 2x + 4 ⇒ x = 16. Fraction = 10/16 = 5/8.

∴ The correct answer is 5/8.

Alternate Method

Let the denominator be x.

Then, the numerator = x − 6.

Original Fraction = (x − 6) / x

According to the question, if 2 is added to both:

⇒ ((x − 6) + 2) / (x + 2) = 2/3

⇒ (x − 4) / (x + 2) = 2/3

⇒ 3 × (x − 4) = 2 × (x + 2)

⇒ 3x − 12 = 2x + 4

⇒ 3x − 2x = 4 + 12

⇒ x = 16

⇒ Numerator = 16 − 6 = 10

⇒ Fraction = 10 / 16

⇒ Simplest form = 5 / 8

∴ The correct answer is 5/8.

Additional Information

Simplest Form of a Fraction

A fraction is in its simplest form when the Highest Common Factor (HCF) of the numerator and denominator is 1.

Proper vs Improper Fractions

A proper fraction has a numerator less than the denominator (e.g., 5/8), while an improper fraction has a numerator greater than or equal to the denominator.

Ratio and Proportion

Fractional equality a/b = c/d can be solved using cross-multiplication: a × d = b × c.

Shortcut Trick

Total Profit = (Profit in Transaction 1) + (Profit or Loss in Transaction 2)

Transaction 1: CP₁ = 24,000 + 3,000 = 27,000; SP₁ = 30,000 ⇒ Profit₁ = 3,000

Transaction 2: Loss₂ = 1,200

Net Profit = 3,000 − 1,200 = 1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given: Initial purchase price = ₹24,000, Repair cost = ₹3,000, First selling price = ₹30,000, Buy back price = ₹28,000, Loss on second sale = ₹1,200.

Formulas: Total CP = Purchase Price + Overheads | Profit = SP − CP | Loss = CP − SP | Net Profit = Total SP − Total CP

⇒ Total Cost Price for Transaction 1 = 24,000 + 3,000 = 27,000

⇒ Profit from Transaction 1 = 30,000 − 27,000 = 3,000

⇒ Cost Price for Transaction 2 = 28,000

⇒ Selling Price for Transaction 2 = CP − Loss = 28,000 − 1,200 = 26,800

⇒ Total CP = 27,000 + 28,000 = 55,000

⇒ Total SP = 30,000 + 26,800 = 56,800

⇒ Overall Profit = 56,800 − 55,000 = 1,800

∴ The correct answer is ₹1,800.

Additional Information

Treatment of Overhead Expenses

Any money spent on repairs, transportation, or labor after buying an item is added to the purchase price to calculate the Effective Cost Price.

Net Profit/Loss in Multiple Transactions

The Overall Profit or Loss is the algebraic sum of results from each individual transaction. Profit is taken as positive (+), and Loss is taken as negative (−).

Shortcut Trick

Number 648XYZ is divisible by 10 ⇒ Z = 0.

Divisible by 4 ⇒ YZ must be divisible by 4. Since Z = 0, smallest Y = 0 (00 is divisible by 4).

Divisible by 9 ⇒ 6 + 4 + 8 + X + 0 + 0 = 18 + X must be divisible by 9 ⇒ X = 0 or 9.

Using X = 0, Y = 0, Z = 0: X + Z − Y = 0 + 0 − 0 = 0.

∴ The correct answer is 0.

Alternate Method

Given: Number 648XYZ is divisible by 4, 9, and 10. Y is the smallest possible digit.

Calculations:

⇒ Divisibility by 10: The last digit must be 0. ∴ Z = 0.

⇒ Divisibility by 4: The last two digits (YZ) must be divisible by 4. Since Z = 0, Y can be 0, 2, 4, 6, 8.

⇒ Smallest Y: The question states Y is the smallest possible number. ∴ Y = 0.

⇒ Divisibility by 9: Sum of digits (6 + 4 + 8 + X + Y + Z) must be divisible by 9.

⇒ 18 + X + 0 + 0 = 18 + X.

⇒ For 18 + X to be divisible by 9, X can be 0 or 9.

⇒ If X = 0, then X + Z − Y = 0 + 0 − 0 = 0.

⇒ If X = 9, then X + Z − Y = 9 + 0 − 0 = 9 (Not in options).

∴ The correct answer is 0.

Additional Information

Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4 or are 00.

Divisibility by 10

A number is divisible by 10 if its unit digit is strictly 0.

Shortcut Trick

Arrange the Basketball scores in ascending order: 2, 5, 7, 8, 10, 10, 10, 11, 12, 14, 17, 18, 24, 25, 27.

Total number of match scores (n) = 15 (Odd).

For odd n, Median = ((n + 1) ÷ 2)^th term.

Median = ((15 + 1) ÷ 2)^th term = 8^th term.

The 8^th value in the sorted list is 11.

∴ The correct answer is 11 points.

Alternate Method

Given Data: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 10, 8, 10, 12, 11.

Formula Used: Median = ((n + 1) ÷ 2)^th term (where n is odd).

⇒ Total observations (n) = 15

⇒ Sorted Data in ascending order: 2, 5, 7, 8, 10, 10, 10, 11, 12, 14, 17, 18, 24, 25, 27

⇒ Median index = (15 + 1) ÷ 2 = 8^th term

⇒ Median value = Value of 8^th term

⇒ Median = 11

∴ The correct answer is 11 points.

Additional Information

Median for Even n

If the number of observations (n) is even, the median is the average of the (n ÷ 2)^th and ((n ÷ 2) + 1)^th terms.

Mode of Data

The mode is the most frequently occurring value. In this set, 10 occurs three times, so Mode = 10.

Arithmetic Mean

Mean = (Sum of all values) ÷ (Total number of values). It represents the central balance point of the data.

Shortcut Trick

Let the pipes be A, B, and C (waste pipe).

Total Capacity (LCM of 100, 25, 40) = 200 units.

Efficiency of A = 2, B = 8, and A+B−C = 5.

Efficiency of C = (2 + 8) − 5 = 5 units/min.

Time taken by pipe C = 200 ÷ 5 = 40 min.

∴ The correct answer is 40 min.

Alternate Method

Calculations:

⇒ Let the efficiency of the waste pipe (C) be x.

⇒ Total efficiency = Efficiency of A + Efficiency of B − Efficiency of C

⇒ 5 = 2 + 8 − x

⇒ 5 = 10 − x

⇒ x = 10 − 5 = 5 units/min

⇒ Time taken by waste pipe to empty the tank = Total capacity ÷ Efficiency of C

⇒ Time = 200 ÷ 5 = 40 min

∴ The correct answer is 40 min.

Additional Information

Pipes and Cisterns Logic

Inlet pipes have positive efficiency, while outlet (leak/waste) pipes have negative efficiency.

Total Work Formula

Total Work = Efficiency × Time. Using the LCM method is the fastest way to solve multiple pipe problems.

Fractional Method

If a pipe fills a tank in 'x' minutes, its work in 1 minute is 1/x of the total tank volume.

Shortcut Trick

Net compound interest rate for 2 years at 20% = 20 + 20 + (20 × 20) ÷ 100 = 44%.

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

Amount = 144% × 2400 = 1.44 × 2400 = ₹3456.

∴ The correct answer is ₹3456.

Alternate Method

Given: Principal (P) = ₹2,400, Rate (R) = 20% per annum, Time (n) = 2 years.

Formula Used: Amount (A) = P(1 + R/100)ⁿ

⇒ Amount (A) = 2400 × (1 + 20 ÷ 100)²

⇒ Amount (A) = 2400 × (1 + 1 ÷ 5)²

⇒ Amount (A) = 2400 × (6 ÷ 5) × (6 ÷ 5)

⇒ Amount (A) = 2400 × 36 ÷ 25

⇒ Amount (A) = 96 × 36

⇒ Amount (A) = 3456

∴ The correct answer is ₹3456.

Additional Information

Compound Interest (CI)

It is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.

Difference between CI and SI

For 2 years, the difference between Compound Interest and Simple Interest is given by the formula: D = P × (R ÷ 100)².

Effective Rate for Half-Yearly Compounding

When interest is compounded half-yearly, the annual rate (R) is divided by 2, and the time period (n) is multiplied by 2.

Shortcut Trick

Total Amount = ₹1,287

Total Deductions = 36 + 22 + 29 = ₹87

Remaining Amount = 1,287 − 87 = ₹1,200

Remaining share ratio = 17 : 7 : 16 (Total parts = 40)

Monika's remaining share = (7 ÷ 40) × 1,200 = ₹210

Actual share = 210 + 22 = ₹232

∴ The correct answer is ₹232.

Alternate Method

Given:

Total amount = ₹1,287

Individual deductions: Rahul = ₹36, Monika = ₹22, Yuvraj = ₹29

Ratio after deduction = 17 : 7 : 16

Calculations:

⇒ Let the shares after deduction be 17x, 7x, and 16x.

⇒ (17x + 36) + (7x + 22) + (16x + 29) = 1287

⇒ 40x + 87 = 1287

⇒ 40x = 1287 − 87

⇒ 40x = 1200

⇒ x = 1200 ÷ 40 = 30

⇒ Monika's share after deduction = 7 × 30 = ₹210

⇒ Monika's actual share = 210 + 22 = ₹232

∴ The correct answer is ₹232.

Additional Information

Ratio and Proportion

If a total sum S is divided into a ratio a : b : c, then the share of b is calculated as (b ÷ (a + b + c)) × S.

Deduction Concept

When fixed amounts are deducted from shares to obtain a new ratio, subtract the total deduction from the original sum before applying the ratio parts.

Shortcut Trick

Speed ratio (Train : Car) = 160 : 100 = 8 : 5.

Time ratio (Train : Car) = 5 : 8 (since distance is constant).

Difference in time units = 8 − 5 = 3 units.

Given 3 units = 20 minutes ⇒ 8 units (Car's time) = (20 ÷ 3) × 8 = 160/3 minutes = 8/9 hours.

Speed of car = Distance ÷ Time = 60 ÷ (8/9) = 540 ÷ 8 = 67.5 km/h.

∴ The correct answer is 67.5 km/h.

Alternate Method

Given: Distance = 60 km, Train speed is 60% more than Car, Time delay = 20 mins.

Formula: Time = Distance ÷ Speed

⇒ Let Speed of car = x km/h.

⇒ Speed of train = 1.6x km/h.

⇒ Time taken by car − Time taken by train = 20/60 hours.

⇒ 60/x − 60/1.6x = 1/3

⇒ 60/x × (1 − 1/1.6) = 1/3

⇒ 60/x × (0.6/1.6) = 1/3

⇒ 60/x × (3/8) = 1/3

⇒ 22.5/x = 1/3

⇒ x = 22.5 × 3 = 67.5 km/h.

∴ The correct answer is 67.5 km/h.

Additional Information

Inverse Proportion of Speed and Time

When distance is constant, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Relative Speed

In same direction, relative speed = S₁ − S₂. In opposite direction, relative speed = S₁ + S₂.

Average Speed

For constant distance segments, Average Speed = 2S₁S₂ ÷ (S₁ + S₂).

Shortcut Trick

Ratio of Income (Ashish : Kiran) = 5 : 8

Ratio of Expenditure (Ashish : Kiran) = 2 : 4 = 1 : 2

Since Savings (Income − Expenditure) are same (₹75,000), make the ratio difference equal: (5−1) ≠ (8−2). Adjust expenditure ratio by ×3 → 3 : 6. Now (5−3) = (8−6) = 2 units.

2 units = ₹75,000 ⇒ 1 unit = ₹37,500.

Income of Ashish = 5 units = 5 × ₹37,500 = ₹1,87,500.

∴ The correct answer is ₹1,87,500.

Alternate Method

Given: Ratio of incomes = 5 : 8, Ratio of expenditures = 2 : 4, Constant savings = ₹75,000.

Formula Used: Income − Savings = Expenditure

⇒ (5x − 75000) / (8x − 75000) = 2 / 4

⇒ (5x − 75000) / (8x − 75000) = 1 / 2

⇒ 2 × (5x − 75000) = 1 × (8x − 75000)

⇒ 10x − 1,50,000 = 8x − 75,000

⇒ 2x = 1,50,000 − 75,000

⇒ 2x = 75,000

⇒ x = 37,500

⇒ Monthly Income of Ashish = 5x = 5 × 37,500 = ₹1,87,500

∴ The correct answer is ₹1,87,500.

Additional Information

Income, Expenditure, and Savings Relationship

The fundamental equation for these problems is Income = Expenditure + Savings.

Ratio Balancing Method

When the difference (Savings) is constant, you can solve by making the vertical difference between the ratio parts equal for both individuals.

Cross-Multiplication Method

For different savings amounts, use the formula (I₁ × E₂ − I₂ × E₁) units = (S₁ × E₂ − S₂ × E₁).

Shortcut Trick

Identify the base value (A's marks) and the comparing value (B's marks).

Required percentage = (B ÷ A) × 100

Required percentage = (50 ÷ 75) × 100 = 2/3 × 100 = 66.666...

Rounding to two decimal places, we get 66.67%.

∴ The correct answer is 66.67%.

Alternate Method

Given:

Marks obtained by A = 75

Marks obtained by B = 50

Maximum Marks = 90 (Not required for this specific ratio calculation)

Formula Used:

Percentage = (Value / Base Value) × 100

Calculations:

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

⇒ Percentage = (50 / 75) × 100

⇒ Percentage = (2 / 3) × 100

⇒ Percentage = 200 / 3

⇒ Percentage = 66.6666...

⇒ Rounded Percentage ≈ 66.67%

∴ The correct answer is 66.67%.

Additional Information

Percentage Change

To find the percentage by which A is more than B: [(A − B) / B] × 100. Always divide by the 'than' value.

Fraction to Percentage Conversion

Common fractions: 1/2 = 50%, 1/3 = 33.33%, 2/3 = 66.67%, 1/4 = 25%, 1/5 = 20%.

Base Value Identification

The value following the word 'of' or 'than' is typically treated as the denominator (base value) in percentage problems.

Shortcut Trick

Area of a regular polygon with n sides inscribed in a circle of radius R is given by: Area = (n/2) × R² × sin(360°/n).

For Hexagon (n = 6): Area₆ = (6/2) × R² × sin(60°) = 3 × R² × (√3/2) = (3√3/2)R².

For Pentagon (n = 5): Area₅ = (5/2) × R² × sin(72°) = (5/2) × R² × (√(10 + 2√5)/4) = (5√(10 + 2√5)/8)R².

Ratio = (3√3/2) : (5√(10 + 2√5)/8) ⇒ Multiply by 8 ⇒ 12√3 : 5√(10 + 2√5).

∴ The correct answer is 12√3 : 5√(10 + 2√5).

Alternate Method

Given:

Radius of the circumcircle = R

Number of sides of first polygon (Hexagon) = 6

Number of sides of second polygon (Pentagon) = 5

Formula Used:

Area of regular n-gon = (n/2) × R² × sin(360°/n)

Calculations:

⇒ Area of Hexagon (A₁) = (6/2) × R² × sin(360°/6)

Shortcut Trick

For two successive discounts a% and b%, use the successive discount formula.

Equivalent Discount = a + b − (a × b) ÷ 100

Equivalent Discount = 12 + 13 − (12 × 13) ÷ 100

Equivalent Discount = 25 − 156 ÷ 100 = 25 − 1.56

Equivalent Discount = 23.44%

∴ The correct answer is 23.44%.

Alternate Method

Given: First discount (d₁) = 12%, Second discount (d₂) = 13%.

Formulas: Final SP = MP × (100 − d₁)% × (100 − d₂)% | Equivalent Discount% = (MP − SP) ÷ MP × 100

⇒ Let the Marked Price (MP) be 100.

⇒ Price after 1^st discount = 100 × (100 − 12) ÷ 100 = 88.

⇒ Price after 2^nd discount (Final SP) = 88 × (100 − 13) ÷ 100 = 88 × 0.87 = 76.56.

⇒ Single Equivalent Discount = MP − Final SP = 100 − 76.56 = 23.44%.

∴ The correct answer is 23.44%.

Additional Information

Successive Discount Concept

When a series of discounts are applied on the Marked Price, each subsequent discount is calculated on the reduced value obtained after the previous discount.

Net Discount for 3 values

For three successive discounts a, b, and c, calculate the equivalent of 'a' and 'b' first, then use that result with 'c'.

Shortcut Trick

Total Expenditure = 1120 + 910 = ₹2030

Total Books purchased = 52 + 47 = 99

Average Price = Total Cost ÷ Total Books = 2030 ÷ 99

Average Price ≈ ₹20.51

∴ The correct answer is ₹20.51.

Alternate Method

Given:

Number of books from Shop 1 = 52 | Cost = ₹1,120

Number of books from Shop 2 = 47 | Cost = ₹910

Formula Used:

Average = (Total Sum of Observations) ÷ (Total Number of Observations)

Calculations:

⇒ Total Cost Price = 1120 + 910 = ₹2030

⇒ Total Quantity (Books) = 52 + 47 = 99

⇒ Average price per book = 2030 ÷ 99

⇒ Average price per book = 20.50505...

⇒ Rounding off to two decimal places, we get ₹20.51

∴ The correct answer is ₹20.51.

Additional Information

Weighted Average

When items have different frequencies or counts, the average is calculated as (n₁x₁ + n₂x₂) ÷ (n₁ + n₂).

Rounding Off Rules

If the third decimal digit is 5 or more, the second decimal digit is increased by 1 (e.g., 20.505 → 20.51).

Arithmetic Mean

It represents the central value of a data set, calculated by dividing the sum of all values by the total count of values.

Shortcut Trick

Directly calculate the Total Surface Area (TSA) ratio.

TSA of one small cube = 6 × 4² = 6 × 16 = 96 cm².

TSA of original cuboid = 2 × (18×12 + 12×8 + 18×8) = 2 × (216 + 96 + 144) = 912 cm².

Required Ratio = 96 : 912 = 1 : 9.5 = 2 : 19.

∴ The correct answer is 2 : 19.

Alternate Method

Given:

Dimensions of cuboid: Length (L) = 18 cm, Width (B) = 12 cm, Height (H) = 8 cm.

Edge of small cube (a) = 4 cm.

Formula Used:

TSA of cuboid = 2(LB + BH + LH)

TSA of cube = 6a²

Calculations:

⇒ Total Surface Area of the cuboid = 2 × (18 × 12 + 12 × 8 + 8 × 18)

⇒ TSA of cuboid = 2 × (216 + 96 + 144) = 2 × (456) = 912 cm²

⇒ Total Surface Area of one small cube = 6 × (4)²

⇒ TSA of cube = 6 × 16 = 96 cm²

⇒ Required Ratio = TSA of cube : TSA of cuboid

⇒ Ratio = 96 : 912

⇒ Dividing both by 48: 96 ÷ 48 = 2 and 912 ÷ 48 = 19

⇒ Ratio = 2 : 19

∴ The correct answer is 2 : 19.

Additional Information

Volume of Solids

The volume of a cuboid is L × B × H and a cube is a³. When melting and recasting, the total volume remains constant: n × a³ = L × B × H.

Lateral Surface Area (LSA)

LSA of a cuboid = 2H(L + B), and LSA of a cube = 4a². LSA only considers the area of the four side walls.

Diagonal of Solids

Diagonal of a cuboid = √(L² + B² + H²), and diagonal of a cube = a√3.

Given:

Hitesh starts from Point A and drives 62 km East.

He turns left → 36 km North.

He turns left → 78 km West.

He turns left → 71 km South.

He takes a final left → 16 km East.

 

So,To reach Point A again, he must travel 35 km North.

Hence, the correct answer is Option 3.

The logic followed here is:

For, VHEP → OAXI

Now, TFCN → MYVG

Now, applying the same logic for YKHS:

So, YKHS → RDAL

Hence, the correct answer is Option 4.

Given Statements:

All mints are lamps.

All mints are jams.

All mints are kites.

The least possible Venn diagram for the given statements is:

Conclusions:

I. All lamps are jams – Does not follow (Only mints are common to both lamps and jams; lamps may exist outside jams).

II. Some jams are kites – Follows (Since all mints are jams and all mints are kites, at least some jams are definitely kites).

So, only conclusion II follows.

Hence, the correct answer is Option 2.

Given: S, P, D, M, and R are sitting around a circular table facing the centre.

Only two people sit between R and M when counted from the left of R.

D is an immediate neighbour of M and R.

P is sitting to the immediate left of R.

 

 

Therefore, the immediate neighbours of S are P and M.

Hence, the correct answer is Option 2.

Given:

Rank from top = 56

Rank from bottom = 71

Total number of students = (Rank from top + Rank from bottom) − 1

= (56 + 71) − 1

= 127 − 1

= 126

Here, there are 126 students in his class.

Hence, the correct answer is "Option 4".

The logic followed here is:

For, NJL → OKM

Now, CYA → DZB

Now, checking the options:

Option 1) IEG → IFG

Option 2) IDE → JFH

Option 3) IEG → JFH

Option 4) IDE → IEG

Hence, the correct answer is Option 3.

The logic followed here is:

thus, "63" will replace the question mark in the given series.

Hence, the correct option is "Option 3"

The logic followed here is:

So, 'ZAW 76' will replace the question mark in the given series.

Hence, the correct answer is "Option 1".

BODMAS Table: 

Given equation: 92 A 2 D (35 A 7) B 4 C 30 A 3 C 5 B 11 = ?

Now, replacing the symbols as:

 

⇒ 92 ÷ 2 − (35 ÷ 7) × 4 + 30 ÷ 3 + 5 × 11 = ?

⇒ 46 − 5 × 4 + 10 + 5 × 11 = ?

⇒ 46 − 20 + 10 + 55 = ?

⇒ 26 + 10 + 55 = ?

⇒ 36 + 55 = ?

⇒ 91

Hence, the correct answer is Option 3.

Family Chart:

Given: C # N × K % T + R

It means: C is the daughter of N, N is the wife of K, K is the father of T, T is the mother of R.

Here, C is related to R as Mother’s sister

Hence, the correct answer is Option 4.

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

Now,

We need to identify the unique single-digit numbers present in the series and count those that are exactly divisible by 3.

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

Now, identifying unique digits: 1, 2, 3, 4, 5, 6, 7, 8, 9

Now, only valid unique digits: {3, 6, 9}

Total unique single-digit numbers divisible by 3 = 3

Hence, the correct answer is "Option 3".