RRB Group D Official Paper (Held On: 29 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is The Poles.

Key Points

• The acceleration due to gravity (g) is the acceleration gained by an object due to the gravitational force of the Earth. Its value on the surface of the Earth is determined by the formula g = GM / R², where G is the universal gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.
• The Earth is not a perfect sphere but an oblate spheroid, meaning it is flattened at the Poles and bulges at the Equator. This shape is caused by the Earth's rotation.
• Because of this shape, the equatorial radius (approximately 6,378 km) is about 21 km longer than the polar radius (approximately 6,357 km).
• Since the value of g is inversely proportional to the square of the distance from the center of the Earth (R), the value of g is highest where the radius is the smallest. Therefore, g is maximum at the Poles.
• Quantitatively, the value of g at the Poles is approximately 9.832 m/s², which is greater than the value at the Equator. Hence, option 1 is correct.

Additional Information

• Effect of Rotation (Centrifugal Force): The rotation of the Earth creates a centrifugal force that acts outward, away from the Earth's axis. This force is strongest at the Equator and zero at the Poles. This outward force partially counteracts the pull of gravity, further reducing the effective value of g at the Equator.
• The Equator: At the Equator (0 degrees latitude), the combination of the maximum radius and the maximum centrifugal force results in the minimum value of acceleration due to gravity, which is approximately 9.780 m/s². Hence, option 4 is incorrect.
• Tropics: The Tropic of Cancer (23.5 degrees North) and the Tropic of Capricorn (23.5 degrees South) are located at intermediate latitudes. The value of g at these locations is higher than at the Equator but significantly lower than at the Poles. Hence, options 2 and 3 are incorrect.
• Weight vs. Mass: While the mass of an object remains constant regardless of location, its weight (W = mg) changes. An object will weigh the most at the Poles and the least at the Equator because of the variation in g.

Important Points

• Variation with Altitude: As one moves higher above the Earth's surface (increasing altitude), the value of g decreases because the distance from the center of the Earth increases.
• Variation with Depth: If one goes below the surface of the Earth (increasing depth), the value of g also decreases. This is because the mass of the Earth contributing to the gravitational pull decreases. At the Center of the Earth, the value of g becomes zero.
• Standard Gravity: For most physics calculations, a standard gravity value of 9.80665 m/s² is used, which represents an average value at 45 degrees latitude at sea level.
• Gravitational Constant: The value of G is 6.674 x 10^-11 Nm²/kg², which was first measured by Henry Cavendish in 1798.

The correct answer is Only a.

Key Points

• The primary reason for spreading clothes in the open sky is that increasing the surface area directly leads to an increase in the rate of evaporation. Since evaporation is a surface phenomenon, a larger area allows more water molecules to acquire enough kinetic energy to escape into the atmosphere simultaneously. Hence, statement a is correct.
• Spreading the cloth actually increases its air resistance, making it more likely to be carried away by a strong breeze if not secured. The claim that it prevents the cloth from flying due to wind speed is logically and physically inconsistent with the act of spreading. Hence, statement b is incorrect.
• While spreading clothes may help in reducing wrinkles or creases as they dry, it is a secondary aesthetic benefit and not the fundamental physical reason related to the efficiency of the drying process in the open sky. Hence, statement c is incorrect.
• In the context of physics and thermal properties, the time taken for a liquid to change into vapor at a temperature below its boiling point is minimized when the liquid is spread over a maximum area.
• Therefore, the correct answer is Only a.

Additional Information

• Evaporation:
• It is the process by which water changes from a liquid to a gas or vapor. It is a vital part of the water cycle and occurs at all temperatures, unlike boiling which occurs at a fixed temperature (e.g., 100 degrees Celsius for water at sea level).
• Energy in the form of heat (often from the Sun) is required to break the molecular bonds of the liquid.
• Factors Affecting the Rate of Evaporation:
• Surface Area: As seen in this example, a larger exposed area facilitates faster vaporization because more particles are at the surface.
• Temperature: An increase in ambient temperature provides more heat energy to the water molecules, increasing their velocity and the rate of evaporation.
• Humidity: The rate of evaporation is inversely proportional to the humidity of the surrounding air. If the air is already saturated with water vapor, the drying process slows down significantly.
• Wind Speed: Higher wind speeds help move the saturated air away from the surface of the wet cloth, replacing it with dry air, which accelerates the process.
• Practical Applications:
• This principle is used in cooling towers, salt production (from seawater in large shallow ponds), and even in the biological cooling mechanism of sweating in humans.

p data-end="57" data-start="0">The correct answer is Black.

 

Key Points

• Copper(II) oxide, represented by the chemical formula CuO, is a black solid powder. It is the higher oxide of copper.
• When black copper oxide is added to dilute sulphuric acid (H₂SO₄), a chemical reaction occurs where the copper oxide dissolves in the acid.
• The reaction results in the formation of Copper(II) sulphate (CuSO₄) and water (H₂O). The resulting solution appears blue-green or blue in color because of the formation of copper(II) ions in the aqueous solution.
• The chemical equation for this reaction is: CuO (s) + H₂SO₄ (aq) → CuSO₄ (aq) + H₂O (l). In this reaction, the black solid reacts with the colorless acid to produce a colored salt solution.
• This type of reaction is classified as a neutralization reaction because a metal oxide (which is basic in nature) reacts with an acid to form a salt and water.

Additional Information

• Nature of Copper Oxides:
• There are two main oxides of copper: Copper(II) oxide (CuO), which is black, and Copper(I) oxide (Cu₂O), which is red or brownish-red.
• Copper(II) oxide is used as a pigment in ceramics to produce blue, green, or red glazes. It is also used as a catalyst in various chemical reactions and in the production of superconductors.
• Copper(II) Sulphate (CuSO₄):
• Copper(II) sulphate is most commonly found as the pentahydrate (CuSO₄·5H₂O), which is bright blue. This form is often called blue vitriol.
• It has wide applications as a fungicide in agriculture (like Bordeaux mixture), in electroplating, and as a mordant in dyeing processes.
• Chemical Indicators and Reactions:
• Most metal oxides are basic. When they react with dilute acids, they typically form soluble salts and water.
• For example, if Copper(II) oxide reacts with Hydrochloric acid (HCl), it forms Copper(II) chloride (CuCl₂), which gives a blue-green color to the solution.
• The specific blue-green color mentioned in the question is characteristic of the Cu²⁺ ions surrounded by water molecules (coordinated water) in the solution.

Important Points

• Physical Properties:
• Copper(II) oxide is insoluble in water but soluble in acids. It has a melting point of approximately 1326 °C.
• The molar mass of CuO is approximately 79.54 g/mol.
• Safety and Occurrence:
• In nature, Copper(II) oxide occurs as the mineral tenorite. It is often formed on the surface of copper metal when heated in the presence of oxygen.
• In the laboratory, it can be prepared by heating copper(II) nitrate, copper(II) hydroxide, or copper(II) carbonate.

The correct answer is remains constant.

Key Points

• Uniform Circular Motion (UCM) is defined as the motion of an object traveling along a circular path at a constant speed. This means the object covers equal distances along the circumference in equal intervals of time.
• In UCM, the magnitude of the velocity (which is the speed) does not change. Therefore, the speed remains constant throughout the duration of the motion. Hence, option 4 is correct.
• While the speed is constant, the direction of motion changes at every single point along the circular track. Since velocity is a vector quantity (possessing both magnitude and direction), the velocity is considered to be changing continuously even if the speed is fixed. Hence, statement 2 is incorrect regarding speed.
• The speed of the body depends on the frequency and radius (v = 2πrn), but the characteristic of the motion itself is that the speed is maintained as a constant value, not that it "depends on radius only." Hence, statement 1 is incorrect.
• In any active motion, if the body is moving, the speed cannot be zero at every point; a zero speed would imply the body is at rest. Hence, statement 3 is incorrect.

Additional Information

• Velocity and Direction: In a circular path, the direction of the velocity vector at any point is always tangential to the circle at that specific location. Even though the numerical value of the speed (e.g., 5 meters per second) stays the same, the change in direction necessitates an acceleration.
• Centripetal Acceleration: An object in UCM experiences an acceleration directed toward the center of the circle. This is known as centripetal acceleration (a = v² / r). It is responsible for changing the direction of the velocity without changing its magnitude.
• Angular Velocity (ω): This refers to the rate at which the central angle is swept out by the radius vector. In uniform circular motion, the angular velocity remains constant and is measured in radians per second. The relationship between linear speed (v) and angular velocity is v = rω.
• Work Done: In UCM, the work done by the centripetal force is always 0. This is because the centripetal force acts towards the center (radially), while the displacement is always tangential, creating a 90-degree angle between force and displacement.
• Kinetic Energy: Since kinetic energy (K.E. = 1/2 mv²) depends on the square of the speed, and the speed is constant in UCM, the kinetic energy of the body remains constant during the motion.

Important Points

• Time Period (T): The time taken by the body to complete one full revolution is called the Time Period. It can be calculated using the formula T = 2πr / v, where r is the radius and v is the constant speed.
• Frequency (f): This is the number of revolutions completed by the object in one second. It is the reciprocal of the time period (f = 1 / T) and is measured in Hertz (Hz).
• Real-world Examples: Common examples of Uniform Circular Motion include the motion of the Moon orbiting the Earth (approximated), the movement of the tips of clock hands, and a satellite orbiting at a constant altitude.
• Non-Uniform Circular Motion: If the speed of the object also changes while moving in a circle, it is called non-uniform circular motion. In this case, there is both centripetal acceleration (changing direction) and tangential acceleration (changing speed).

The correct answer is calcium hydroxide with chlorine gas.

Key Points

• Bleaching powder is prepared by the action of chlorine gas on dry slaked lime, which is chemically known as calcium hydroxide.
• The chemical reaction for this preparation is: Ca(OH)2 + Cl2 → CaOCl2 + H2O. This clearly shows that calcium hydroxide and chlorine are the necessary reactants. Hence, statement 3 is correct.
• The reaction between sodium hydroxide and chlorine gas (Option 1) typically produces sodium hypochlorite (liquid bleach) and sodium chloride, rather than the solid bleaching powder. Hence, statement 1 is incorrect.
• Calcium carbonate reacting with chlorine (Option 2) does not yield bleaching powder; calcium carbonate is a stable salt often found in limestone and marble. Hence, statement 2 is incorrect.
• The reaction of calcium hydroxide with hydrochloric acid (Option 4) is a neutralization reaction that produces calcium chloride and water, not bleaching powder. Hence, statement 4 is incorrect.

Additional Information

• Chemical Identity and Formula:
• The chemical name of bleaching powder is calcium oxychloride or calcium hypochlorite.
• Its chemical formula is CaOCl2, although the actual composition is quite complex.
• Physical Properties:
• It is a yellowish-white powder that has a strong smell of chlorine.
• It is soluble in cold water, though the lime portion present in it usually remains as an insoluble residue.
• Industrial Applications:
• Textile Industry: It is used for bleaching cotton and linen to remove natural colors.
• Paper Industry: It is utilized for bleaching wood pulp.
• Water Treatment: One of its most vital functions is disinfecting drinking water to make it potable by killing harmful germs.
• Chemical Industry: It serves as a powerful oxidizing agent in many chemical reactions and is also used in the manufacture of chloroform.

Important Points

• Manufacturing Plants: In large-scale industrial production, bleaching powder is manufactured using the Hasenclever plant or the Bachmann’s plant.
• Storage: Bleaching powder should be stored in a cool, dry place because it undergoes auto-oxidation and loses its available chlorine content when exposed to air and moisture.
• Reaction with Acids: When bleaching powder reacts with an excess of dilute acids (like sulfuric acid), it releases chlorine gas, which is responsible for its bleaching action.

The correct answer is It is found in both lower and higher plants.

Key Points

• Asexual reproduction in plants is a process by which new individuals are produced from a single parent without the involvement of gamete formation or fusion.
• Statement 1: It occurs rapidly during unfavourable conditions. Asexual reproduction typically occurs when environmental conditions are favourable and stable, allowing the plant to colonize an area quickly. During unfavourable conditions, many plants prefer sexual reproduction or produce dormant structures like seeds or spores to ensure survival. Hence, statement 1 is incorrect.
• Statement 2: It involves male and female flowers. Flowers are the reproductive organs specifically designed for sexual reproduction, involving pollination and fertilization. Asexual reproduction bypasses the floral mechanism entirely. Hence, statement 2 is incorrect.
• Statement 3: It is found in both lower and higher plants. This is the defining characteristic. Lower plants like Algae, Fungi, and Bryophytes reproduce asexually through fragmentation, budding, or zoospores. Higher plants (Angiosperms) frequently reproduce asexually through vegetative propagation using roots, stems, or leaves. Hence, statement 3 is correct.
• Statement 4: It involves formation of gametes. The formation and subsequent fusion of haploid gametes (sperm and egg) is the hallmark of sexual reproduction. Asexual methods rely on mitotic division of somatic cells. Hence, statement 4 is incorrect.

Additional Information

• Vegetative Propagation in Higher Plants:
• This is a form of asexual reproduction where new plants are produced from vegetative parts.
• Stem Tubers: Examples include the Potato, where "eyes" act as nodes for new growth.
• Rhizomes: Underground stems found in Ginger and Turmeric that spread horizontally.
• Runners/Stolons: Found in Strawberry plants, these stems run along the ground to establish new plantlets.
• Bulbs: Short underground stems with fleshy leaves, such as in Onions and Garlic.
• Adventitious Buds: The Bryophyllum plant produces buds along its leaf margins that drop off to form new plants.
• Asexual Methods in Lower Plants:
• Fragmentation: In Spirogyra (an alga), the parent body breaks into smaller pieces, each growing into a new individual.
• Budding: Common in Yeast, where a small bulb-like projection (bud) develops and detaches.
• Spore Formation: Non-flowering plants like Ferns and Mosses produce spores in structures called sporangia which germinate under suitable conditions.
• Advantages of Asexual Reproduction:
• The offspring are genetically identical (clones) to the parent, ensuring the preservation of desirable traits.
• It is a faster and more energy-efficient process as it does not require the plant to produce flowers, nectar, or find pollinators.
• It allows for the rapid multiplication of a species in a stable habitat.

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

Key Points

• In a domestic electric circuit, all electrical appliances like bulbs, fans, and sockets are connected in parallel. This configuration ensures that every appliance receives the same voltage (220 V in India) and can be operated independently using its own switch. Hence, statement (i) is correct.
• The live wire (also known as the phase wire) is the wire through which current enters the circuit and is usually color-coded with red or brown insulation. The green insulation is strictly reserved for the earth wire, which serves as a safety mechanism. Hence, statement (ii) is incorrect.
• An electric fuse is a vital safety device designed to protect appliances. It consists of a wire with a low melting point; if the current becomes excessive (due to short-circuiting or overloading), the fuse wire melts and breaks the circuit to stop the flow of electricity. Hence, statement (iii) is incorrect.
• The neutral wire completes the electrical circuit by providing a return path for the current and is maintained at near-zero potential. In standard domestic wiring, the neutral wire is covered with black or blue insulation. Hence, statement (iv) is correct.
• Therefore, the correct statements regarding domestic circuits are (i) and (iv).

Additional Information

• Standard Voltage and Frequency: In India, the domestic power supply is Alternating Current (AC) with a potential difference of 220 V and a frequency of 50 Hz. This means the direction of current changes 100 times per second.
• Earthing System: The earth wire (green insulation) is connected to a metal plate buried deep in the ground. It is used as a safety measure for appliances with metallic bodies (like refrigerators or toasters) to ensure that any leakage of current is transferred safely to the ground, preventing electric shocks.
• Short-Circuiting: This phenomenon occurs when the live wire and neutral wire come into direct contact. This leads to a massive decrease in resistance and a sudden surge in current, which can cause sparks or fires.
• Overloading: Overloading happens when the total power rating of appliances connected to a single circuit exceeds the current-carrying capacity of the wires. This usually occurs when too many high-power devices are plugged into one socket.
• MCB (Miniature Circuit Breaker): In modern domestic circuits, MCBs are often used instead of traditional fuses. They automatically switch off the electrical supply during an overload or short circuit and can be easily reset manually once the fault is fixed.

Important Points

• Parallel vs. Series: Domestic circuits never use series connections for appliances because in a series circuit, if one appliance fails or is switched off, the entire circuit breaks and all other appliances stop working. Furthermore, the voltage would be divided among the appliances, causing them to function inefficiently.
• Wire Color Coding: According to the new international convention, the Live wire is Brown, the Neutral wire is Light Blue, and the Earth wire is Green or Yellow. However, the traditional Indian coding of Red (Live), Black (Neutral), and Green (Earth) is still widely recognized.
• Two Separate Circuits: Domestic wiring usually consists of two separate circuits: a 5 A circuit for low-power devices (bulbs, fans) and a 15 A circuit for high-power devices (geysers, air conditioners).

The correct answer is Both (A) and (R) are true but (R) is not the correct explanation of (A).

Key Points

• Assertion (A) states that the onset of puberty in females occurs earlier than in males. Biologically, girls typically enter puberty between the ages of 8 and 13, while boys usually begin between 9 and 14 years. Despite the difference in start times, both genders generally attain their final adult height by the age of 18–20 years when the growth plates fuse. Hence, statement (A) is correct.
• Reason (R) mentions that the growth spurt in males is more rapid. During the peak of adolescence, males experience a more intense height velocity (approximately 9.5 cm per year) compared to females (approximately 8.3 cm per year). Hence, statement (R) is correct.
• While both statements are factually accurate, the rapidity of the male growth spurt does not explain why females start puberty earlier or why both finish growing by age 20. The timing of puberty is controlled by hormonal signals from the hypothalamus, while the cessation of growth is due to the fusion of epiphyseal plates caused by sex steroids.
• Therefore, Reason (R) is a true biological fact but it does not serve as a logical explanation for the specific timings mentioned in Assertion (A).
• Hence, the correct answer is Both (A) and (R) are true but (R) is not the correct explanation of (A).

Additional Information

• Puberty and Hormones:
• In females, the primary hormone responsible for pubertal changes and eventually the closure of growth plates is Estrogen.
• In males, Testosterone drives the growth spurt and the development of secondary sexual characteristics like facial hair and deepening of the voice.
• Growth Plates (Epiphyseal Plates):
• These are areas of active cartilage at the ends of long bones. Height increases as long as these plates stay open.
• By the end of adolescence (usually by age 18-20), hormonal changes cause these plates to calcify and turn into bone, marking the end of vertical growth.
• Peak Height Velocity (PHV):
• This refers to the period where the maximum rate of growth occurs. Females typically reach PHV about two years earlier than males, often around age 11-12, whereas males reach it around age 13-14.
• Because males have two additional years of pre-pubertal growth and a more vigorous growth spurt, they generally end up taller than females on average.

Important Points

• Individual Variation: While the standard range is 18-20 years, factors like nutrition, genetics, and health conditions can influence the exact age at which an individual stops growing.
• Sexual Dimorphism: The differences in growth timing and intensity contribute significantly to sexual dimorphism (physical differences between males and females of the same species) in human height and skeletal structure.

The correct answer is packaging and secretion.

Key Points

• The Golgi complex, also known as the Golgi apparatus or Golgi body, is a vital organelle in eukaryotic cells responsible for the modification, sorting, and packaging of proteins and lipids.
• Its primary function is to act as a packaging center. It receives materials synthesized in the Endoplasmic Reticulum (ER), processes them, and then directs them to their final destinations, such as the plasma membrane or lysosomes.
• If a cell lacks a functional Golgi complex, the proteins and lipids synthesized by the cell would not be packaged into vesicles or secreted properly. This would lead to a failure in delivering essential molecules to where they are needed inside or outside the cell. Hence, the option packaging and secretion is correct.
• The organelle is composed of stacked, flattened, membrane-bound sacs called cisternae. These structures are involved in glycosylation (adding sugar chains to proteins) and the formation of glycoproteins and glycolipids.
• The Golgi complex is also essential for the formation of lysosomes and the acrosome in sperm cells, which is necessary for fertilization.

Additional Information

• Protein Synthesis:
• Protein synthesis is primarily the function of ribosomes and the Rough Endoplasmic Reticulum (RER).
• While the Golgi complex modifies proteins after they are made, it does not actually synthesize them. Therefore, a lack of Golgi would not stop the synthesis of proteins, only their transport and final processing. Hence, statement 1 is incorrect.
• Controlling Cell Division:
• The regulation and control of cell division are managed by the nucleus (which contains DNA) and structures like centrioles in animal cells.
• The Golgi complex is not the primary controller of the cell cycle or mitosis. Hence, statement 2 is incorrect.
• Formation of ATP:
• The formation of ATP (Adenosine Triphosphate) occurs through cellular respiration in the mitochondria.
• Mitochondria are known as the powerhouse of the cell. The Golgi complex does not produce energy for the cell. Hence, statement 4 is incorrect.
• Discovery:
• The Golgi complex was discovered by the Italian biologist Camillo Golgi in 1898. He used a specialized silver staining technique called the black reaction to identify the organelle's structure.
• His work on the nervous system and the discovery of this organelle earned him the Nobel Prize in Physiology or Medicine in 1906.

Important Points

• Vesicular Transport: The Golgi complex functions by receiving transport vesicles from the ER at its cis face and releasing secretory vesicles from its trans face.
• Plant Cells: In plant cells, the Golgi apparatus is often referred to as dictyosomes, which are smaller and more scattered compared to animal cells. They play a major role in synthesizing hemicellulose and pectin for the cell wall.
• Secretion: Cells that are highly active in secretion (like pancreatic cells secreting enzymes or goblet cells secreting mucus) have a very high number of Golgi complexes.

The correct answer is Light from stars undergoes atmospheric refraction.

Key Points

• The twinkling of a star is due to the atmospheric refraction of star light. As the light from a star enters the Earth’s atmosphere, it undergoes refraction continuously before it reaches the Earth.
• Since the stars are very distant, they approximate point-sized sources of light. As 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. Hence, statement 1 is correct.
• The atmospheric refraction occurs in a medium of gradually changing refractive index. The Earth’s atmosphere is not stationary; its density and temperature change continuously, leading to changes in the refractive index of different layers of air.
• While stars may have intrinsic variability, the "twinkling" we observe is primarily an optical illusion caused by the atmosphere rather than the stars changing their brightness continuously. Hence, statement 2 is incorrect.
• The distance between stars and Earth is vast, and any minor orbital movement or variation in distance is too insignificant to cause the rapid flickering effect known as twinkling. Hence, statement 3 is incorrect.
• Although stars do move at high velocities, their proper motion is not perceivable over short timeframes to the naked eye and does not cause the twinkling effect. Hence, statement 4 is incorrect.

Additional Information

• Why Planets do not Twinkle:
• Unlike stars, planets are much closer to the Earth and are thus seen as extended sources (a collection of many point-sized sources) rather than a single point source.
• The total variation in the amount of light entering our eye from all the individual point-sized sources in a planet will average out to zero, thereby nullifying the twinkling effect.
• Atmospheric Refraction Phenomena:
• Advance Sunrise and Delayed Sunset: The Sun is visible to us about 2 minutes before the actual sunrise and about 2 minutes after the actual sunset because of atmospheric refraction.
• Apparent Position: Due to the bending of light, stars appear slightly higher (above their actual position) when viewed near the horizon.
• Refractive Index: The refractive index of the atmosphere generally increases as we move closer to the Earth's surface because the air becomes denser. This causes the light to bend toward the normal.

Important Points

• Scintillation: This is the scientific term for the twinkling of stars. It is caused by small-scale fluctuations in the air density resulting from turbulence in the atmosphere.
• Astronomy: To avoid the blurring and twinkling caused by the atmosphere, modern large telescopes (like the Hubble Space Telescope) are placed in outer space, or use adaptive optics to cancel out the atmospheric distortion.
• Medium Consistency: If the Earth had no atmosphere, stars would not twinkle and would appear as steady points of light against a black sky.

The correct answer is Fluid content inside the plasma membrane containing organelles.

Key Points

• The cytoplasm is the gel-like substance that fills the interior of a cell, situated between the nucleus and the plasma membrane.
• It serves as the primary arena for cellular activities and chemical reactions that keep the cell alive.
• Statement 1: It defines cytoplasm as the fluid content inside the plasma membrane that houses various specialized organelles. Each of these organelles performs a specific function for the cell. Hence, statement 1 is correct.
• Statement 2: A rigid layer outside the plasma membrane that provides support and protection is known as the cell wall. This is found in plants, fungi, and bacteria, but not in animal cells. Hence, statement 2 is incorrect.
• Statement 3: A network of tubes involved in the transport of proteins and lipids within the cell refers to the Endoplasmic Reticulum (ER). Hence, statement 3 is incorrect.
• Statement 4: Genetic material present in the form of chromatin is found within the nucleus, which is separate from the general cytoplasm in eukaryotic cells. Hence, statement 4 is incorrect.
• Hence, the correct answer is Fluid content inside the plasma membrane containing organelles.

Additional Information

• Composition of Cytoplasm:
• The cytoplasm is composed of the cytosol, organelles, and cytoplasmic inclusions.
• The cytosol is the aqueous part containing water (about 80%), salts, and organic molecules like enzymes.
• In prokaryotic cells, which lack a defined nucleus, the genetic material (DNA) is located directly in the cytoplasm in a region called the nucleoid.
• Functions of Cytoplasm:
• It facilitates cytoplasmic streaming (cyclosis), which helps in the distribution of nutrients and movement of organelles within the cell.
• It acts as a site for metabolic pathways such as glycolysis (the first step of cellular respiration) and protein synthesis in the ribosomes.
• The cytoplasm provides turgidity to the cell, maintaining its shape and preventing it from collapsing.
• Cell Organelles:
• Organelles like Mitochondria, Golgi Apparatus, and Lysosomes are suspended in the cytoplasm.
• These membrane-bound structures are essential for energy production, waste disposal, and molecular packaging.

Important Points

• Historical Context: The term "cytoplasm" was introduced by Rudolf von Kolliker in 1863 as a synonym for protoplasm.
• Cytoskeleton: Within the cytoplasm lies a complex network of microtubules and microfilaments called the cytoskeleton. It provides mechanical support and assists in cell division.
• Ectoplasm and Endoplasm: In some cells, like the Amoeba, the cytoplasm is divided into an outer clear layer called ectoplasm and an inner granular layer called endoplasm.

The correct answer is Hybrid crops are unaffected by unpredictable weather.

Key Points

• Climate-resistant hybrid crops are engineered or crossbred to possess traits that allow them to survive and thrive under extreme environmental conditions. This resilience is crucial because traditional varieties often fail during sudden climatic shifts.
• One of the primary benefits is that these crops are relatively unaffected by unpredictable weather, such as sudden droughts, unseasonal rainfall, or extreme heatwaves. This ensures a consistent food supply even in volatile years. Hence, statement 4 is correct.
• The role of the farmer is actually intensified rather than reduced, as they must manage specialized agronomic practices to maximize the potential of these high-tech seeds. Hence, statement 1 is incorrect.
• While some hybrids might have uniform aesthetic qualities, their importance for future farming is rooted in biological survival and yield security, not visual appeal. Hence, statement 2 is incorrect.
• Storage characteristics depend on moisture content and post-harvest technology; climate resistance specifically refers to the plant's growth cycle in the field, not its shelf life. Hence, statement 3 is incorrect.
• These hybrids are designed to combat abiotic stress, which includes non-living factors like salinity, chilling, and high temperatures that are becoming more frequent due to global warming.

Additional Information

• Climate-Smart Agriculture (CSA):
• This is an integrated approach to managing landscapes—cropland, livestock, forests, and fisheries—that addresses the interlinked challenges of food security and accelerating climate change.
• The Food and Agriculture Organization (FAO) defines CSA through three pillars: increasing productivity, enhancing resilience, and reducing greenhouse gas emissions.
• Types of Resistance:
• Drought Tolerance: Varieties developed to require less water or to enter a dormant state during dry spells to prevent wilting.
• Flood/Submergence Tolerance: For example, Swarna-Sub1 rice, which can survive underwater for up to 14 days, a trait vital for regions like Eastern India and Bangladesh.
• Salinity Tolerance: Essential for coastal farming where rising sea levels increase soil salt concentration.
• Economic Impact:
• By 2050, the global population is expected to reach nearly 10 billion. Climate-resistant hybrids help stabilize market prices by preventing mass crop failures.
• The Indian Council of Agricultural Research (ICAR) has released several climate-resilient varieties of wheat (HD 3385) and rice to help Indian farmers adapt to rising temperatures.

Important Points

• Hybridization vs. GMO: It is important to distinguish between hybrid crops (produced via natural cross-pollination) and Genetically Modified Organisms (GMOs) (produced via laboratory gene splicing), although both can be used to achieve climate resilience.
• Global Food Security: According to the IPCC Reports, agricultural yields could decline by 10-25% per degree of warming without the adoption of these resistant varieties.
• Sustainable Development Goals (SDG): The use of these crops directly supports SDG 2 (Zero Hunger) and SDG 13 (Climate Action) by ensuring sustainable food production systems.

The correct answer is Sugar.

Key Points

• A compound is a substance formed when two or more chemical elements are chemically bonded together. The elements in a compound are always present in fixed ratios by mass.
• Sugar, specifically table sugar known as Sucrose, is a chemical compound with the molecular formula C12H22O11. It is composed of 12 carbon atoms, 22 hydrogen atoms, and 11 oxygen atoms. Hence, this option is correct.
• Mercury is a chemical element represented by the symbol Hg and has the atomic number 80. It is a pure substance consisting of only one type of atom and cannot be broken down into simpler substances. Hence, this option is incorrect.
• Hydrogen is the simplest and lightest chemical element with the symbol H and atomic number 1. While it often exists as a diatomic molecule (H2), it is still classified as an element, not a compound, because it consists of only one kind of atom. Hence, this option is incorrect.
• Iron is a metallic element with the symbol Fe and atomic number 26. It is a transition metal found in Group 8 of the periodic table. As a pure metal, it is an element. Hence, this option is incorrect.

Additional Information

• Characteristics of Mercury (Hg):
• It is the only common metal that exists in a liquid state at standard room temperature (approximately 25 degrees Celsius).
• It has a high density and is used in scientific instruments like thermometers, barometers, and manometers due to its uniform thermal expansion.
• In nature, it is primarily found in the ore Cinnabar (mercuric sulfide).
• Characteristics of Iron (Fe):
• It is the fourth most abundant element in the Earth's crust and makes up a significant portion of the Earth's core.
• It is highly magnetic and is the primary raw material used to produce Steel, an alloy essential for construction and industrial machinery.
• Biologically, it is a key component of hemoglobin, the protein in red blood cells responsible for carrying oxygen throughout the body.
• Characteristics of Hydrogen (H):
• It is the most abundant element in the universe, making up roughly 75% of all baryonic mass.
• At standard temperature and pressure, it is a colorless, odorless, and tasteless gas.
• It is used in the Haber process for the production of ammonia and is being explored as a clean renewable energy fuel.
• Characteristics of Sugar (Sucrose):
• It is a disaccharide, meaning it is formed by the combination of two simple sugars: glucose and fructose.
• The covalent bonds holding the atoms together give the compound unique physical properties, such as being a white crystalline solid that is highly soluble in water.
• It is primarily extracted from sugarcane (Saccharum officinarum) and sugar beets for commercial use.

Important Points

• Pure Substances vs. Mixtures: Matter can be classified as pure substances (elements and compounds) or mixtures. A compound like sugar is a pure substance because its composition is chemically uniform.
• Chemical Symbols and Formulas: Elements are represented by one or two-letter symbols (e.g., Fe for Iron, H for Hydrogen), while compounds are represented by chemical formulas (e.g., H2O for Water, NaCl for Salt).
• Decomposition: Compounds can be broken down into their constituent elements only by chemical methods (such as electrolysis or thermal decomposition), whereas elements cannot be broken down further by any chemical means.
• Law of Constant Proportions: This fundamental law states that a chemical compound always contains its component elements in a fixed ratio by mass, regardless of its source or method of preparation.

The correct answer is Joule.

Key Points

• Work is said to be done when a force acts upon an object to cause a displacement. The mathematical expression for work done is W = F × s × cosθ, where F is the force, s is the displacement, and θ is the angle between the force and displacement vectors.
• The SI unit of work is the Joule (J). It is named in honor of the English physicist James Prescott Joule, who studied the nature of heat and its relationship to mechanical work in the mid-19th century.
• One Joule is defined as the amount of work done when a constant force of one Newton displaces an object through a distance of one meter in the direction of the force. Mathematically, 1 J = 1 N · m.
• Work is a scalar quantity, meaning it has only magnitude and no direction, even though the force and displacement used to calculate it are vector quantities.
• In the CGS (Centimetre-Gram-Second) system, the unit of work is the Erg. The relationship between the two is that 1 Joule is equal to 10⁷ Ergs.

Additional Information

• Pascal (Pa):
• The Pascal is the SI derived unit of pressure. It is defined as one Newton per square meter (N/m²).
• It was named after the French polymath Blaise Pascal in 1971. It is commonly used in meteorology to measure atmospheric pressure and in engineering for material strength.
• Newton (N):
• The Newton is the SI unit of force. It is defined as the force needed to accelerate one kilogram of mass at the rate of one meter per second squared (1 kg·m/s²).
• Named after Sir Isaac Newton in recognition of his work on classical mechanics and the laws of motion published in 1687.
• Coulomb (C):
• The Coulomb is the SI unit of electric charge. It is the quantity of electricity transported in one second by a current of one ampere.
• It is named after Charles-Augustin de Coulomb, a French physicist known for Coulomb's Law, which describes the electrostatic force between charges.

Important Points

• The dimensional formula for Work is [ML²T^-2], which is the same as the dimensions for energy and torque.
• Work done can be positive, negative, or zero depending on the angle θ. If the force and displacement are in the same direction (θ = 0°), work is positive. If they are in opposite directions (θ = 180°), work is negative.
• Zero Work: No work is done if the displacement is zero, or if the force is perpendicular to the displacement (θ = 90°), such as a person carrying a load on their head while walking horizontally.
• In atomic physics, work and energy are often measured in a smaller unit called the electronvolt (eV), where 1 eV ≈ 1.602 × 10^-19 Joules.

The correct answer is Poor conductor of electricity.

Key Points

• Non-metals are elements that generally lack the characteristics of metals. One of their most defining physical properties is that they are poor conductors of electricity and heat. This is because non-metals do not have free-moving electrons (delocalized electrons) in their structure to carry an electric charge.
• Most non-metals have low melting points and low boiling points compared to metals. This is due to the weak intermolecular forces (like Van der Waals forces) that hold their molecules together, which require less energy to overcome.
• Non-metals are brittle in their solid state. Unlike metals, they cannot be beaten into thin sheets (malleability) or drawn into wires (ductility). If subjected to stress, they tend to shatter or break into pieces.
• They are non-sonorous, meaning they do not produce a deep, ringing sound when struck. This is a sharp contrast to metals, which are used in bells and musical instruments for their sonorous property.
• In terms of chemical behavior, non-metals are electronegative. They have a tendency to gain electrons during chemical reactions to form negative ions (anions) or share electrons to form covalent bonds.

Additional Information

• Physical State: Unlike metals, which are mostly solids at room temperature, non-metals can exist in all three states of matter. For example, Oxygen and Nitrogen are gases, Carbon and Sulfur are solids, and Bromine is the only non-metal that is a liquid at room temperature.
• Lustre: Non-metals generally have a dull appearance and do not reflect light well (they are non-lustrous). However, Iodine and Graphite are notable exceptions as they exhibit a metallic-like shine or lustre.
• Exceptions to Conductivity: While non-metals are typically insulators, Graphite (an allotrope of Carbon) is a good conductor of electricity. This unique property allows it to be used in electrodes and dry cells.
• Hardness: Most non-metals are relatively soft compared to metals. However, Diamond (another allotrope of Carbon) is the hardest known natural substance and has an extremely high melting point.
• Oxides: Non-metals react with oxygen to form acidic or neutral oxides. For instance, Sulfur dioxide (SO2) is acidic, whereas Carbon monoxide (CO) and Water (H2O) are neutral oxides.

Important Points

• Ductility: This is a property of metals (like Gold and Copper) that allows them to be drawn into thin wires. Non-metals lack this property.
• Density: Non-metals usually have low densities and are lighter than metals. They do not have the high mass-to-volume ratio characteristic of transition metals.
• Valence Electrons: Non-metals generally have 4 to 8 electrons in their outermost shell. This configuration drives their reactivity toward gaining electrons to achieve a stable noble gas configuration.
• Uses: Non-metals are essential for life. Oxygen is vital for respiration, Nitrogen is used in fertilizers, Chlorine is used for water purification, and Phosphorus is used in matchsticks.

The correct answer is R = 2f.

Key Points

• A spherical mirror is a mirror whose reflecting surface is a part of a hollow sphere of glass. The center of this sphere is known as the Center of Curvature (C).
• The Radius of Curvature (R) is defined as the distance between the Pole (P) of the mirror and the Center of Curvature (C).
• The Principal Focus (F) is a point on the principal axis where all rays parallel to the axis converge (in a concave mirror) or appear to diverge from (in a convex mirror) after reflection.
• The Focal Length (f) is the distance from the Pole (P) to the Principal Focus (F).
• For spherical mirrors with a small aperture, the Principal Focus (F) lies exactly at the midpoint of the distance between the Pole (P) and the Center of Curvature (C).
• Because the focus is the midpoint, the Radius of Curvature (R) is twice the Focal Length (f). Mathematically, this is expressed as R = 2f.

Additional Information

• Concave Mirror: This is a converging mirror where the reflecting surface is curved inwards. Its focal length is considered negative in the standard Cartesian sign convention. It is commonly used in solar cookers, torches, and by dentists to see larger images of teeth.
• Convex Mirror: This is a diverging mirror where the reflecting surface is curved outwards. Its focal length is considered positive. These mirrors are primarily used as rear-view mirrors in vehicles because they provide a wider field of view and always form an erect, virtual, and diminished image.
• Aperture: The aperture of a spherical mirror is the diameter of its reflecting surface. It represents the size of the mirror that faces the incident light.
• Sign Convention: According to the New Cartesian Sign Convention, the pole is treated as the origin. Distances measured in the direction of incident light are positive, while those measured against the direction of incident light are negative.

Important Points

• The relationship R = 2f is only strictly accurate for paraxial rays. These are rays that strike the mirror very close to the Pole (P) and make small angles with the principal axis.
• If the aperture of the mirror is large, rays falling near the edges of the mirror (marginal rays) do not meet at the same focal point as the paraxial rays. This defect in image formation is known as spherical aberration.
• The Mirror Formula, which relates focal length, object distance (u), and image distance (v), is given by 1/f = 1/v + 1/u. This formula is derived using the R = 2f assumption for small apertures.
• In year 1672, Isaac Newton used the properties of spherical and parabolic mirrors to construct the first reflecting telescope, significantly advancing the field of optics.

The correct answer is Salt, water, carbon dioxide.

Key Points

• When a metal carbonate reacts with an acid, the chemical reaction consistently produces three distinct products: a salt, water, and carbon dioxide gas. Hence, option 1 is correct.
• The general chemical equation for this interaction can be represented as: Metal Carbonate + Acid → Salt + Water + Carbon Dioxide.
• The salt produced in the reaction is formed by the cation (positive ion) from the metal carbonate and the anion (negative ion) from the acid. For instance, if hydrochloric acid is used, the salt will be a chloride.
• The evolution of carbon dioxide is a characteristic feature of this reaction, typically observed as effervescence or brisk bubbling within the solution. Hence, options 2 and 3 are incorrect as they mention carbon monoxide.
• Water is formed through the combination of hydrogen ions from the acid and oxygen atoms released from the carbonate group. Hence, option 4 is incorrect as it omits the salt formation.

Additional Information

• Common Examples:
• A well-known example is the reaction between Sodium Carbonate (Na2CO3) and Hydrochloric Acid (HCl), which yields Sodium Chloride (NaCl), water (H2O), and carbon dioxide (CO2).
• Another example is Calcium Carbonate (CaCO3), often found in limestone or marble, reacting with Sulfuric Acid (H2SO4) to form Calcium Sulfate (CaSO4), water, and carbon dioxide.
• Metal Hydrogen Carbonates:
• Similar to metal carbonates, metal hydrogen carbonates (also known as bicarbonates, such as Sodium Bicarbonate or NaHCO3) also react with acids to produce salt, water, and carbon dioxide.
• This is the fundamental principle behind baking powder, which releases CO2 gas to make cakes and bread rise.
• Testing for Carbon Dioxide:
• The carbon dioxide gas produced can be tested by passing it through lime water (Calcium Hydroxide solution).
• The lime water turns milky due to the formation of a white precipitate of Calcium Carbonate. This is a standard analytical test in chemistry labs.
• Applications:
• This reaction is utilized in fire extinguishers, where an acid reacts with a carbonate/bicarbonate to release a large volume of CO2 to displace oxygen and extinguish flames.
• It is also the basis for antacid tablets containing Calcium Carbonate, which neutralize excess stomach acid (HCl).

Important Points

• Neutralization Aspect: While this reaction produces salt and water (like a standard neutralization reaction between an acid and a base), the addition of carbon dioxide distinguishes carbonate reactions from simple oxide or hydroxide reactions.
• Gas Evolution: Unlike the reaction of metals with acids, which produces hydrogen gas, the reaction of metal carbonates with acids produces carbon dioxide.
• Stoichiometry: The valency of the metal and the basicity of the acid determine the specific ratio of reactants and the chemical formula of the salt produced.
• Safety: In laboratory settings, these reactions must be handled carefully as the rapid evolution of gas can cause splashing of the acidic solution.

The correct answer is Direction of magnetic field around a current-carrying conductor.

Key Points

• The Right-hand thumb rule, often attributed to Maxwell, is a fundamental principle in electromagnetism used to determine the direction of magnetic field lines produced by a straight current-carrying conductor.
• According to this rule, if a current-carrying conductor is held in the right hand such that the thumb points in the direction of the electric current, then the fingers will curl around the conductor in the direction of the magnetic field lines.
• This rule visualizes the magnetic field as concentric circles surrounding the wire. If the current flows upwards, the magnetic field moves in an anticlockwise direction (when viewed from above). Hence, statement 3 is correct.
• The Right-hand thumb rule does not determine the magnitude of the current; the magnitude is usually measured using an ammeter in Amperes (A). Hence, statement 1 is incorrect.
• The direction of force on a conductor is determined by Fleming's Left-Hand Rule, not the thumb rule. Hence, statement 2 is incorrect.
• The resistance of a wire is a property of the material and its dimensions, calculated using Ohm's Law (V=IR) or the resistivity formula, and is independent of the thumb rule. Hence, statement 4 is incorrect.

Additional Information

• Maxwell’s Corkscrew Rule:
• This is another version of the Right-hand thumb rule. It states that if a right-handed corkscrew is rotated such that it moves in the direction of the current, the direction of rotation represents the direction of the magnetic field.
• Magnetic Field Properties:
• The strength of the magnetic field (B) produced by a straight wire is directly proportional to the magnitude of the current (I) and inversely proportional to the distance (r) from the wire.
• The S.I. unit of the magnetic field is the Tesla (T).
• Circular Loops and Solenoids:
• The Right-hand thumb rule is also applicable to circular loops. If the fingers curl in the direction of the current in the loop, the thumb points in the direction of the magnetic field at the center of the loop.
• In a solenoid, this rule helps identify the North and South poles created by the flow of electricity.

Important Points

• Fleming's Left-Hand Rule: Specifically used for Electric Motors. The thumb indicates Motion/Force, the forefinger indicates the Magnetic Field, and the middle finger indicates the Current.
• Fleming's Right-Hand Rule: Specifically used for Electric Generators to find the direction of induced current. The thumb indicates the motion of the conductor, the forefinger indicates the magnetic field, and the middle finger indicates the induced current.
• Discovery: The relationship between electricity and magnetism was first observed by Hans Christian Oersted in 1820, when he noticed a compass needle deflect near a wire carrying electric current.
• Applications: Understanding these rules is essential for the design and functioning of electromagnets, transformers, speakers, and various industrial sensors.

p data-end="57" data-start="0">The correct answer is waves and particles.

 

Key Points

• The dual character of electromagnetic radiation means that it exhibits both wave-like and particle-like properties. This concept was fundamental in the transition from classical physics to quantum mechanics.
• The wave nature is evidenced by phenomena such as interference and diffraction. These properties suggest that radiation travels through space as oscillating electric and magnetic fields. Hence, the first part of the option is correct.
• The particle nature (also known as the corpuscular nature) is demonstrated through Black Body Radiation and the Photoelectric Effect. These phenomena could not be explained by classical wave theory and required the concept of discrete energy packets. Hence, the second part of the option is correct.
• Niels Bohr incorporated these dualistic ideas into his 1913 Atomic Model. He proposed that electrons move in fixed orbits and that the energy emitted or absorbed during transitions between these orbits occurs in discrete units or photons, which aligns with the particle theory.
• By accepting that light behaves as both waves and particles, scientists were able to resolve the inconsistencies between the continuous spectra predicted by classical physics and the line spectra observed in elements like hydrogen.

Additional Information

• Quantum Theory of Radiation:
• Max Planck proposed in 1900 that atoms and molecules could emit or absorb energy only in discrete quantities called quanta. The energy of a quantum is proportional to its frequency.
• The relationship is given by the equation E = hν, where h is Planck’s constant, valued at approximately 6.626 x 10⁻³⁴ Joules-second (J·s).
• The Photoelectric Effect:
• In 1905, Albert Einstein applied Planck's theory to light, suggesting that light itself consists of particles called photons.
• This discovery provided the strongest evidence for the particle nature of electromagnetic radiation and earned Einstein the Nobel Prize in Physics in 1921.
• Wave Nature of Light:
• James Clerk Maxwell established in the 1860s that light is an electromagnetic wave traveling at a constant speed of 3 x 10⁸ meters per second in a vacuum.
• Characteristics of these waves include wavelength (λ) and frequency (ν), related by the formula c = λν.
• De Broglie Hypothesis:
• In 1924, Louis de Broglie extended the dual nature of light to all matter, proposing that particles like electrons also possess wave-like characteristics, a concept known as matter waves.

Important Points

• Key Characteristics of Photons:
• A photon has zero rest mass but possesses momentum and energy. This allows it to exert pressure, known as radiation pressure.
• The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.
• Electromagnetic Spectrum:
• The spectrum includes Gamma rays, X-rays, Ultraviolet, Visible light, Infrared, Microwaves, and Radio waves. High-frequency waves like Gamma rays exhibit more pronounced particle-like behavior.
• Low-frequency radiations like Radio waves exhibit more pronounced wave-like behavior, making them ideal for communication over long distances.

The correct answer is Growth of a moustache.

Key Points

• Puberty is the stage of life during which adolescents reach sexual maturity and become capable of reproduction. In boys, this transition typically begins between the ages of 10 and 14.
• The growth of a moustache and other facial hair is a prominent secondary sexual characteristic in males. It marks the transition toward physical adulthood. Hence, the correct answer is Growth of a moustache.
• This change is triggered by the production of androgens, specifically testosterone, which is secreted by the testes under the influence of the pituitary gland.
• Other physiological changes include the deepening of the voice due to the enlargement of the larynx (Adam's apple) and the broadening of the shoulders.
• The appearance of hair is not limited to the face; it also includes the development of axillary (underarm) hair and pubic hair, which follow a predictable pattern known as Tanner stages.

Additional Information

• Hormonal Mechanism: The onset of puberty is initiated by the hypothalamus, which secretes Gonadotropin-releasing hormone (GnRH). This stimulates the anterior pituitary to release Luteinizing Hormone (LH) and Follicle-Stimulating Hormone (FSH).
• Testosterone Functions: In males, testosterone is responsible for the enlargement of reproductive organs, stimulation of spermatogenesis (sperm production), and the development of muscular strength.
• Physical Growth: Boys often experience a significant growth spurt during puberty, leading to an increase in height and bone density. This is often accompanied by an increase in the activity of sebaceous and sweat glands, which can lead to acne.
• Primary vs. Secondary Characteristics: Primary sexual characteristics refer to the internal and external reproductive organs (testes, penis) present at birth that mature during puberty. Secondary sexual characteristics are features that appear during puberty but are not directly part of the reproductive system, such as facial hair and body shape changes.

Important Points

• Growth of Pineal Gland: The pineal gland, located in the brain, is responsible for secreting melatonin, which regulates circadian rhythms. It does not grow as a result of puberty; rather, it often undergoes calcification as an individual ages. Hence, statement 1 is incorrect.
• Growth of hairs on head: The growth of hair on the scalp is a continuous biological process that occurs from birth and is influenced by genetics and nutrition. It is not a specific marker for the onset of puberty. Hence, statement 2 is incorrect.
• Growth of liver: The liver is a vital organ involved in metabolism and detoxification. While it increases in size as the body grows during childhood and adolescence, its growth is part of general physical development and is not a specific characteristic of sexual maturation. Hence, statement 4 is incorrect.
• Psychological Changes: Along with physical markers like the growth of a moustache, puberty also involves cognitive and emotional changes, including increased independence and the development of sexual orientation.

The correct answer is It brings the object in motion.

Key Points

• An unbalanced force is defined as a force where the net resultant force acting on an object is not zero. When multiple forces act on a body and do not cancel each other out, the object experiences a change in its state of motion.
• According to Newton’s First Law of Motion, an object will maintain its state of rest or uniform motion in a straight line unless compelled to change that state by an external unbalanced force.
• Statement 1: "It brings the object in motion." If an unbalanced force is applied to a stationary object, it overcomes the object's inertia and causes it to start moving in the direction of the resultant force. Hence, statement 1 is correct.
• Statement 2: "It has no effect on the object’s state of motion." This description applies to balanced forces, where the forces are equal in magnitude but opposite in direction, resulting in a net force of zero. Hence, statement 2 is incorrect.
• Statement 3: "It keeps the object at rest." An unbalanced force, by definition, causes acceleration. If an object remains at rest despite forces being applied, it means the forces are balanced (such as static friction balancing a push). Hence, statement 3 is incorrect.
• Statement 4: "It always increases the mass of the object." Mass is an intrinsic property representing the amount of matter in an object and its inertia. In classical mechanics, applying a force does not change the mass of the object. Hence, statement 4 is incorrect.

Additional Information

• Newton’s Second Law of Motion: This law provides a quantitative definition of force, stating that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The relationship is expressed by the formula F = ma.
• Acceleration and Velocity: An unbalanced force always results in acceleration, which means a change in the velocity of the object. This can involve an increase in speed, a decrease in speed (deceleration), or a change in the direction of motion.
• Vector Nature of Force: Force is a vector quantity, characterized by both magnitude and direction. The SI unit of force is the Newton (N), named after Sir Isaac Newton for his work on classical mechanics in 1687.
• Inertia: This is the tendency of objects to resist any change in their state of motion. The mass of an object is a measure of its inertia; a heavier object requires a larger unbalanced force to achieve the same acceleration as a lighter object.
• Types of Forces: Forces can be categorized into contact forces (like friction, tension, and normal force) and non-contact forces (like gravitational force, magnetic force, and electrostatic force). Any of these can act as an unbalanced force depending on the system.

The correct answer is changing.

Key Points

• An Amoeba is a unicellular organism that belongs to the Kingdom Protista and is characterized by its lack of a definite shape.
• The shape of an amoeba is constantly changing because of the movement of its cytoplasm and the formation of pseudopodia.
• Pseudopodia, which literally means "false feet," are temporary protrusions of the cell membrane that the organism extends and pulls back to facilitate locomotion.
• In addition to movement, these pseudopodia are used for capturing food through a process known as phagocytosis, where the organism wraps its body around a food particle to form a food vacuole.
• Because it lacks a cell wall or a pellicle (a stiff outer covering found in some other protists), the amoeba remains irregular and flexible throughout its life cycle.

Additional Information

• Slipper Shape (Paramecium):
• Unlike the amoeba, the Paramecium has a fixed, slipper-like shape.
• This fixed shape is maintained by a stiff but flexible outer membrane called the pellicle.
• The Paramecium uses tiny hair-like structures called cilia for movement, which cover its entire body surface.
• Biological Classification:
• Amoebas were first discovered by August Johann Rösel von Rosenhof in 1755.
• They are typically found in freshwater environments like ponds, lakes, and slow-moving streams, though some species live in saltwater or damp soil.
• The most well-known species is Amoeba proteus, which can grow up to 700 to 800 micrometers in length.
• Cellular Components:
• The body of an amoeba consists of a plasma membrane, cytoplasm (divided into ectoplasm and endoplasm), and a nucleus.
• It contains a contractile vacuole, which is vital for osmoregulation, helping the cell maintain water balance by expelling excess liquid.
• It reproduces asexually through a process called binary fission, where a single cell divides into two identical daughter cells.
• Comparison of Shapes:
• Star shape: While not characteristic of amoebas, some cells in the human body, such as astrocytes (a type of glial cell in the brain), exhibit a star-shaped morphology.
• Fixed shape: Most plant cells and many bacteria have fixed shapes due to the presence of a rigid cell wall.
• Sleeper shape: This is a common descriptive term used exclusively for Paramecium in biological studies to help students identify its distinct asymmetrical form.

Important Points

• The ability to change shape is a survival mechanism that allows the amoeba to navigate through microscopic crevices in its environment.
• The movement of the amoeba is specifically referred to as amoeboid movement, which is one of the most primitive forms of animal locomotion.
• Some species of amoeba can be pathogenic; for example, Entamoeba histolytica is the organism responsible for amoebic dysentery in humans.

The correct answer is Spinal cord.

Key Points

• A reflex arc is a neural pathway that controls a reflex action. In higher animals, most sensory neurons do not pass directly into the brain, but synapse in the spinal cord.
• The reflex arc is formed in the spinal cord to allow for faster reflex actions by activating spinal motor neurons without the delay of routing signals through the brain.
• While the information input also goes on to reach the brain, the immediate response is coordinated by the spinal cord to ensure a quick reaction to potentially harmful stimuli.
• The process starts when a receptor detects a stimulus; the signal then travels through a sensory neuron to the spinal cord, where it is processed by an interneuron (in polysynaptic reflexes) and sent back via a motor neuron to an effector.
• This mechanism is an evolutionary adaptation that helps organisms survive by responding nearly instantaneously to environmental hazards, such as heat or sharp objects.

Additional Information

• Components of a Reflex Arc:
• Receptor: The specialized cell or organ that perceives the stimulus (e.g., skin).
• Sensory Neuron: Carries the impulse from the receptor to the Central Nervous System (CNS).
• Integration Center: Located within the spinal cord, where the sensory impulse is converted into a motor impulse.
• Motor Neuron: Transmits the impulse from the spinal cord to the effector organ.
• Effector: The muscle or gland that responds to the motor impulse.
• Forebrain:
• The Forebrain is the main thinking part of the brain. It consists of the cerebrum, thalamus, and hypothalamus.
• It deals with voluntary actions, memory, learning, and the processing of sensory information like sight and sound.
• Hindbrain:
• The Hindbrain consists of the pons, cerebellum, and medulla.
• It controls involuntary actions such as blood pressure, salivation, and vomiting, and coordinates body posture and balance.
• Muscle (The Effector):
• In the context of a reflex arc, the muscle acts as the effector that performs the physical movement (contraction) to pull away from a stimulus.
• Types of Reflexes:
• Spinal Reflexes: Reflexes where the spinal cord is the primary control center (e.g., the knee-jerk reflex).
• Cranial Reflexes: Reflexes where the brain is the control center (e.g., blinking of eyes in response to bright light).

The correct answer is Sum of the number of protons and neutrons.

Key Points

• The mass number (represented by the symbol A) of an element is defined as the total sum of the number of protons and neutrons present in the nucleus of an atom.
• Protons and neutrons are collectively known as nucleons because they are situated in the nucleus of the atom.
• The relationship between the mass number and the identity of an atom is expressed by the formula: Mass Number (A) = Atomic Number (Z) + Number of Neutrons (n).
• While the atomic number identifies the number of protons (and thus the element itself), the mass number accounts for the bulk of the atom's physical mass.
• Since the mass of an electron is approximately 1/1840th of the mass of a proton, it is considered negligible and is not included when calculating the mass number. Hence, the correct answer is the sum of protons and neutrons.

Additional Information

• Atomic Number (Z):
• This refers strictly to the number of protons in the nucleus of an atom.
• In a neutral atom, the atomic number is also equal to the number of electrons.
• It determines the chemical properties of an element and its position in the Modern Periodic Table, a concept established by Henry Moseley in 1913.
• Isotopes:
• These are atoms of the same element that have the same atomic number but different mass numbers.
• This occurs because isotopes have the same number of protons but a different number of neutrons.
• For example, Carbon-12 and Carbon-14 are isotopes used in radiocarbon dating.
• Isobars:
• Isobars are atoms of different chemical elements that have the same mass number but different atomic numbers.
• An example includes Argon-40, Potassium-40, and Calcium-40, all of which have a mass number of 40 but different numbers of protons.
• Atomic Mass Unit (amu):
• The mass number is closely related to the atomic mass, which is measured in unified atomic mass units (u).
• 1 amu is defined as 1/12th of the mass of a single Carbon-12 atom.

Important Points

• Nuclear Composition: The nucleus contains almost 99.9% of the total mass of an atom. Because protons and neutrons each have a mass of approximately 1.007 u and 1.008 u respectively, their sum provides the integer mass number.
• Notation: In scientific notation, the mass number (A) is written as a superscript and the atomic number (Z) as a subscript to the left of the element's chemical symbol (e.g., ¹²₆C).
• Isotones: These are different elements that share the same number of neutrons. For instance, Boron-12 and Carbon-13 both contain 7 neutrons.
• Neutron Calculation: To find the number of neutrons in an atom, subtract the atomic number from the mass number (n = A - Z).

The correct answer is Multiple reflection of sound.

Key Points

• The stethoscope is an essential medical instrument used by healthcare professionals to listen to the internal sounds of an animal or human body, primarily those originating from the heart or lungs.
• It functions based on the principle of multiple reflection of sound. When the diaphragm of the stethoscope is placed on the patient's chest, the internal sounds (like heartbeats) create vibrations that turn into longitudinal sound waves.
• These sound waves pass through the hollow plastic or rubber tubes of the stethoscope. Inside these tubes, the sound waves strike the inner walls and undergo repeated reflections (multiple reflections) before reaching the doctor's ears.
• This process of multiple reflection ensures that the sound energy is confined within the tube and directed toward the ear tips without significant loss of intensity, allowing the doctor to hear even faint sounds clearly.
• The chest piece usually contains a diaphragm and a bell. The diaphragm is designed to capture high-frequency sounds, while the bell is used for low-frequency sounds.

Additional Information

• Refraction of Sound: This is the bending of sound waves when they pass from one medium to another with a different density or temperature. While sound refracts in the atmosphere, it is not the mechanism used in a stethoscope.
• Diffraction of Sound: This refers to the spreading out or bending of sound waves around obstacles or through narrow openings. While sound waves naturally diffract, this property is not utilized for the amplification or transmission of sound in medical auscultation devices.
• Dispersion of Sound: This is the phenomenon where sound waves of different frequencies travel at different speeds, causing a pulse to spread out. This is generally not a factor in the functional design of a standard acoustic stethoscope.
• Invention: The stethoscope was invented in France in 1816 by René Laennec. The original version was a simple wooden tube, which was far more effective and hygienic than the previous method of placing the ear directly on the patient's chest.
• Acoustic vs. Electronic: While traditional acoustic stethoscopes rely entirely on physical reflection, modern electronic stethoscopes amplify the sound electronically and can even convert the acoustic sound into digital data for further analysis.

Important Points

• Reflection Laws: The reflection of sound follows the same laws as the reflection of light: the angle of incidence is equal to the angle of reflection, and the incident wave, reflected wave, and the normal all lie in the same plane.
• Sound Propagation: Sound requires a material medium (solid, liquid, or gas) to travel. In a stethoscope, the sound travels through the air column inside the tubing, reflecting off the solid walls of the tube.
• Applications: Beyond the heart and lungs, stethoscopes are used to listen to the intestines (bowel sounds) and blood flow in arteries and veins (especially during blood pressure measurement using a sphygmomanometer).
• Design Features: The tubing length is usually between 18 to 27 inches. Shorter tubing may provide better sound quality but requires the doctor to be closer to the patient. The material is typically PVC, which provides the necessary flexibility and internal smoothness for efficient reflection.

Shortcut Trick

Effective rate for 2 years at 20% Compound Interest = 20 + 20 + (20 × 20)/100 = 44%.

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

Amount = 1.44 × 5,000.

Amount = ₹7,200.

∴ The correct answer is ₹7,200.

Alternate Method

Given:

⇒ Principal (P) = ₹5,000

⇒ Annual Rate (R) = 20%

⇒ Time (T) = 2 years

Formula Used:

⇒ Amount (A) = P(1 + R/100)^T

Calculations:

⇒ A = 5000 × (1 + 20/100)²

⇒ A = 5000 × (1 + 1/5)²

⇒ A = 5000 × (6/5) × (6/5)

⇒ A = 5000 × 36/25

⇒ A = 200 × 36

⇒ A = ₹7,200

∴ The correct answer is ₹7,200.

Additional Information

Compound Interest Formula

Interest alone is calculated as CI = P[(1 + R/100)ⁿ − 1].

CI vs SI for 2 Years

The difference (D) between CI and SI for 2 years is given by D = P(R/100)².

Half-Yearly Compounding

If compounded half-yearly, the rate is halved (R/2) and the time is doubled (2T).

Shortcut Trick

Age ratio (Father : Son) 7 years ago = 5 : 1 (Difference = 4)

Age ratio (Father : Son) 7 years hence = 3 : 1 (Difference = 2)

Equalize difference: New ratio 7 years hence = 6 : 2 (Difference = 4)

Ratio change in 14 years (7 ago to 7 hence) for son: 2 − 1 = 1 unit → 14 years

Son's age 7 years ago = 1 unit = 14 years ⇒ Present age = 14 + 7 = 21 years

∴ The correct answer is 21 years.

Alternate Method

Given: Ratio 7 years ago = 5 : 1, Ratio 7 years later = 3 : 1.

Formula Used: Age calculation based on linear equations.

⇒ Let the age of Son and Father 7 years ago be x and 5x respectively.

⇒ Present age of Son = x + 7 and Father = 5x + 7.

⇒ 7 years from now: Father's age = (5x + 7) + 7 = 5x + 14.

⇒ 7 years from now: Son's age = (x + 7) + 7 = x + 14.

⇒ According to question: 5x + 14 = 3 × (x + 14)

⇒ 5x + 14 = 3x + 42 ⇒ 2x = 28 ⇒ x = 14.

⇒ Son's present age = x + 7 = 14 + 7 = 21 years.

∴ The correct answer is 21 years.

Additional Information

Ratio Difference Concept

The age difference between two people always remains constant. If the difference in parts is not equal, multiply the ratios by the difference of the other ratio.

Present Age Conversion

If age 'n' years ago is x, present age is x + n. If age 'n' years later is y, present age is y − n.

Shortcut Trick

Mode is simply the value that appears most frequently in the given data set.

Scan the list for repeating numbers: 20 appears 3 times, 32 appears 2 times, and 34 appears 5 times.

Since 34 has the highest count, it is the mode.

∴ The correct answer is 34.

Alternate Method

Given observations: 21, 32, 34, 34, 34, 20, 32, 24, 34, 34, 20, 25, 29, 26, 20

To find the mode, we calculate the frequency (number of occurrences) of each observation:

⇒ Frequency of 20: Found at positions 6, 11, 15 → Count = 3

⇒ Frequency of 21: Found at position 1 → Count = 1

⇒ Frequency of 24: Found at position 8 → Count = 1

⇒ Frequency of 25: Found at position 12 → Count = 1

⇒ Frequency of 26: Found at position 14 → Count = 1

⇒ Frequency of 29: Found at position 13 → Count = 1

⇒ Frequency of 32: Found at positions 2, 7 → Count = 2

⇒ Frequency of 34: Found at positions 3, 4, 5, 9, 10 → Count = 5

⇒ Since the value 34 has the maximum frequency of 5, it is the Mode.

∴ The correct answer is 34.

Additional Information

Empirical Formula

For a moderately skewed distribution, the relationship is: Mode = 3 × Median − 2 × Mean.

Types of Mode

Data with one mode is unimodal, two modes is bimodal, and more than two is multimodal.

Median Calculation

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

Shortcut Trick

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

34,400 × (87 ÷ 100) × ((100 − Z) ÷ 100) = 22,446.

29,928 × (100 − Z) ÷ 100 = 22,446.

100 − Z = (22,446 × 100) ÷ 29,928 ⇒ 100 − Z = 75.

Z = 100 − 75 = 25.

∴ The correct answer is 25.

Alternate Method

Formulas: Price after discount = MP × (100 − D%) ÷ 100 | Second Discount % = (Intermediate SP − Final SP) ÷ Intermediate SP × 100

⇒ Marked Price (MP) = ₹34,400. ⇒ Price after 1^st discount of 13% = 34,400 − (13% of 34,400) = 34,400 − 4,472 = ₹29,928. ⇒ Final Selling Price (SP) after 2^nd discount = ₹22,446. ⇒ Amount of 2^nd discount = 29,928 − 22,446 = ₹7,482. ⇒ Z% = (7,482 ÷ 29,928) × 100 = 0.25 × 100 = 25%.

∴ The correct answer is 25.

Additional Information

Successive Discounts

When two or more discounts are applied one after another, they are called successive discounts. The total effective discount is always less than the sum of individual percentages.

Effective Discount Formula

For two successive discounts a% and b%, the single equivalent discount is given by: (a + b − (a × b ÷ 100))%.

Shortcut Trick

Since the denominators (30) are the same, compare only the numerators.

Numerator 1 = 1, Numerator 2 = 2.

Required percentage = (1 ÷ 2) × 100 = 50%.

∴ The correct answer is 50%.

Alternate Method

Given: First value = 1/30, Second value = 2/30.

Formula: Percentage = (Value / Base) × 100

⇒ Percentage = [(1/30) / (2/30)] × 100

⇒ Percentage = (1/30) × (30/2) × 100

⇒ Percentage = (1/2) × 100

⇒ Percentage = 50%

∴ The correct answer is 50%.

Additional Information

Percentage Change

To find the percentage change between two numbers: [(New − Old) ÷ Old] × 100.

Fraction to Percentage

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

Base Comparison

In questions like 'A is what % of B', B is always the denominator (base) and A is the numerator.

Shortcut Trick

Ratio of investment of Preeti, Qadir and Reena = 2 : 8 : 13.

Total ratio units = 2 + 8 + 13 = 23 units.

Difference between Preeti and Reena = 13 − 2 = 11 units.

Value of 11 units = (Total Profit ÷ 23) × 11 = (87,193 ÷ 23) × 11.

Value = 3,791 × 11 = 41,701.

∴ The correct answer is ₹41,701.

Alternate Method

Given → Investment Ratio = 2 : 8 : 13 | Total Profit = ₹87,193. Formulas: Profit Share ∝ Investment × Time | Profit% = (Profit ÷ CP) × 100

⇒ Since time is the same for all, Profit Ratio = Investment Ratio = 2 : 8 : 13. ⇒ Let the shares be 2x, 8x, and 13x respectively. ⇒ Total Profit = 2x + 8x + 13x = 23x = 87,193. ⇒ x = 87,193 ÷ 23 = 3,791. ⇒ Difference between Preeti and Reena = 13x − 2x = 11x. ⇒ Difference = 11 × 3,791 = 41,701.

∴ The correct answer is ₹41,701.

Additional Information

Partnership Basics

When the time period of investment is the same for all partners, the profit is distributed in the ratio of their investments.

Compound Partnership

If time periods are different, Profit Ratio = (Capital₁ × Time₁) : (Capital₂ × Time₂).

Shortcut Trick

Ratio of Length : Breadth : Height = 3 : 2 : 1. Let common factor be x.

Product of ratios = 3 × 2 × 1 = 6. Given Volume = 10,368 cm³.

x³ = 10,368 ÷ 6 = 1,728 ⇒ x = ∛1,728 = 12 cm.

Total Surface Area (in ratio units) = 2 × (3×2 + 2×1 + 1×3) = 2 × 11 = 22 units.

Total Cost = TSA × Rate = (22 × x²) × 2 = 22 × 144 × 2 = ₹6,336.

∴ The correct answer is ₹6,336.

Alternate Method

Given: Volume of rectangular block = 10,368 cm³. Ratio of L : B : H = 3 : 2 : 1. Polishing rate = ₹2/cm².

Formula Used: Volume = L × B × H | Total Surface Area (TSA) = 2(LB + BH + HL)

⇒ Let length = 3x, breadth = 2x, and height = x.

⇒ Volume = 3x × 2x × x = 6x³

⇒ 6x³ = 10,368

⇒ x³ = 1,728 ⇒ x = 12

⇒ Dimensions: L = 36 cm, B = 24 cm, H = 12 cm

⇒ TSA = 2(36 × 24 + 24 × 12 + 12 × 36)

⇒ TSA = 2(864 + 288 + 432) = 2(1,584) = 3,168 cm²

⇒ Total Cost = Area × Rate = 3,168 × 2 = ₹6,336

∴ The correct answer is ₹6,336.

Additional Information

Volume of a Cuboid

The space occupied by a rectangular prism is calculated as Length × Breadth × Height.

Lateral Surface Area

Area of the four side walls is given by 2 × Height × (Length + Breadth).

Diagonal of a Cuboid

The longest rod that can be placed inside is √(L² + B² + H²).

Surface Area of a Cube

If L=B=H=a, then the Total Surface Area = 6a².

Shortcut Trick

Net percentage change = a + b + (a × b) ÷ 100

Here, markup percentage a = +85% and discount percentage b = −49%.

Net result = 85 − 49 + (85 × (−49)) ÷ 100

Net result = 36 − 41.65 = −5.65%.

Negative sign indicates a loss.

∴ The correct answer is 5.65%.

Alternate Method

Given: Marked Price (MP) is 85% above Cost Price (CP), Discount = 49%.

Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Loss% = (CP − SP) ÷ CP × 100

⇒ Let Cost Price (CP) be 100.

⇒ Marked Price (MP) = 100 + 85% of 100 = 185.

⇒ Selling Price (SP) = 185 × (100 − 49) ÷ 100 = 185 × 51 ÷ 100 = 94.35.

⇒ Loss = CP − SP = 100 − 94.35 = 5.65.

⇒ Loss Percentage = (5.65 ÷ 100) × 100 = 5.65%.

∴ The correct answer is 5.65%.

Additional Information

Relation between CP and MP

The ratio of Marked Price to Cost Price can be expressed as: MP ÷ CP = (100 + Profit%) ÷ (100 − Discount%).

Markup Percentage

Markup% is the percentage by which the Marked Price is higher than the Cost Price: Markup% = (MP − CP) ÷ CP × 100.

Shortcut Trick

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

Square the difference and add twice the product: (9)² + 2(20)

81 + 40 = 121

∴ The correct answer is 121.

Alternate Method

Given:

Difference (a − b) = 9

Product (ab) = 20

Formula Used:

The square of a binomial difference is given by: (a − b)² = a² + b² − 2ab

Calculations:

Substituting the given values into the formula:

⇒ (9)² = a² + b² − 2 × 20

⇒ 81 = a² + b² − 40

⇒ a² + b² = 81 + 40

⇒ a² + b² = 121

∴ The correct answer is 121.

Additional Information

Square of Sum Identity

The value can also be found if the sum is known using: a² + b² = (a + b)² − 2ab.

Relationship between Sum and Difference

(a + b)² − (a − b)² = 4ab. This relates the sum, difference, and product of two numbers.

Difference of Squares

a² − b² = (a + b)(a − b). This is a fundamental factoring identity in algebra.

Shortcut Trick Identify odd numbers between 415 and 428: 417, 419, 421, 423, 425, 427. Eliminate numbers divisible by 3 (sum of digits): 417 (12), 423 (9). Eliminate numbers divisible by 5: 425. Check remaining: 419, 421, 427. Since 427 = 7 × 61, it is not prime. Prime numbers: 419 and 421. ∴ The correct answer is 2.

Alternate Method Given: Range is 415 < n < 428. Numbers to check: 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427.

⇒ 417: 4+1+7 = 12 (Divisible by 3) ⇒ 419: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. (Prime) ⇒ 421: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19. (Prime) ⇒ 423: 4+2+3 = 9 (Divisible by 3) ⇒ 425: Ends in 5 (Divisible by 5) ⇒ 427: 427 ÷ 7 = 61 (Divisible by 7) ∴ The correct answer is 2.

Additional Information Prime Number Definition A natural number greater than 1 that has no positive divisors other than 1 and itself. Primality Test To check if n is prime, test divisibility by primes up to √n. For 421, √421 ≈ 20.5, so check primes up to 19. Divisibility Rule of 7 Double the last digit and subtract it from the rest: 427 → 42 − (7 × 2) = 28. Since 28 is divisible by 7, 427 is divisible by 7.

Shortcut Trick

In melting problems, Total Volume remains constant: R³ = r₁³ + r₂³ + r₃³.

⇒ 1.5³ = 0.75³ + 1³ + r₃³

⇒ 3.375 = 0.421875 + 1 + r₃³

⇒ r₃³ = 3.375 − 1.421875 = 1.953125 ⇒ r₃ = 1.25 cm.

⇒ Diameter of the third ball = 2 × 1.25 = 2.5 cm.

∴ The correct answer is 2.5 cm.

Alternate Method

Given:

Radius of large iron ball (R) = 1.5 cm

Radius of first smaller ball (r₁) = 0.75 cm

Radius of second smaller ball (r₂) = 1.0 cm

Formula Used:

Volume of a sphere = 4/3 × π × r³

Let the radius of the third smaller ball be r₃.

⇒ Total Volume = Volume₁ + Volume₂ + Volume₃

⇒ 4/3 × π × (1.5)³ = 4/3 × π × (0.75)³ + 4/3 × π × (1.0)³ + 4/3 × π × (r₃)³

⇒ 1.5³ = 0.75³ + 1.0³ + r₃³

⇒ 3.375 = 0.421875 + 1 + r₃³

⇒ 3.375 = 1.421875 + r₃³

⇒ r₃³ = 3.375 − 1.421875

⇒ r₃³ = 1.953125

⇒ r₃ = 1.25 cm

⇒ Diameter of the third ball = 2 × r₃ = 2 × 1.25 = 2.5 cm

∴ The correct answer is 2.5 cm.

Additional Information

Surface Area of Sphere

The total surface area of a sphere is given by 4 × π × r².

Properties of Melting

When one solid object is melted and recast into other shapes, the total volume remains invariant (constant), assuming no material loss.

Hemisphere Formulas

Volume of hemisphere = 2/3 × π × r³ and Total Surface Area = 3 × π × r².

Shortcut Trick

Total deduction = 19 + 9 + 13 = ₹41.

Remaining amount = 8200 − 41 = ₹8159.

Ratio of remaining shares = 12 : 19 : 10 (Total units = 41).

1 unit = 8159 ÷ 41 = ₹199.

Difference (B − C) = (19 − 10) units + (deduction difference) = 9 × 199 + (9 − 13) = 1791 − 4 = ₹1787.

∴ The correct answer is ₹1787.

Alternate Method

Given: Total = ₹8,200. Deductions for A, B, C are ₹19, ₹9, ₹13 respectively. Ratio of remaining parts = 12 : 19 : 10.

⇒ Let remaining shares be 12x, 19x, and 10x.

⇒ (12x + 19) + (19x + 9) + (10x + 13) = 8200

⇒ 41x + 41 = 8200

⇒ 41x = 8159

⇒ x = 8159 ÷ 41 = 199

⇒ B's original share = 19x + 9 = 19 × 199 + 9 = 3781 + 9 = ₹3790

⇒ C's original share = 10x + 13 = 10 × 199 + 13 = 1990 + 13 = ₹2003

⇒ Difference = 3790 − 2003 = ₹1787

∴ The correct answer is ₹1787.

Additional Information

Ratio Distribution with Constants

When specific amounts are deducted or added to shares, first adjust the total sum and then divide the adjusted sum in the given ratio.

Difference between Shares

Difference = (Ratio Difference × Value per unit) ± (Difference between constants).

Shortcut Trick

Relative Speed when moving towards each other = Sum of speeds = 23 + 89 = 112 km/hr.

Time = Distance ÷ Relative Speed = 28 ÷ 112 = 1/4 hours.

In minutes = 1/4 × 60 = 15 minutes.

∴ The correct answer is 15 min.

Alternate Method

Given:

Distance between engines = 28 km

Speed of engine 1 (S₁) = 23 km/hr

Speed of engine 2 (S₂) = 89 km/hr

Formula Used:

Time = Distance ÷ Relative Speed

Relative Speed (Opposite Direction) = S₁ + S₂

Calculations:

⇒ Relative Speed = 23 + 89 = 112 km/hr

⇒ Time (in hours) = 28 ÷ 112

⇒ Time = 1/4 hours

⇒ Time (in minutes) = (1/4) × 60

⇒ Time = 15 min

∴ The correct answer is 15 min.

Additional Information

Relative Speed Concept

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

Time Conversion

To convert time from hours to minutes, multiply the value by 60. To convert km/hr to m/s, multiply by 5/18.

Meeting Time Rule

The Meeting Time is the initial distance between the objects divided by the relative speed, provided they start at the same time.

Shortcut Trick

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

∴ ∠TES = (∠Q + ∠R) ÷ 2

⇒ ∠TES = (74° + 39°) ÷ 2

⇒ ∠TES = 113° ÷ 2 = 56.5°

∴ The correct answer is 56.5°.

Alternate Method

Given: In quadrilateral QRST, ∠Q = 74° and ∠R = 39°. SE and TE are angle bisectors of ∠S and ∠T respectively.

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

In quadrilateral QRST:

⇒ ∠Q + ∠R + ∠S + ∠T = 360°

⇒ 74° + 39° + ∠S + ∠T = 360°

⇒ 113° + ∠S + ∠T = 360°

⇒ ∠S + ∠T = 360° − 113° = 247°

Since SE and TE are bisectors:

Shortcut Trick

Total Work = LCM of 12, 20, and 36 = 180 units.

Efficiency of A + B + C = 180 ÷ 12 = 15 units/day.

Efficiency of A = 180 ÷ 20 = 9 units/day | Efficiency of C = 180 ÷ 36 = 5 units/day.

Efficiency of B = 15 − (9 + 5) = 1 unit/day.

Time taken by B = 180 ÷ 1 = 180 days.

∴ The correct answer is 180 days.

Alternate Method

Given: Time taken by (A+B+C) = 12 days, A = 20 days, C = 36 days.

Formula Used: Efficiency = Total Work ÷ Time taken

⇒ Efficiency of (A + B + C) = Efficiency of A + Efficiency of B + Efficiency of C

⇒ 15 = 9 + Efficiency of B + 5

⇒ Efficiency of B = 15 − 14 = 1 unit/day

⇒ Time taken by B alone = Total Work ÷ Efficiency of B

⇒ Time taken by B = 180 ÷ 1 = 180 days

∴ The correct answer is 180 days.

Additional Information

Combined Efficiency Rule

The total work done by multiple individuals working together is the sum of their individual efficiencies. Total Efficiency = E₁ + E₂ + E₃.

Work & Time Relation

Time is inversely proportional to efficiency when the work remains constant. Time = Work ÷ Efficiency.

Pipes & Cisterns Analogy

This problem follows the same logic as three pipes filling a tank, where the combined rate is the sum of individual rates.

Shortcut Trick

Option Verification Method:

Check which pair satisfies Sum = 54 and LCM = 195.

Option (2): 39 + 15 = 54. Factors of 39 = 3 × 13; Factors of 15 = 3 × 5.

LCM(39, 15) = 3 × 13 × 5 = 195. This satisfies both conditions.

∴ The correct answer is 39, 15.

Alternate Method

Given: Sum of numbers = 54, LCM = 195.

Formula Used: HCF of two numbers is always a factor of their sum and their LCM.

⇒ Let the numbers be 3a and 3b, where a and b are co-prime.

⇒ Sum: 3a + 3b = 54 ⇒ a + b = 18

⇒ LCM: 3 × a × b = 195 ⇒ ab = 65

⇒ We need two numbers whose sum is 18 and product is 65. The factors of 65 are 1, 5, 13, 65.

⇒ Clearly, a = 13 and b = 5 (since 13 + 5 = 18 and 13 × 5 = 65).

⇒ First number = 3 × 13 = 39.

⇒ Second number = 3 × 5 = 15.

∴ The correct answer is 39, 15.

Additional Information

Product Rule

For any two numbers x and y, x × y = HCF(x,y) × LCM(x,y).

HCF property

The Highest Common Factor (HCF) must divide both the sum and the Least Common Multiple (LCM) of the numbers.

Co-prime Numbers

Numbers are co-prime if their HCF is 1. In the method above, a and b must be co-prime to ensure 3 is the highest factor.

Shortcut Trick

Time taken by A, B, and C working together to complete the original work = 3 ÷ 3 = 1 day.

To complete twice the same work, the time taken will be doubled.

Time taken = 1 day × 2 = 2 days.

∴ The correct answer is 2 days.

Alternate Method

Given:

Time taken by A = 3 days

Time taken by B = 3 days

Time taken by C = 3 days

Formula Used:

Efficiency = Total Work ÷ Time Taken

Combined Efficiency = Efficiency of A + Efficiency of B + Efficiency of C

Time Taken = Total Work ÷ Combined Efficiency

Calculations:

⇒ Total original work = 3 units

⇒ Combined efficiency of A, B and C = 1 + 1 + 1 = 3 units/day

⇒ New work to be completed = 2 × Total original work = 2 × 3 = 6 units

⇒ Time taken to do twice the work = 6 ÷ 3 = 2 days

∴ The correct answer is 2 days.

Additional Information

Time and Work Relation

Time taken is directly proportional to the amount of work done if the efficiency remains constant. If work is multiplied by n, the time taken is also multiplied by n.

Combined Efficiency Formula

For three individuals with time periods T₁, T₂, and T₃, their combined work rate is (1/T₁) + (1/T₂) + (1/T₃).

Shortcut Trick

Excess runs scored in 16^th inning = 111 − 94 = 17.

This excess is distributed over the total 16 innings to find the increase in average.

Increase in average = 17 ÷ 16 = 1.0625.

Rounding 1.0625 to the nearest whole number gives 1.

∴ The correct answer is 1.

Alternate Method

Given: Average of 15 innings = 94 | Runs in 16^th inning = 111.

Formula Used: Total runs = Average × Number of innings.

⇒ Total runs after 15 innings = 15 × 94 = 1410.

⇒ Total runs after 16 innings = 1410 + 111 = 1521.

⇒ New average for 16 innings = 1521 ÷ 16 = 95.0625.

⇒ Increase in average = New average − Old average

⇒ Increase in average = 95.0625 − 94 = 1.0625.

⇒ Rounded to nearest whole number = 1.

∴ The correct answer is 1.

Additional Information

Net Deviation Method

Increase in average = (New value − Old average) ÷ Total number of items including the new one.

Weighted Average

The combined average of two groups with n₁ and n₂ items and averages a₁ and a₂ is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂).

Batting Average Formula

Batting Average = (Total runs scored) ÷ (Number of times the batsman got out/Total innings played).

Shortcut Trick

Total votes = 14,000

Valid votes = 90% of 14,000 = 12,600

Candidate A got 75% of valid votes → Candidate B got 25% of valid votes

Votes for Candidate B = 12,600 × 1/4 = 3,150

∴ The correct answer is 3150.

Alternate Method

Given:

Total votes = 14,000

Invalid percentage = 10%

Candidate A valid vote share = 75%

Calculations:

⇒ Total Valid Votes = 14,000 − (10% of 14,000)

⇒ Total Valid Votes = 14,000 − 1,400 = 12,600

⇒ Candidate B share % = 100% − 75% = 25%

⇒ Candidate B votes = 25% of 12,600

⇒ Candidate B votes = 12,600 ÷ 4 = 3,150

∴ The correct answer is 3150.

Additional Information

Valid vs Invalid Votes

In election problems, winners and losers are always calculated based on Valid Votes unless stated otherwise. Valid = Total − Invalid.

Successive Percentage Method

Votes for B = Total × (Valid% / 100) × (B% / 100). Here, 14,000 × 0.90 × 0.25 = 3,150.

Voter Turnout

If 'voters who did not cast vote' is mentioned, subtract them first from the Total Registered Voters to find total votes polled.

Shortcut Trick

Express given values as powers of 5: 125 = 5³ and 3125 = 5⁵.

By equating the exponents, we get: x − y = 3 and x + y = 5.

Add the two equations directly: (x − y) + (x + y) = 3 + 5.

⇒ 2x = 8 ⇒ x = 4.

∴ The correct answer is 4.

Alternate Method

Given:

5^{(x − y)} = 125

5^{(x + y)} = 3125

Concept Used:

If a^m = aⁿ, then m = n.

Calculations:

⇒ 125 = 5 × 5 × 5 = 5³

⇒ 3125 = 5 × 5 × 5 × 5 × 5 = 5⁵

⇒ 5^{(x − y)} = 5³

⇒ x − y = 3       ---- (1)

⇒ 5^{(x + y)} = 5⁵

⇒ x + y = 5       ---- (2)

⇒ Adding equation (1) and (2):

⇒ (x − y) + (x + y) = 3 + 5

⇒ 2x = 8

⇒ x = 8 ÷ 2

⇒ x = 4

∴ The correct answer is 4.

Additional Information

Product Rule of Exponents

When multiplying terms with the same base, you add their exponents: a^m × aⁿ = a^m+n

Quotient Rule of Exponents

When dividing terms with the same base, you subtract their exponents: a^m ÷ aⁿ = a^m−n

Power of a Power Rule

When raising a power to another power, you multiply the exponents: (a^m)ⁿ = a^m×n

Zero Exponent Rule

Any non-zero base raised to the power of zero is equal to one: a⁰ = 1

Shortcut Trick

For an angle of elevation of 45°, the horizontal distance (Base) is always equal to the perpendicular height above eye level.

Height above eye level = Distance from base = 28.7 m.

Total Height of Chimney = Height above eye level + Eye level height = 28.7 + 1.3 = 30 m.

∴ The correct answer is 30 m.

Alternate Method

Given:

Distance from base (D) = 28.7 m

Angle of elevation (θ) = 45°

Height of observer (h) = 1.3 m

Formula Used:

tan(θ) = Perpendicular / Base

Total Height = Perpendicular + height of observer

Calculations:

⇒ tan(45°) = (Height of chimney − 1.3) / 28.7

⇒ 1 = (Height − 1.3) / 28.7

⇒ Height − 1.3 = 28.7

⇒ Height = 28.7 + 1.3

⇒ Height = 30 m

∴ The correct answer is 30 m.

Additional Information

Angle of Elevation

The angle formed by the line of sight with the horizontal when looking up at an object.

Trigonometric Ratios for Standard Angles

tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3 are frequently used in height and distance problems.

Observer Height Rule

If the observer has a significant height, it must be added to the calculated vertical distance from the eye level to find the total height from the ground.

Shortcut Trick

Average speed for equal distances = 2uv ÷ (u + v)

Original Avg Speed = (2 × 8 × 6) ÷ (8 + 6) = 96 ÷ 14 = 48/7 km/hr

New Speed = (48/7) × 70% = (48/7) × 0.7 = 4.8 km/hr

Time = Distance ÷ New Speed = 120 ÷ 4.8 = 25 hr

∴ The correct answer is 25 hr.

Alternate Method

Given: Total Distance = 120 km, Speed₁ = 8 km/hr, Speed₂ = 6 km/hr, Speed Reduction = 30%

Formula: Average Speed = Total Distance ÷ Total Time

⇒ Time for 1^st half = 60 ÷ 8 = 7.5 hr

⇒ Time for 2^nd half = 60 ÷ 6 = 10 hr

⇒ Original Average Speed = 120 ÷ (7.5 + 10) = 120 ÷ 17.5 = 1200 ÷ 175 = 48/7 km/hr

⇒ Reduced overall speed = (48/7) × (100 − 30)/100 = (48/7) × 0.7 = 4.8 km/hr

⇒ Required Time = 120 ÷ 4.8 = 25 hr

∴ The correct answer is 25 hr.

Additional Information

Average Speed (Equal Distances)

When a journey is divided into two equal distances with speeds u and v, the average speed is given by (2uv) / (u + v).

Percentage Speed Change

If speed decreases by x%, the new speed is (100 − x)% of the original speed. Time taken is inversely proportional to speed for a fixed distance.

General Average Speed Formula

Average Speed = (Total Distance) ÷ (Total Time taken), regardless of the number of segments or distances involved.

Shortcut Trick

Using the Alligation or Weighted Average method for Income (I), Expenditure (E), and Savings (S).

Ratio of E : S = 250 : 100 = 5 : 2.

Net change in Income = [(Expenditure % change × E ratio) + (Savings % change × S ratio)] ÷ Total ratio

Net change = [(−3 × 5) + (21.5 × 2)] ÷ 7 = [−15 + 43] ÷ 7 = 28 ÷ 7 = 4%.

∴ The correct answer is 4%.

Alternate Method

Given:

Let Savings = 100

Expenditure = 100 + 150% of 100 = 250

Income = Expenditure + Savings = 250 + 100 = 350

Calculations:

⇒ New Expenditure = 250 − (3% of 250) = 250 − 7.5 = 242.5

⇒ New Savings = 100 + (21.5% of 100) = 100 + 21.5 = 121.5

⇒ New Income = 242.5 + 121.5 = 364

⇒ Increase in Income = 364 − 350 = 14

⇒ % Increase = (14 ÷ 350) × 100

⇒ % Increase = (1 ÷ 25) × 100 = 4%

∴ The correct answer is 4%.

Additional Information

Income Equation

The fundamental relation in financial arithmetic is Income = Expenditure + Savings.

Percentage Change in Sum

If Z = X + Y, the % change in Z is given by (ΔX + ΔY)/Z_initial × 100, which is the weighted average of individual changes.

Base Change Property

When a value is increased by x%, it becomes (100+x)/100 times its original value.

Shortcut Trick

2 Chairs + 1 Table = ₹350

Multiplying by 2: 4 Chairs + 2 Tables = ₹700

Given: 3 Chairs + 2 Tables = ₹600

Difference (1 Chair) = ₹700 − ₹600 = ₹100

Cost of 4 Chairs = 4 × 100 = ₹400

∴ The correct answer is ₹400.

Alternate Method

Given:

Cost of 3 chairs and 2 tables = ₹600

Cost of 2 chairs and 1 table = ₹350

Formula Used:

Linear equations in two variables: ax + by = c

Calculations:

Let the cost of one chair be C and one table be T.

⇒ 3C + 2T = 600 --- (i)

⇒ 2C + T = 350 --- (ii)

⇒ From (ii), T = 350 − 2C

⇒ Substitute T in (i): 3C + 2(350 − 2C) = 600

⇒ 3C + 700 − 4C = 600

⇒ −C = 600 − 700

⇒ C = 100

⇒ Cost of 4 chairs = 4 × 100 = 400

∴ The correct answer is ₹400.

Additional Information

Elimination Method

A process of solving simultaneous equations by multiplying equations by suitable constants so that adding or subtracting them eliminates one variable.

Substitution Method

Solving for one variable in terms of the other from one equation and substituting it into the second equation.

Unitary Method

Finding the value of a single unit first (1 Chair) and then multiplying by the required number of units (4 Chairs).

Logic followed here is:

In each pair, the first letter of the second cluster is obtained by subtracting 1 from the first letter of the first cluster, and the second letter of the second cluster is obtained by subtracting 1 from the second letter of the first cluster.

Option (1) KI − JH

Option (2) FD − EC

 

Option (3) AY − YZ

Option (4) WU − VT

 

 

In options (1), (2), and (4), the pattern followed is (Cluster 1) → (-1, -1) → (Cluster 2). However, in option (3), this pattern is not followed.

Therefore, the letter-cluster pair that does not belong to the group is AY − YZ.

Hence, the correct answer is "Option 3".

Statements:

All markers are pens.

All markers are sketches.

All markers are chalks.

Venn Diagram Logic:

Conclusions:

(I) All pens are chalks → Does not follows (Because only markers are common to both pens and chalks. It is not given that all pens are chalks.)

(II) Some chalks are sketches → Follows (Because all markers are chalks and all markers are sketches, therefore some chalks are definitely sketches.)

Therefore, only conclusion (II) follows.

Hence, the correct answer is "Option 3".

Logic followed here is: Each cluster in the series follows a specific pattern for its letters and its number. Each letter decreases by 2 in positional value, and the number increases by 4.

Therefore, the code that replaces the question mark is JMK 52.

Hence, the correct answer is "Option 3".

Logic followed here is:( Second digit of Highest Number) + (Second digit of Lowest Number).

The given numbers are:

515, 758, 761, 726, 579

Highest number is 761 →

The second digit of the highest number (761) is 6.

 

Lowest number is 515 →

The second digit of the lowest number (515) is 1.

 

Similarly, let's find the resultant:

Resultant = 6 + 1 = 7

 

Therefore, the resultant is 7.

Hence, the correct answer is "Option 2".

Logic followed here is: The difference between the positional values of consecutive letters in each cluster follows a specific pattern of -5 and -7.

1. DYR:

2. VQJ:

 

3. UPI:

 

4. AWO:

 

The pattern observed in DYR, VQJ, and UPI is a consistent decrease of -5 and -7. However, 'AWO' follows a different pattern of -4 and -8.

Therefore, the letter-cluster that does not belong to the group is AWO.

Hence, the correct answer is "Option 4".

Question:

Seven friends, L, M, N, O, Q, R and S, are sitting in a straight line facing north.

→ Only three people sit to the left of M.

→ Only two people sit between M and L.

→ Only three people sit between S and N.

→ S sits to the immediate left of L.

→ O sits somewhere to the left of R but somewhere to the right of Q.

Q is at position 1.

R is at position 5.

People between them are:

→ N

→ O

→ M

Total people between Q and R = 3

Hence, the correct answer is "Option 4 — Three".

Given:

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

Logic followed here is: We need to find odd digits that are immediately preceded by an odd digit and immediately followed by an even digit.

Pattern: Odd Number → Odd Number → Even Number

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

The required odd digits are:

→ 9 in (5 9 2)

→ 5 in (9 5 2)

→ 5 in (1 5 2)

Therefore, there are 3 such odd digits.

Hence, the correct answer is "Option 2".

Logic followed here is: The series follows an alternating addition pattern where consecutive terms are obtained by adding 5 and 6 alternately.

 

Therefore, the number that should come in place of the question mark is 29.

Hence, the correct answer is "Option 1".

Signs interchange:

Given equation:

13 − 54 × 9 ÷ 7 + 32 = ?

After interchanging the signs:

⇒ 13 + 54 ÷ 9 × 7 − 32

⇒ 13 + 6 × 7 − 32

⇒ 13 + 42 − 32

⇒ 55 − 32

⇒ 23

Therefore, the value in place of the question mark is 23.

Hence, the correct answer is "Option 2".

Given:

‘book news leaf’ is coded as ‘rk qh aa’

‘leaf gum boat’ is coded as ‘vf rk xl’

Therefore:

→ leaf = rk

Hence, the correct answer is "Option 4".

Logic followed here is:

The series is formed by adding the squares of consecutive natural numbers, starting from 5², to the previous term to obtain the next number.

 

Therefore, the number that should come in place of the question mark is 266.

Hence, the correct answer is "Option 1".

Given:

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

Logic followed here is: We need to find even numbers that are immediately preceded by a perfect square and immediately followed by an odd number.

Perfect square single-digit numbers are:

→ 1

→ 4

→ 9

Pattern: Perfect Square → Even Number → Odd Number

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

The required even numbers are:

→ 8 in (9 8 5)

→ 6 in (4 6 7)

→ 8 in (4 8 3)

→ 6 in (9 6 5)

Therefore, there are 4 such even numbers.

Hence, the correct answer is "Option 1 — More than three".

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

Logic followed here is: We need to find even digits that are immediately preceded by an odd digit and also immediately followed by an even digit.

Pattern: Odd Number → Even Number → Even Number

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

The required even digits are:

→ 6 in (3 6 4)

→ 8 in (3 8 4)

Therefore, there are 2 such even digits.

Hence, the correct answer is "Option 3".