RRB Group D Official Paper (Held On: 08 Jan, 2026 Shift 2) (Previous Year Paper)

The correct answer is Because they make food from inorganic substances.

Key Points

• Autotrophs are organisms that can produce their own food using light, water, carbon dioxide, or other chemicals; they are essentially the primary producers of an ecosystem.
• They utilize inorganic substances from the environment, such as carbon dioxide (CO₂) and water (H₂O), to synthesize organic molecules like glucose through the process of photosynthesis.
• Because they are the only organisms capable of converting solar energy into chemical energy, they form the foundation of all biotic communities.
• In any food chain or ecological pyramid, autotrophs are placed at the first trophic level because they provide the initial energy source that sustains all other levels.
• Common examples of autotrophs include vascular plants on land and phytoplankton or algae in aquatic environments.

 Additional Information

• Trophic Levels: These represent the feeding positions in a food chain, where energy flows from producers to various levels of consumers.
• Herbivores: These are primary consumers that occupy the second trophic level because they feed directly on plants.
• Heterotrophs: Unlike autotrophs, these organisms cannot manufacture their own food and must rely on the consumption of other organisms for nutrition.
• Decomposers: Organisms such as bacteria and fungi that break down dead organic matter and organic waste, playing a vital role in nutrient cycling.
• Saprophytes: A specific group of decomposers that obtain nutrients by absorbing organic matter from decaying organisms through extracellular digestion.
• Energy Flow: According to Lindeman's 10 percent law, only about 10% of the energy available at one trophic level is transferred to the next, which is why producers must be the most abundant group.

The correct answer is Dissolution of salt in water.

Key Points

• A physical change is a type of change in which the physical properties of a substance, such as its shape, size, volume, or state, are altered without changing its chemical composition.
• During the dissolution of salt in water, the sodium chloride (NaCl) crystals break down into sodium (Na+) and chloride (Cl-) ions which disperse throughout the water (H2O). No new chemical substance is formed in this process. Hence, statement 2 is correct.
• One of the primary characteristics of a physical change is that it is often reversible. Salt can be recovered from the solution through physical methods like evaporation or boiling, where the water turns into vapor, leaving the solid salt behind.
• In this specific homogeneous mixture, the solute (salt) and solvent (water) retain their chemical identities even though they are mixed at the molecular level.
• Common examples of physical changes include the melting of ice, boiling of water, crushing a can, and shredding paper.

Additional Information

• Curdling of milk: This happens when an acid (like citric acid from lemon juice) is added to milk, causing the casein protein to denature and coagulate. This is a chemical change because a new substance (curd) is formed and the process is irreversible. Hence, statement 1 is incorrect.
• Digestion of food: This is a biochemical process involving enzymes, hydrochloric acid, and chemical reactions that break down complex organic molecules into simpler nutrients. Since new substances are formed and the original food cannot be recovered, it is a chemical change. Hence, statement 3 is incorrect.
• Rusting of iron: This is an oxidation reaction where iron (Fe) reacts with oxygen and moisture to form Hydrated Iron (III) Oxide. The presence of a salt solution acts as an electrolyte and accelerates this chemical change. Hence, statement 4 is incorrect.
• Chemical Changes: Unlike physical changes, these involve the breaking and forming of chemical bonds. Indicators of a chemical change include evolution of gas, change in color, formation of a precipitate, and release of heat or light.

Important Points

• A solution is defined as a homogeneous mixture of two or more substances. The substance present in the larger quantity is the solvent, while the one in the smaller quantity is the solute.
• The process of dissolution can sometimes be exothermic (releasing heat) or endothermic (absorbing heat), but it is still classified as a physical change as long as the chemical identities of the components remain intact.
• In industrial applications, physical changes are utilized for purification processes such as fractional distillation and crystallization, which rely on the reversibility of physical states.
• Understanding the difference between physical and chemical changes is fundamental in thermodynamics and material science to predict how substances will behave under different temperature and pressure conditions.

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

Key Points

• Acid rain is caused when atmospheric pollutants like sulfur dioxide and nitrogen oxides react with water to form acids, lowering the pH of rainwater to below 5.6. This acidic runoff reaches rivers and lakes, making the habitat lethal for various aquatic organisms. Hence, Acid rain matches with When pH < 5.6, harms aquatic life.
• Indigestion occurs when the stomach secretes an excessive amount of hydrochloric acid (HCl) to digest food, leading to a burning sensation and pain. To treat this, antacids like Magnesium hydroxide (Milk of Magnesia) are used to perform a neutralization reaction. Hence, Indigestion matches with Antacids neutralize excess acid.
• Tooth decay is initiated by the production of acids from the bacterial degradation of sugars and food particles in the oral cavity. When the pH level in the mouth drops below 5.5, it triggers the demineralization of the tooth enamel. Hence, Tooth decay matches with Occurs when the pH of the mouth is < 5.5.
• A Bee sting delivers methanoic acid (formic acid) into the skin, causing acute pain and swelling. Application of a mild alkaline substance like baking soda (sodium hydrogen carbonate) helps in neutralizing the acidic venom and alleviating the pain. Hence, Bee sting matches with Neutralized by baking soda.

 Additional Information

• pH Scale and Indicators:
• The pH scale ranges from 0 to 14, where 7 is neutral; values below 7 indicate acidity and values above 7 indicate alkalinity.
• The human body generally functions within a narrow pH range of 7.0 to 7.8.
• Chemical Components:
• Tooth Enamel: Composed of calcium hydroxyapatite, which is a crystalline form of calcium phosphate and is the hardest substance in the human body.
• Baking Soda: Known chemically as Sodium Hydrogen Carbonate (NaHCO₃), it is used as a common household leavening agent and a mild base.
• Methanoic Acid: Also known as formic acid (HCOOH), it is the simplest carboxylic acid and is found in the stings of ants and bees.

The correct answer is 0.5 A.

Key Points

• In a series circuit, the total resistance (also known as equivalent resistance) is the sum of the individual resistances connected in the path. The formula used is Req = R1 + R2 + R3 + R4 + R5.
• For the given problem, the individual resistances are 0.1 Ohm, 0.4 Ohm, 0.5 Ohm, 0.6 Ohm, and 12.4 Ohm. Adding these values: 0.1 + 0.4 + 0.5 + 0.6 + 12.4 equals a total resistance of 14.0 Ohm.
• According to Ohm's Law, the relationship between voltage (V), current (I), and resistance (R) is given by the formula V = I x R. To find the current, we rearrange the formula to I = V / R.
• Given a battery of 7 V and a total calculated resistance of 14 Ohm, the total current in the circuit is 7 / 14, which equals 0.5 A.
• A fundamental characteristic of series connections is that the current remains constant through every component in the circuit. Therefore, the current flowing through the 12.4 Ohm resistor is exactly the same as the total circuit current, which is 0.5 A. Hence, the correct answer is 0.5 A.

Additional Information

• Ohm's Law:
• Discovered by German physicist Georg Simon Ohm in 1827, this law states that the current through a conductor between two points is directly proportional to the voltage across the two points.
• It is the most basic law of electricity and is used to calculate the power, efficiency, and safety of electrical circuits.
• Series vs. Parallel Circuits:
• In a series circuit, components are connected end-to-end, creating a single path for current. If one component fails, the entire circuit is broken.
• In a parallel circuit, components are connected across the same two nodes, providing multiple paths for current. In this setup, the voltage remains constant across all branches, but the current divides.
• Units of Measurement:
• Voltage is measured in Volts (V), representing the electrical potential difference.
• Current is measured in Amperes (A), representing the rate of flow of electric charge.
• Resistance is measured in Ohms (Ω), representing the opposition to the flow of electric current.

Important Points

• Voltage Drop in Series: While the current is the same in a series circuit, the voltage is divided among the resistors. The voltage drop across the 12.4 Ohm resistor can be calculated as V = I x R = 0.5 A x 12.4 Ohm = 6.2 V.
• Total Resistance Property: In a series combination, the equivalent resistance is always greater than the largest individual resistance in the circuit (14 Ohm > 12.4 Ohm).
• Application: Series circuits are often used in simple devices like flashlights or decorative string lights, though modern string lights often use parallel sections to prevent the entire string from going out if one bulb fails.

The correct answer is There is a fixed balance and characteristics.

Key Points

• A compound is a substance formed when two or more chemical elements are chemically bonded together. The most defining characteristic of a compound is that the elements are always present in fixed ratios by mass.
• This fixed ratio is governed by the Law of Definite Proportions (also known as Proust's Law), which states that a chemical compound always contains its component elements in a fixed ratio and does not depend on its source or method of preparation.
• Unlike mixtures, the formation of a compound involves a chemical change, resulting in a product with entirely different properties from its constituent elements. For example, Sodium (a reactive metal) and Chlorine (a toxic gas) combine to form Sodium Chloride (common table salt).
• Compounds possess a fixed melting and boiling point, which serves as a benchmark for their purity. Because they are held together by chemical bonds (ionic or covalent), they cannot be separated by simple physical methods like filtration or evaporation.

 Additional Information

• Mixtures:
• Components are physically combined and can be present in any proportion.
• They retain the original properties of their constituents (e.g., in a mixture of iron and sulfur, iron remains magnetic).
• They can be homogeneous (uniform composition like Alloys) or heterogeneous (non-uniform like Soil).
• Separation is possible through physical processes such as distillation, sublimation, or magnetism.
• Separation of Compounds:
• Compounds can only be broken down into their individual elements through chemical reactions or electrochemical reactions like electrolysis.
• States of Matter:
• The option Always solid is incorrect because compounds can exist in all three states of matter: solid (Calcium Carbonate), liquid (Water), and gas (Carbon Dioxide).

The correct answer is The displacement of the object during that time interval.

Key Points

• The area under a velocity–time (v-t) graph represents the displacement of an object over a specific time interval. Since velocity is defined as the rate of change of displacement ($v = dx/dt$), the integral of velocity over time ($\int v dt$) yields the total change in position. Hence, option 1 is correct.
• In a graph where the velocity is plotted on the y-axis and time is plotted on the x-axis, the product of the two dimensions (m/s × s) results in meters (m), which is the SI unit for displacement.
• For an object moving with uniform velocity, the graph is a straight horizontal line. The area formed under this line up to the time axis is a rectangle, where the Area = Velocity × Time.
• If the object moves with uniform acceleration, the graph is a sloping straight line. The area under this curve (often a triangle or a trapezium) still represents the net displacement.
• It is important to note that displacement is a vector quantity. Therefore, areas above the time axis (positive velocity) represent positive displacement, while areas below the axis (negative velocity) represent negative displacement.

Additional Information

• Acceleration:
• The slope (gradient) of a velocity-time graph represents the acceleration of the object.
• A steeper slope indicates a higher magnitude of acceleration, while a zero slope (horizontal line) indicates constant velocity and zero acceleration. Hence, option 2 is incorrect as the area does not represent acceleration.
• Speed and Distance:
• The area under a speed-time graph represents the total distance traveled, which is a scalar quantity.
• Unlike displacement, distance is always positive. To find distance from a velocity-time graph, one must take the absolute value of the areas (treating negative areas as positive) and sum them up. Hence, option 3 is incorrect.
• Position-Time Graph:
• In a position-time (s-t) graph, the slope at any point represents the instantaneous velocity of the object.
• A curved line on a position-time graph indicates that the velocity is changing, meaning the object is accelerating.
• Time Taken:
• Time is represented by the independent variable on the x-axis. It is not represented by the area but rather by the interval along the horizontal axis. Hence, option 4 is incorrect.

Important Points

• Equations of Motion: The geometric derivation of the three equations of motion ($v = u + at$, $s = ut + \frac{1}{2}at^2$, and $v^2 = u^2 + 2as$) is often performed using the area and slope properties of the velocity-time graph.
• Non-Uniform Motion: For non-uniformly accelerated motion, the velocity-time graph is a curve. In such cases, calculus (integration) is required to calculate the area under the curve to find the displacement.
• Directionality: If an object returns to its starting point, the total area above the x-axis will equal the total area below the x-axis, resulting in a net displacement of zero, even though the distance traveled is non-zero.

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

 

Key Points

• Infrasound (or infrasonic sound) refers to sound waves with a frequency lower than the lower limit of human audibility, which is generally 20 Hz.
• Large mammals such as Rhinoceroses, Whales, and Elephants utilize infrasound to communicate over vast distances. Hence, the correct answer is Infrasound.
• Rhinoceroses are known to communicate using infrasound of frequency as low as 5 Hz.
• Elephants use low-frequency sounds to coordinate herd movements and find mates over distances of several kilometers, as these waves are less likely to be scattered or absorbed by dense vegetation.
• Whales, specifically Baleen whales, use infrasonic calls to communicate across hundreds or even thousands of kilometers in the ocean, taking advantage of the efficient propagation of low-frequency waves in water.
• Natural events like earthquakes, volcanic eruptions, and ocean waves also generate infrasonic waves, which some animals can perceive as a warning signal before the event occurs.

Additional Information

• Audible Sound:
• This is the frequency range detectable by the human ear, typically spanning from 20 Hz to 20,000 Hz (or 20 kHz).
• Sounds below 20 Hz are felt as vibrations rather than heard, while sounds above 20 kHz are completely silent to humans.
• Ultrasound:
• Sound waves with frequencies higher than 20,000 Hz are classified as ultrasound.
• Animals like Bats, Dolphins, and Porpoises produce ultrasound for echolocation to navigate and hunt prey in the dark or underwater.
• In medical science, ultrasound is used in Sonography (ultrasound imaging) to view internal organs and during pregnancy to monitor fetal growth.
• Supersonic Sound:
• The term Supersonic does not refer to a frequency range but to a speed. It describes an object moving faster than the speed of sound in air, which is approximately 343 meters per second (at 20 degrees Celsius).
• When an aircraft travels at supersonic speeds, it creates shock waves known as a sonic boom.

Important Points

• Sound Propagation: Sound is a mechanical longitudinal wave that requires a material medium (solid, liquid, or gas) to travel. It cannot travel through a vacuum.
• Frequency and Pitch: The frequency of a sound wave determines its pitch. High-frequency waves (ultrasound) have a high pitch, while low-frequency waves (infrasound) have a very low pitch.
• Infrasound Detection: Specialized instruments called microbarometers are used to detect and monitor infrasound for scientific research and nuclear explosion monitoring under the Comprehensive Nuclear-Test-Ban Treaty (CTBT).
• Age and Hearing: As humans age, the upper limit of the audible range often decreases, making it harder to hear high-frequency sounds near 15-20 kHz.

The correct answer is electrolytic reduction.

Key Points

• Metals high in the activity series, such as Potassium (K), Sodium (Na), Calcium (Ca), Magnesium (Mg), and Aluminium (Al), are very reactive and cannot be obtained from their compounds by heating with carbon.
• This is because these metals have a higher affinity for oxygen than carbon does, making carbon an ineffective reducing agent for their oxides.
• Electrolytic reduction is the process used for their extraction, where electric current is passed through the molten state of their chlorides or oxides. Hence, the correct answer is electrolytic reduction.
• During this process, the metal is deposited at the cathode (the negatively charged electrode), and the non-metal, like chlorine, is liberated at the anode (the positively charged electrode). Hence, statement 3 is correct.
• Only heating is a method used for metals very low in the activity series, such as Mercury from its ore Cinnabar (HgS). Hence, statement 1 is incorrect.
• Calcination is a process used for carbonate ores of metals with medium reactivity, where the ore is heated strongly in limited supply of air to convert it into an oxide. Hence, statement 2 is incorrect.
• Heating with carbon (reduction) is used for metals in the middle of the activity series, such as Iron (Fe), Zinc (Zn), and Lead (Pb). Hence, statement 4 is incorrect.

Additional Information

• Reactivity Series: It is a vertical arrangement of metals in decreasing order of their chemical reactivity. Metals at the top are the most electropositive and lose electrons easily.
• Extraction of Sodium: It is obtained by the electrolysis of molten sodium chloride (NaCl). At the cathode, Na+ ions gain electrons to form Sodium metal, while at the anode, Cl- ions lose electrons to form Chlorine gas.
• Hall-Heroult Process: This is a specific industrial electrolytic process used for the extraction of Aluminium from Alumina (Al2O3). The alumina is dissolved in molten cryolite (Na3AlF6) to lower the melting point and increase conductivity.
• Roasting: Unlike calcination, roasting is the process of heating sulphide ores strongly in the presence of excess air to convert them into metal oxides.
• Native State: Metals at the bottom of the activity series, like Gold (Au), Silver (Ag), and Platinum (Pt), are often found in their native (free) state in nature because they are the least reactive.

Important Points

• Affinity for Oxygen: The choice of extraction method is primarily determined by the thermodynamic stability of the metal oxide. Metals high in the series form very stable oxides that require massive energy (electricity) to break.
• Reducing Agents: While carbon (coke) is a cheap and common reducing agent for Iron and Zinc, highly reactive metals like Aluminium are themselves sometimes used as reducing agents in thermite reactions to extract metals like Manganese (Mn) or Chromium (Cr) from their oxides.
• Electrolyte Purity: For electrolytic reduction, the salt must be in a molten (fused) state or in a solution to allow the ions to move freely and conduct electricity.

The correct answer is Gain of electrons.

Key Points

• In chemistry, a reduction reaction is defined as a process where a chemical species (atom, ion, or molecule) gains one or more electrons.
• When an atom or ion gains electrons, its oxidation state or oxidation number decreases (becomes more negative or less positive). Hence, statement 4 is correct.
• The process of losing electrons is known as oxidation, which is the exact opposite of reduction. Hence, statement 2 is incorrect.
• Changes involving neutrons or protons are considered nuclear reactions rather than chemical reactions. Loss of neutrons usually occurs during radioactive decay or nuclear fission. Hence, statement 1 is incorrect.
• Gain of protons would change the atomic number of the element, essentially turning it into a different chemical element altogether, which does not happen in standard chemical reactions. Hence, statement 3 is incorrect.
• A popular mnemonic to remember this is OIL RIG, which stands for Oxidation Is Loss (of electrons) and Reduction Is Gain (of electrons).

Additional Information

• Redox Reactions:
• Reduction and oxidation almost always occur simultaneously in a chemical process known as a Redox reaction.
• In these reactions, electrons are transferred from one substance to another. The substance that loses electrons is oxidized, and the substance that gains electrons is reduced.
• Oxidizing and Reducing Agents:
• A reducing agent (or reductant) is a substance that donates electrons to another substance, thereby undergoing oxidation itself.
• An oxidizing agent (or oxidant) is a substance that accepts electrons from another substance, thereby undergoing reduction itself.
• Classical Definitions:
• While the electronic definition is the most accurate, reduction was classically defined as the loss of oxygen or the gain of hydrogen.
• Conversely, oxidation was defined as the gain of oxygen or the loss of hydrogen.
• Practical Examples:
• In the extraction of iron from its ore (Hematite - Fe₂O₃), carbon monoxide acts as a reducing agent to remove oxygen from the iron oxide, leaving behind metallic iron.
• In electroplating, metal ions in a solution gain electrons at the cathode to form a solid metal coating on an object.

Important Points

• Electrochemical Cells: In a battery or galvanic cell, reduction always occurs at the cathode, while oxidation occurs at the anode.
• Biological Significance: Reduction reactions are vital in photosynthesis and cellular respiration. For instance, in photosynthesis, carbon dioxide is reduced to form glucose.
• Oxidation Number Rules:
• The oxidation number of a free element is always zero.
• For a monatomic ion, the oxidation number is equal to its charge.
• Oxygen usually has an oxidation number of -2, except in peroxides where it is -1.
• Hydrogen usually has an oxidation number of +1 when bonded to non-metals and -1 when bonded to metals.

The correct answer is Prostate gland and seminal vesicles.

Key Points

• The male reproductive system consists of various organs and accessory glands that work together to produce, maintain, and transport sperm. Sperm motility refers to the ability of sperm to move efficiently.
• Seminal Vesicles: These are a pair of pouch-like glands located at the base of the urinary bladder. They secrete a thick, alkaline fluid that makes up approximately 60% to 70% of the total volume of semen.
• The fluid from the seminal vesicles contains fructose, which provides the essential energy source (ATP) for sperm to swim. Without this energy, sperm remain immobile.
• Prostate Gland: This is a single, doughnut-shaped gland located just below the urinary bladder. It secretes a thin, milky, slightly alkaline fluid that contains citrate, zinc, and various enzymes.
• The secretions of the prostate gland play a crucial role in the activation of sperm and significantly enhance their motility and survival by neutralizing the acidic environment of the male urethra and the female reproductive tract.
• If a patient exhibits low sperm motility (a condition often termed asthenozoospermia), it indicates that the sperm lack the necessary biochemical environment or energy to move, pointing directly to the malfunction of the prostate gland and seminal vesicles. Hence, the correct answer is Prostate gland and seminal vesicles.

Additional Information

• Testis and Epididymis:
• The testes are the primary reproductive organs responsible for the production of sperm (spermatogenesis) and the hormone testosterone.
• The epididymis is a coiled tube where sperm are stored and undergo functional maturation, gaining the potential for motility, although they only become fully active upon mixing with accessory gland secretions.
• Bulbourethral Glands (Cowper's Glands):
• These are small glands located below the prostate. They secrete an alkaline mucus that lubricates the urethra and neutralizes any residual acidic urine before ejaculation.
• Analysis of Incorrect Options:
• Pancreas and Adrenal Glands: The pancreas regulates blood glucose levels via insulin and glucagon, while adrenal glands produce adrenaline and cortisol for stress response. Neither is directly responsible for sperm motility.
• Thyroxine and Pineal Glands: Thyroxine (produced by the thyroid) regulates basal metabolic rate, and the pineal gland produces melatonin to regulate the circadian rhythm (sleep-wake cycle).
• Bladder: The urinary bladder is part of the excretory system and serves only as a storage vessel for urine; it has no secretory role in sperm health.
• Semen Composition:
• Semen is a mixture of sperm and seminal plasma. The seminal plasma is formed by the combined secretions of the seminal vesicles, prostate, and bulbourethral glands, providing a transport medium, nutrients, and protection to the sperm cells.

Important Points

• Fertilization Requirement: For successful fertilization, sperm must travel from the vagina through the cervix and uterus to the fallopian tubes. This requires high motility, which is dependent on the chemical composition of the fluid provided by the accessory glands.
• Hormonal Regulation: The functioning of the prostate and seminal vesicles is maintained by the male sex hormone, testosterone, which is secreted by the Leydig cells in the testes under the influence of Luteinizing Hormone (LH).
• Clinical Relevance: Disorders such as prostatitis (inflammation of the prostate) or seminal vesiculitis can lead to changes in the pH or nutrient content of semen, resulting in infertility due to reduced sperm movement.

The correct answer is potential energy.

Key Points

• Potential Energy is defined as the energy possessed by an object by virtue of its position relative to others, stresses within itself, its electric charge, or other factors.
• It is "stored" energy that has the capacity to do work when the object's position or configuration changes. For example, water stored in a dam possesses potential energy due to its height.
• There are two primary forms of potential energy: Gravitational Potential Energy and Elastic Potential Energy.
• Gravitational Potential Energy depends on the mass of the object, the acceleration due to gravity (approx 9.8 m/s² on Earth), and the height of the object from a reference point. The formula is expressed as P.E. = mgh.
• Elastic Potential Energy is stored in objects that can be compressed or stretched, such as a rubber band, a spring, or a drawn bow. This energy is a result of the change in the configuration or shape of the object.

Additional Information

• Kinetic Energy:
• This is the energy an object possesses due to its motion. Unlike potential energy, which is stored, kinetic energy is active.
• The amount of kinetic energy depends on the mass of the object and the square of its velocity. The standard formula is K.E. = 1/2 mv².
• Examples include a rolling ball, a flying airplane, or flowing water. Hence, option 2 is incorrect as it relates to motion, not position.
• Thermal Energy:
• This refers to the energy contained within a system that is responsible for its temperature. It is the internal energy of an object due to the kinetic energy of its atoms and molecules.
• Heat is the transfer of thermal energy between systems. For instance, boiling water contains significant thermal energy. Hence, option 1 is incorrect.
• Sound Energy:
• Sound energy is a form of mechanical energy that moves through substances (like air or water) in longitudinal waves.
• It is produced when an object vibrates, causing the surrounding medium to vibrate as well. Examples include music or a human voice. Hence, option 3 is incorrect.

Important Points

• Law of Conservation of Energy: This fundamental law of physics states that energy cannot be created or destroyed; it can only be transformed from one form to another. In a closed system, the sum of Potential Energy and Kinetic Energy (Mechanical Energy) remains constant in the absence of friction.
• SI Unit: The Standard International (SI) unit for measuring all forms of energy, including potential energy, is the Joule (J).
• Work-Energy Theorem: Work is done when a force acts on an object to cause a displacement. When work is done on an object against gravity, it is stored as Gravitational Potential Energy.
• Chemical Potential Energy: This is another form of stored energy found within the chemical bonds of substances, such as food, batteries, or gasoline, which is released during a chemical reaction.

The correct answer is It increases the chances of survival by enabling adaptation to environmental changes.

Key Points

• Sexual reproduction in plants involves the fusion of male and female gametes (pollen and ovules), which results in offspring that possess a unique combination of genetic material from two different parents.
• Variation is the cornerstone of evolution and natural selection. It ensures that a population contains individuals with diverse traits, some of which may be advantageous during sudden environmental shifts such as global warming, drought, or outbreaks of disease.
• By having a varied genetic pool, the species as a whole has a higher probability of survival; if all plants were genetically identical, a single unfavorable change could lead to the extinction of the entire population. Hence, statement 3 is correct.
• Contrary to limiting adaptability, variation actually expands the range of conditions in which a species can thrive. Hence, statement 1 is incorrect.
• Variation specifically prevents uniformity. Uniformity is a characteristic of asexual reproduction (cloning), where offspring are genetically identical to the parent. Hence, statement 2 is incorrect.
• Sexual reproduction increases genetic diversity rather than decreasing it, providing the raw material upon which evolutionary forces act over thousands of years. Hence, statement 4 is incorrect.

Additional Information

• Mechanisms of Variation:
• Meiosis: This is a type of cell division that reduces the chromosome number by half. During this process, crossing over occurs, where homologous chromosomes exchange genetic segments, creating new gene combinations.
• Random Fertilization: The random fusion of any one sperm (pollen) with any one egg (ovule) further increases the mathematical possibilities of unique genetic outcomes.
• Significance in Agriculture:
• In crop science, genetic variation is utilized by breeders to develop hybrid varieties that are resistant to pests or high salinity.
• The Green Revolution in the 1960s utilized genetic variation to create high-yielding varieties of wheat and rice, significantly increasing food security.
• Natural Selection:
• Proposed by Charles Darwin in his 1859 book "On the Origin of Species," natural selection suggests that individuals with favorable phenotypes are more likely to survive to reproductive age.
• Variation provides the different "options" from which the environment "selects" the most fit individuals.
• Biodiversity and Ecosystem Stability:
• High genetic variation within a plant species contributes to the overall biodiversity of an ecosystem. This makes the ecosystem more resilient to disturbances and helps maintain ecological balance.

Important Points

• Asexual vs. Sexual Reproduction: While asexual reproduction is faster and requires less energy, it produces clones. This lack of variation makes such plants highly vulnerable to rapid environmental changes or specific pathogens.
• DNA Copying: During gamete formation, DNA replication is not 100% accurate. These small "errors" or mutations are a secondary source of variation that can introduce entirely new traits into a population over generations.
• Niche Specialization: Variation allows different individuals of the same species to occupy slightly different ecological niches, reducing direct competition for resources like sunlight, water, and nutrients.

The correct answer is Sweet-smelling.

Key Points

• Esters are functional organic compounds derived from carboxylic acids and alcohols, characterized by a carbon atom double-bonded to an oxygen and single-bonded to an alkoxy group (R-COOR').
• One of the most defining physical properties of low-molecular-weight esters is their pleasant, fruity, or sweet-smelling fragrance, which makes them highly distinguishable from the pungent odors of their parent acids.
• They are synthesized via a reaction called esterification, typically involving the heating of a carboxylic acid with an alcohol in the presence of a strong acid catalyst like concentrated sulfuric acid (H₂SO₄).
• In nature, esters are ubiquitous and are responsible for the natural aroma and  of various fruits and flowers, such as isoamyl acetate (banana smell) and ethyl butyrate (pineapple smell).

 Additional Information

• Industrial and Commercial Uses:
• Because of their volatile nature and pleasant scents, they are extensively used as synthetic flavoring agents in the food industry and as primary components in perfumes and cosmetic products.
• Esters like ethyl acetate are utilized as powerful organic solvents in nail polish removers, paints, and adhesives due to their ability to dissolve various organic substances.
• Chemical Properties:
• Under basic conditions, esters undergo a process called saponification (alkaline hydrolysis), which results in the formation of soap (salts of fatty acids) and an alcohol.
• Unlike carboxylic acids, esters are neutral compounds and do not react with sodium bicarbonate to release carbon dioxide, nor do they turn blue litmus paper red.

The correct answer is Only (iii).

Key Points

• A concave mirror does not always form a real and inverted image. When an object is placed very close to the mirror, specifically between the Principal Focus (F) and the Pole (P), it forms a virtual, erect, and magnified image behind the mirror. Hence, statement (i) is incorrect.
• When an object is placed between the Center of Curvature (C) and the Principal Focus (F), the concave mirror forms an image beyond C. This image is real, inverted, and magnified (enlarged), not diminished. Hence, statement (ii) is incorrect.
• If an object is at infinity, the incident rays are parallel to the principal axis. These rays, after reflection, converge at the Principal Focus (F). The resulting image is real, inverted, and highly diminished (point-sized). Hence, statement (iii) is correct.
• The nature and size of the image formed by a concave mirror depend entirely on the distance of the object from the pole of the mirror.
• Hence, the correct answer is Only (iii).

Additional Information

• Definition of Concave Mirror: A spherical mirror whose reflecting surface is curved inwards (faces towards the center of the sphere) is called a concave mirror. It is also known as a converging mirror because it converges a parallel beam of light after reflection.
• Image Formation Summary:
• Object at C: The image is formed at C, and it is real, inverted, and same size as the object.
• Object beyond C: The image is formed between F and C, and it is real, inverted, and diminished.
• Object at F: The image is formed at infinity, and it is real, inverted, and highly magnified.
• Common Uses:
• Shaving mirrors: Used to see a magnified and erect image of the face when held close.
• Dentists: They use concave mirrors to see large images of the teeth of patients.
• Reflectors: Used in torches, search-lights, and vehicle headlights to get powerful parallel beams of light.
• Solar Furnaces: Large concave mirrors are used to concentrate sunlight to produce heat in solar energy applications.

Important Points

• Real Image vs. Virtual Image: A real image is formed when light rays actually intersect at a point and can be obtained on a screen. A virtual image is formed when light rays appear to diverge from a point and cannot be caught on a screen.
• Sign Convention: According to the New Cartesian Sign Convention, the focal length of a concave mirror is always taken as negative, while that of a convex mirror is positive.
• Principal Focus: For a concave mirror, the Principal Focus is a point on the principal axis where all rays traveling parallel to the axis actually meet after reflection.

The correct answer is To provide a low-resistance path to ensure leakage current keeps the appliance potential to that of the earth, preventing severe shocks.

Key Points

• The earth wire, which typically has green insulation, is a vital safety measure in domestic electric circuits, especially for appliances with a metallic body like electric irons, toasters, and refrigerators.
• Its primary function is to provide a low-resistance conducting path for the electric current. If the insulation of the live wire inside an appliance fails and touches the metallic casing, the earth wire ensures that the current flows directly into the ground.
• By directing the leakage current to the earth, the wire maintains the electric potential of the appliance's metallic body at zero volts (the same as the earth), ensuring that a user touching the appliance does not receive a severe electric shock. Hence, statement 4 is correct.
• The low-resistance path is critical because electricity always follows the path of least resistance. Since the human body has higher resistance compared to the copper earth wire, the current prefers the wire over the person. Hence, statement 1 is incorrect as it suggests a high-resistance path.
• The earth wire does not regulate or stabilize voltage; voltage stabilization is the function of devices like Voltage Stabilizers or Transformers. Hence, statement 2 is incorrect.
• The earth wire is not designed to replace the neutral wire. While both are connected to the ground at some point in the power system, the neutral wire completes the circuit for normal operation, whereas the earth wire is strictly for safety and protection. Hence, statement 3 is incorrect.

Additional Information

• Standard Color Coding: In a typical domestic circuit in India, three types of wires are used. The Live wire (positive) usually has red or brown insulation, the Neutral wire (negative) has black or blue insulation, and the Earth wire has green or yellow insulation.
• Domestic Power Supply: In India, the potential difference (voltage) between the live and neutral wires is 220 V, and the frequency of the alternating current (AC) is 50 Hz.
• Circuit Types: Domestic circuits are often divided into two types: a 5 A current rating circuit for low-power bulbs and fans, and a 15 A current rating circuit for high-power appliances like air conditioners and geysers.
• Safety Devices: To prevent damage from short-circuiting or overloading, devices such as Electric Fuses and Miniature Circuit Breakers (MCB) are installed. A fuse works on the heating effect of current and breaks the circuit if the current exceeds a safe limit.

Important Points

• Earthing Mechanism: The earth wire is connected to a metal plate buried deep in the earth near the house. This setup ensures that any excess charge or leakage is instantly discharged into the ground.
• Short Circuiting: This occurs when the live wire and the neutral wire come into direct contact due to damaged insulation or a fault in the appliance, causing a sudden, massive surge in current.
• Overloading: This happens when too many high-power appliances are connected to a single socket, drawing more current than the wires can safely carry, leading to potential fire hazards.
• Three-Pin Plugs: The top pin of a three-pin plug is the earth pin. It is intentionally made longer and thicker than the live and neutral pins to ensure the appliance is grounded first before the power is connected.

The correct answer is 60°.

Key Points

• The Laws of Reflection state that when a ray of light strikes a reflective surface, the angle of incidence (i) is always equal to the angle of reflection (r).
• The angle of incidence is defined as the angle between the incident ray and the normal (a perpendicular line to the surface). In the given problem, this angle is 30°.
• According to the law (i = r), since the angle of incidence is 30°, the angle of reflection must also be 30°.
• The question asks for the total angle between the incident ray and the reflected ray. This is calculated by adding the angle of incidence and the angle of reflection (i + r).
• Calculation: 30° + 30° = 60°. Therefore, the angle between the two rays is 60°. Hence, the correct answer is 60°.

Additional Information

• Plane Mirror Characteristics:
• A plane mirror is a flat reflective surface that forms a virtual, erect, and laterally inverted image.
• The size of the image formed is exactly the same as the size of the object.
• The distance of the image behind the mirror is equal to the distance of the object in front of the mirror.
• Types of Reflection:
• Regular Reflection: Also known as specular reflection, it occurs on smooth surfaces like mirrors where all reflected rays are parallel.
• Diffuse Reflection: Occurs on rough surfaces where light is reflected in many different directions, allowing us to see non-shiny objects.
• Normal Line: In optics, the normal is an imaginary line drawn perpendicular (90°) to the surface of the mirror at the point of incidence. All angles in the laws of reflection are measured relative to this line.

Important Points

• Glancing Angle: This is the angle between the incident ray and the mirror surface itself. If the angle with the normal is 30°, the glancing angle is 90° - 30° = 60°.
• Angle of Deviation (δ): This is the angle by which the light ray is turned from its original path. For a plane mirror, the deviation is calculated as 180° - (i + r). In this case, 180° - 60° = 120°.
• First Law of Reflection: It specifies that the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.
• Universal Application: These rules of reflection are not limited to plane mirrors; they apply to all curved surfaces (concave and convex) and spherical mirrors as well.

The correct answer is 122.5 m.

Key Points

• The motion of the stone is a case of vertical motion under gravity. When an object is thrown upwards, it experiences a constant downward acceleration due to gravity (g), which is approximately 9.8 m/s².
• The initial velocity (u) is given as 49 m/s. As the stone moves upward, its velocity decreases until it reaches the maximum height, where the final velocity (v) becomes 0 m/s for a brief moment.
• To find the maximum height, we use the third equation of motion: v² = u² + 2as. In this context, s represents the maximum height (H) and a represents -g (since gravity acts in the opposite direction of the initial motion).
• Substituting the known values into the equation: 0² = (49)² + 2(-9.8)H. This simplifies to 0 = 2401 - 19.6H.
• Rearranging the formula to solve for H: 19.6H = 2401, which leads to H = 2401 / 19.6 = 122.5 m. Hence, the correct option is 122.5 m.

Additional Information

• Equations of Motion: There are three primary kinematic equations used for objects under uniform acceleration:
• First Equation: v = u + at, used to find velocity or time.
• Second Equation: s = ut + (1/2)at², used to find displacement when time is known.
• Third Equation: v² = u² + 2as, used to find displacement or velocity when time is unknown.
• Time to Reach Maximum Height: The time taken (t) can be calculated using the first equation v = u + at. Setting v = 0 and a = -9.8, we get 0 = 49 - 9.8t, resulting in t = 5 seconds.
• Total Time of Flight: In the absence of air resistance, the time taken to go up is equal to the time taken to come back down. Therefore, the total time of flight (T) is 2 × 5 = 10 seconds.
• Velocity on Return: When the stone returns to the point of projection, its final velocity will be equal in magnitude to the initial velocity but in the opposite direction (-49 m/s), assuming no energy is lost.

Important Points

• Sign Convention: In physics problems involving vertical motion, it is crucial to maintain a consistent sign convention. Usually, upward direction is taken as positive (+) and downward direction (gravity) is taken as negative (-).
• Acceleration due to Gravity (g): This value varies slightly across the Earth's surface (higher at poles, lower at the equator), but for most calculations, 9.8 m/s² or 10 m/s² is used as a standard numerical value.
• Mass Independence: According to Galileo's principle, the height reached or the time taken by an object under gravity is independent of its mass, provided air resistance is neglected.
• Momentary Rest: The point of maximum height is characterized by a velocity of zero, but the acceleration at that point is still 9.8 m/s² downward.

The correct answer is Cuboidal epithelium.

Key Points

• Cuboidal epithelium is a type of epithelial tissue characterized by cells that are roughly cube-shaped, having a similar height and width.
• It is found in the lining of kidney tubules (specifically the nephrons) and the ducts of salivary glands.
• The primary functions of cuboidal epithelium are secretion and absorption.
• In the kidney, it plays a vital role in the reabsorption of water and nutrients from the filtrate back into the bloodstream.
• In glandular ducts, it facilitates the transport and secretion of substances like saliva or enzymes.
• Additionally, this tissue provides mechanical support to the organs and ducts it lines.

Additional Information

• Ciliated Epithelium:
• These cells possess cilia, which are hair-like projections on their outer surface.
• They are found in the respiratory tract (to move mucus) and fallopian tubes (to move the egg).
• Their function is to move particles or mucus in a specific direction over the epithelium.
• Columnar Epithelium:
• Consists of tall and slender cells resembling pillars with nuclei located at the base.
• Found in the inner lining of the stomach and intestine.
• It is primarily involved in absorption and secretion of digestive juices.
• Stratified (Compound) Epithelium:
• Composed of multiple layers of cells rather than a single layer.
• The skin is a classic example of stratified squamous epithelium, where layers prevent wear and tear.
• Its main function is to provide protection against chemical and mechanical stresses.
• Simple Squamous Epithelium:
• Extremely thin and flat cells forming a delicate lining.
• Located in the alveoli of lungs, blood vessels, and esophagus.
• Facilitates diffusion and filtration across the membrane.

Important Points

• Basement Membrane: All epithelial tissues are usually separated from the underlying tissue by an extracellular fibrous basement membrane.
• Microvilli: In the Proximal Convoluted Tubule (PCT) of the nephron in the kidney, cuboidal cells have microvilli to increase the surface area for reabsorption.
• Glandular Epithelium: Sometimes a portion of the epithelial tissue folds inward, and a multicellular gland is formed; this is known as glandular epithelium, often modified from cuboidal or columnar cells.
• Cell Junctions: Epithelial cells are held together by tight junctions, adhering junctions, and gap junctions to facilitate communication and structural integrity.

The correct answer is Electron Clouds.

Key Points

• The size of an atom is almost entirely determined by the volume of its Electron Clouds. Although electrons have negligible mass, they move rapidly in regions of space around the nucleus, creating a "cloud" that defines the atom's outer boundary.
• While the nucleus contains more than 99.9% of the atom's total mass, it occupies an extremely small fraction of the atom's volume. If an atom were expanded to the size of a football stadium, the nucleus would be the size of a small marble in the center, while the electron cloud would fill the rest of the stadium.
• The atomic radius is the physical distance from the center of the nucleus to the outermost stable electron orbital. This radius typically ranges from 30 to 300 picometers (pm) (1 picometer = 10^-12 meters).
• The distribution of electrons in different energy levels (shells) and subshells (s, p, d, f orbitals) dictates how much space the atom takes up. Atoms with more electron shells are generally larger in size.
• The interaction between the positively charged nucleus and the negatively charged electrons determines how tightly the cloud is pulled inward, thus influencing the overall atomic volume.

Additional Information

• Nucleus:
• The nucleus was discovered by Ernest Rutherford in 1911 through his famous Gold Foil Experiment.
• It is the dense, central core of the atom containing protons and neutrons. The diameter of a nucleus is about 10^-15 meters, which is 100,000 times smaller than the atom itself (10^-10 meters).
• Protons and Neutrons:
• Protons are positively charged particles discovered by Ernest Rutherford. The number of protons is known as the Atomic Number (Z) and determines the identity of the chemical element.
• Neutrons are neutral subatomic particles discovered by James Chadwick in 1932. Together with protons, they make up the Atomic Mass of the atom but do not contribute to its volume.
• Trends in Atomic Size:
• In the Periodic Table, the atomic size decreases across a period (from left to right) because the increasing nuclear charge pulls the electron cloud closer to the center.
• The atomic size increases down a group (from top to bottom) because new principal energy levels or shells are added, which are further from the nucleus.
• Measurement Units:
• Atomic sizes are often measured in Angstroms (Å) or Nanometers (nm). One Angstrom is equal to 10^-10 meters, which is roughly the size of a hydrogen atom.

The correct answer is collenchyma.

Key Points

• Collenchyma is a simple permanent tissue that provides mechanical support and flexibility to various plant parts such as stems, tendrils, and leaf stalks.
• It allows these parts to bend easily without breaking, which is vital for the growth and adaptation of climbing plants and young seedlings.
• The cells of collenchyma are living, elongated, and characterized by unevenly thickened cell walls, particularly at the corners.
• The thickening of the cell wall is due to the deposition of cellulose, hemicellulose, and pectin.
• It is generally located in the hypodermis of dicotyledonous stems and the petioles of leaves, but it is typically absent in monocot stems and roots.

Additional Information

• Parenchyma:
• This is the most fundamental and abundant simple permanent tissue in plants.
• It consists of relatively unspecialized, living cells with thin cell walls made of cellulose.
• The cells are usually isodiametric and have large intercellular spaces.
• Its primary functions include storage of food, photosynthesis (as chlorenchyma), and providing buoyancy to aquatic plants (as aerenchyma).
• Sclerenchyma:
• This tissue consists of dead cells at maturity that are long and narrow.
• The cell walls are highly thickened with lignin, a chemical substance that acts as cement and hardens them.
• Its main function is to provide strength and rigidity to plant parts, making them hard and stiff.
• Common examples include the husk of coconut, the hard shells of nuts, and the gritty texture in the pulp of pears.
• Xylem:
• Unlike the others mentioned, Xylem is a complex permanent tissue, meaning it is made of more than one type of cell.
• It is responsible for the conduction of water and minerals upward from the roots to the leaves.
• It consists of four elements: tracheids, vessels, xylem parenchyma, and xylem fibres.
• Except for xylem parenchyma, most xylem cells are dead at maturity.

Important Points

• Simple Permanent Tissue: These are tissues composed of only one type of cell that are similar in structure and function. The three main types are Parenchyma, Collenchyma, and Sclerenchyma.
• Permanent Tissues: These tissues are formed from meristematic tissues once the cells have lost their ability to divide and have attained a permanent shape, size, and function. This process is called differentiation.
• Lignin vs Pectin: While lignin is the characteristic hardening agent in sclerenchyma (leading to rigidity), pectin is the primary thickening agent at the corners of collenchyma (allowing for flexibility).

The correct answer is 18.

Key Points

• The distribution of electrons in different orbits or shells of an atom was suggested by Bohr and Bury in the year 1921. According to this scheme, the maximum number of electrons that can be accommodated in a shell is given by the formula 2n², where 'n' is the orbit number or energy level index.
• The M-shell is the third shell from the nucleus of an atom, which means its principal quantum number or orbit number n is equal to 3.
• By applying the 2n² formula for the M-shell, the calculation is 2 x (3)², which equals 2 x 9 = 18. Therefore, the M-shell can accommodate a maximum of 18 electrons.
• This rule dictates the capacity of each discrete energy level. For the first three shells, the capacities are K-shell (2), L-shell (8), and M-shell (18).
• While the M-shell has a physical capacity for 18 electrons, the Bohr-Bury scheme also specifies that the outermost shell of any atom cannot accommodate more than 8 electrons, even if it has the capacity for more.

Additional Information

• K-Shell (n=1):
• The K-shell is the first shell and is closest to the nucleus.
• Using the formula 2(1)², its maximum capacity is 2 electrons.
• Elements like Hydrogen and Helium fill this shell first.
• L-Shell (n=2):
• The L-shell is the second shell.
• Following the formula 2(2)², it can hold a maximum of 8 electrons.
• Once the K-shell is full, electrons begin to occupy the L-shell.
• N-Shell (n=4):
• The N-shell is the fourth shell from the nucleus.
• According to the 2n² rule, where n=4, the calculation is 2 x (4)² = 2 x 16, resulting in a maximum of 32 electrons.
• Significance of the Bohr Model:
• Proposed by Neils Bohr in 1913, this model introduced the idea that electrons travel in defined circular orbits around the nucleus.
• The model explains that energy is absorbed or emitted only when an electron jumps from one orbit to another.
• The Bohr-Bury scheme refined this by providing the rules for electron configuration and stability.

Important Points

• Octet Rule: Regardless of the shell's maximum capacity (like 18 for M or 32 for N), the outermost shell (valence shell) is considered stable when it contains 8 electrons. This is known as the Octet Rule.
• Step-wise Filling: Electrons are not accommodated in a given shell unless the inner shells are completely filled. This means the shells are filled in a step-wise manner starting from the lowest energy level (K) to higher ones.
• Energy Levels: As the value of n increases, the energy of the shell also increases, and the shell is located farther from the nucleus.
• Valence Electrons: The electrons present in the outermost shell are called valence electrons, and they determine the chemical reactivity and valency of the element.

The correct answer is It involves directional growth in response to a stimulus.

Key Points

• Tropic movements, or Tropisms, are directional growth movements of a plant in response to an external stimulus such as light, gravity, or touch.
• Tendril movement is a specific type of tropism called Thigmotropism. It occurs when the plant part makes contact with a solid object, triggering a growth response that causes the tendril to coil around the support.
• The movement is directional because the growth is oriented toward the stimulus (the contact point). Hence, statement 3 is correct.
• Unlike Nastic movements, which are based on changes in turgor pressure and internal water levels (like the closing of Mimosa pudica leaves), tendril coiling is a result of permanent differential growth. Hence, statement 1 is incorrect.
• Tropic movements are not random; they are highly regulated biological responses to environmental cues intended to aid the plant's survival and positioning. Hence, statement 2 is incorrect.
• While plant growth occurs throughout the 24-hour cycle, it is not restricted to occurring always during night. The stimulus-driven growth of tendrils happens whenever the contact stimulus is present. Hence, statement 4 is incorrect.

Additional Information

• Mechanism of Thigmotropism: When a tendril touches a support, the plant hormone Auxin diffuses toward the side of the tendril that is not in contact with the support. This higher concentration of Auxin causes the outer side to grow much faster than the side touching the object, resulting in the characteristic coiling or curving.
• Types of Tropisms:
• Phototropism: Growth in response to light (e.g., shoots growing toward a window).
• Geotropism (Gravitropism): Growth in response to gravity (e.g., roots growing downward).
• Hydrotropism: Growth of roots toward water or moisture sources.
• Chemotropism: Growth in response to chemical substances, such as the growth of a pollen tube toward the ovule.
• Examples of Tendril-bearing plants: Climbing plants such as Pea (Pisum sativum), Bitter Gourd, and Grapevines use these structures to climb higher for better access to sunlight.

Important Points

• Growth vs. Turgor: It is essential to distinguish between tropic movements (which are growth-dependent and usually irreversible) and nastic movements (which are growth-independent and reversible).
• Auxin's Role: Auxin is the primary phytohormone responsible for mediating the growth rate in almost all tropic movements. In Phototropism, it moves to the shaded side, while in Thigmotropism, it moves to the non-contact side.
• Biological Significance: These movements allow plants to optimize their environment. For instance, tendrils allow weak-stemmed plants to reach heights they could not sustain on their own, ensuring they are not shaded out by larger vegetation.

The correct answer is It is a curved line..

Key Points

• A velocity-time graph (v-t graph) is a graphical representation used in kinematics to show how the velocity of an object changes with respect to time. The y-axis represents velocity, typically measured in meters per second (m/s), and the x-axis represents time, measured in seconds (s).
• The slope or gradient of a velocity-time graph represents the acceleration of the moving object. If the slope is constant, the acceleration is uniform; if the slope varies, the acceleration is non-uniform.
• Non-uniform acceleration occurs when the velocity of an object changes by unequal amounts in equal intervals of time. This implies that the rate of change of velocity is not a constant value throughout the motion.
• In the case of non-uniform acceleration, the slope of the graph changes at every point. Geometrically, a line with a continuously changing slope is represented as a curved line. Hence, statement 1 is correct.
• If the graph were a straight line (sloping upwards or downwards), it would represent uniform acceleration or uniform retardation, where the rate of change of velocity remains fixed. Hence, statement 2 is incorrect.
• A horizontal line on a v-t graph indicates that the velocity is constant over time, meaning the acceleration is exactly zero. Hence, statement 4 is incorrect.

Additional Information

• Types of Motion on v-t Graphs:
• Uniform Velocity: Represented by a straight horizontal line parallel to the time axis. The acceleration in this case is 0 m/s².
• Uniform Acceleration: Represented by a straight line with a positive slope. This means the object's velocity increases at a steady rate.
• Uniform Retardation (Deceleration): Represented by a straight line with a negative slope (sloping downwards). The velocity decreases at a constant rate.
• Non-uniform Acceleration: Represented by curves. A curve bending upwards indicates increasing acceleration, while a curve leveling off indicates decreasing acceleration.
• Calculation of Displacement: One of the most important features of a velocity-time graph is that the area under the curve (between the line and the time axis) represents the total displacement covered by the object. For a curved graph, this area is often calculated using integration in calculus.
• Instantaneous Acceleration: In a non-uniform acceleration scenario, the instantaneous acceleration at any specific time (t) is found by calculating the slope of the tangent drawn to the curved line at that specific point.
• Average Acceleration: For an object moving with non-uniform acceleration between two time intervals, the average acceleration is calculated by finding the change in velocity divided by the total time taken, which corresponds to the slope of the chord connecting the two points on the curve.

Important Points

• Physical Units: Always remember that the SI unit for velocity is m/s and for acceleration is m/s². On a graph, the product of the units of the two axes (m/s * s) yields the unit of displacement, which is meters (m).
• Non-Zero Intercept: A velocity-time graph does not always have to start at the origin (0,0). If it starts at a point on the y-axis, it indicates that the object had an initial velocity (u) at time t = 0.
• Direction of Motion: If the graph crosses below the x-axis into the negative y-region, it signifies that the object has changed direction and is moving in the opposite direction relative to the starting reference.

The correct answer is a - iv, b - i, c - ii, d - iii.

Key Points

• Aerobic respiration is a biological process in which food glucose is converted into energy in the presence of oxygen. This type of respiration is common in most higher organisms, including humans, animals, and plants. Hence, statement a matches with iv.
• Anaerobic respiration takes place in the absence of oxygen. In this process, the incomplete breakdown of food occurs, leading to the production of significantly less energy compared to aerobic respiration. Examples include fermentation in yeast and energy production in human muscles during heavy exercise. Hence, statement b matches with i.
• During aerobic respiration, the pyruvate molecule (produced from glucose) enters the Mitochondria. Here, in the presence of oxygen, it is completely broken down into carbon dioxide, water, and a large amount of ATP. Hence, statement c matches with ii.
• The first step of cellular respiration is glycolysis, which involves the breakdown of glucose (a 6-carbon molecule) into pyruvate (a 3-carbon molecule). This specific stage, as well as anaerobic pathways where oxygen is absent, occurs within the Cytoplasm of the cell. Hence, statement d matches with iii.
• Therefore, the correct matching sequence is a - iv, b - i, c - ii, d - iii.

Additional Information

• Site of Respiration: While glycolysis happens in the cytoplasm, the Krebs cycle and electron transport chain occur in the mitochondria. The mitochondria are often called the powerhouse of the cell because they generate the bulk of ATP.
• Energy Yield: Aerobic respiration is highly efficient, typically yielding approximately 36 to 38 ATP molecules per glucose molecule. In contrast, anaerobic respiration produces only 2 ATP molecules per glucose molecule.
• By-products: In aerobic conditions, the by-products are CO2 and water. In anaerobic conditions in humans, lactic acid is produced, which can cause muscle cramps. In yeast, the anaerobic process produces ethanol and CO2, a process widely used in the brewing and baking industries.
• Role of Pyruvate: Pyruvate acts as a crucial metabolic intermediate. Depending on the availability of oxygen, it either enters the mitochondria for further oxidation or undergoes fermentation in the cytoplasm.

Important Points

• ATP (Adenosine Triphosphate): This is known as the energy currency of the cell. The energy released during respiration is used to synthesize ATP from ADP and inorganic phosphate.
• Muscle Fatigue: During vigorous exercise, when the supply of oxygen to our muscle cells is insufficient, anaerobic respiration takes place, leading to the accumulation of lactic acid. This accumulation is the primary cause of muscle cramps and fatigue.
• Cellular Respiration stages: It involves four main stages: Glycolysis (Cytoplasm), Pyruvate Oxidation (Mitochondria), Citric Acid Cycle/Krebs Cycle (Mitochondria), and Oxidative Phosphorylation (Mitochondria).
• Universal Process: Glycolysis is considered the most ancient metabolic pathway as it occurs in almost all living organisms, regardless of whether they breathe oxygen or not, and it always takes place in the cytosol/cytoplasm.

The correct answer is Involves only one parent.

Key Points

• Asexual reproduction is a biological process by which an organism creates a genetically similar or identical copy of itself without the contribution of genetic material from another individual.
• It involves only one parent, meaning there is no requirement for a partner or the fusion of male and female gametes. Hence, statement 2 is correct and statement 1 is incorrect.
• The offspring produced through this method are genetically identical to the parent and are often referred to as clones. This process does not lead to genetic diversity within the population. Hence, statement 3 is incorrect.
• This mode of reproduction does not typically require gametes (sperm and egg cells), nor does it involve fertilization. It relies primarily on mitotic cell division. Hence, statement 4 is incorrect.
• It is a rapid and energy-efficient method of multiplication, commonly found in unicellular organisms, bacteria, archaea, and many plants and fungi.

Additional Information

• Types of Asexual Reproduction:
• Binary Fission: A parent cell divides into two equal daughter cells. This is common in Amoeba and Bacteria.
• Budding: A small outgrowth or bud develops on the parent body, which eventually detaches to become a new individual. Examples include Yeast and Hydra.
• Fragmentation: The body of the parent breaks into distinct pieces, and each piece grows into a complete organism. This is seen in Spirogyra and Planaria.
• Spore Formation: Organisms produce microscopic reproductive units called spores that germinate under favorable conditions. Example: Rhizopus (Bread Mould).
• Vegetative Propagation:
• This is a form of asexual reproduction in plants where new plants are produced from vegetative parts like roots, stems, or leaves.
• Examples include Potato (tubers), Ginger (rhizomes), and Bryophyllum (leaf buds).
• Sexual Reproduction Comparison:
• In contrast, sexual reproduction involves two parents and the fusion of haploid gametes to form a diploid zygote, resulting in genetic variation which is crucial for evolution and adaptation.

Important Points

• Advantages: The population can be increased rapidly when conditions are favorable. Only one parent is needed, which is more time and energy efficient as it eliminates the need to find a mate.
• Disadvantages: It does not lead to genetic variation in a population. The species may only be suited to one habitat, and disease may affect all individuals in a population because they are all genetically identical.
• Parthenogenesis: This is a special form of asexual reproduction where an unfertilized egg develops into a new individual. This occurs in some invertebrates (like bees and wasps) and a few vertebrates (like certain lizards).

Shortcut Trick

Days for (A+B) = 32/5, (B+C) = 64/9, (A+C) = 16/3.

L.C.M of numerators (32, 64, 16) = 64 units (Total Work).

Efficiencies: (A+B) = 10, (B+C) = 9, (A+C) = 12. Sum = 31.

Efficiency of (A+B+C) = 31 ÷ 2 = 15.5 units/day.

Time = 64 ÷ 15.5 = 128 ÷ 31 = 4(4/31) days.

∴ The correct answer is 4(4/31) days.

Alternate Method

Given: Time for (A+B) = 32/5 days, (B+C) = 64/9 days, (A+C) = 16/3 days.

⇒ Total efficiency of 2(A + B + C) = 10 + 9 + 12 = 31

⇒ Efficiency of (A + B + C) = 31/2

⇒ Time taken by A, B and C together = Total Work ÷ Efficiency

⇒ Time = 64 ÷ (31/2) = 128/31

⇒ Time = 4(4/31) days

∴ The correct answer is 4(4/31) days.

Additional Information

Total Work using LCM

When time is given in fractions, Total Work is the LCM of the numerators of the time values.

Efficiency Formula

Efficiency = Total Work ÷ Time taken. It represents the work done in one unit of time.

Combined Efficiency

If pairs work together, summing their individual pair efficiencies gives twice the total efficiency of all individuals: 2(A+B+C).

Shortcut Trick

Let the Marked Price (MP) be 100%.

Successive discounts are 36% and 10%.

Effective Selling Price (SP) factor = (100 − 36)% × (100 − 10)% = 0.64 × 0.90 = 0.576.

Given SP = ₹7,200 ⇒ 0.576 × MP = 7,200.

MP = 7,200 ÷ 0.576 = 12,500.

∴ The correct answer is ₹12,500.

Alternate Method

Given: Selling Price (SP) = ₹7,200, Discount₁ (d₁) = 36%, Discount₂ (d₂) = 10%.

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

⇒ 7,200 = MP × (64/100) × (90/100)

⇒ 7,200 = MP × (16/25) × (9/10)

⇒ 7,200 = MP × (144/250)

⇒ MP = (7,200 × 250) ÷ 144

⇒ MP = 50 × 250 = 12,500.

∴ The correct answer is ₹12,500.

Additional Information

Successive Discount Formula

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Marked Price Definition

The Marked Price (MP) is the price printed on the label of an item, and discounts are always calculated on this value.

Shortcut Trick

Calculate total days: February (28 − 9 = 19) + March (31) + April (23) = 73 days.

Time in years = 73 ÷ 365 = 1/5 years.

Interest = (Principal × Rate × Time) ÷ 100

Interest = (2000 × 7 × 1/5) ÷ 100 = 20 × 7 ÷ 5 = 4 × 7 = ₹28.

∴ The correct answer is ₹28.

Alternate Method

Given:

Principal (P) = ₹2,000

Rate (R) = 7% per annum

Time Period: 9 February 2023 to 23 April 2023

Formula Used:

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

Calculations:

⇒ Total days = (28 − 9) in Feb + 31 in Mar + 23 in Apr

⇒ Total days = 19 + 31 + 23 = 73 days

⇒ Time (T) in years = 73 ÷ 365 = 1/5 years

⇒ SI = (2000 × 7 × 1/5) ÷ 100

⇒ SI = (400 × 7) ÷ 100

⇒ SI = 2800 ÷ 100

⇒ SI = ₹28

∴ The correct answer is ₹28.

Additional Information

Counting Days

While calculating the period for interest, the day on which money is borrowed is typically excluded, while the day on which money is returned is included.

Leap Year Check

For a non-leap year (like 2023), February has 28 days. For a leap year, February has 29 days. Always check if the year is divisible by 4.

Time Conversion

To convert days into years, divide by 365 (standard year). Common periods include 73 days (1/5 year), 146 days (2/5 year), and 219 days (3/5 year).

Shortcut Trick

For two objects moving towards each other, Relative Speed = Sum of individual speeds.

Total Distance = Sum of lengths of both trains = 496 + 490 = 986 m.

Relative Speed = 12 + 22 = 34 m/s.

Time taken = Total Distance ÷ Relative Speed = 986 ÷ 34 = 29 sec.

∴ The correct answer is 29 sec.

Alternate Method

Given:

Length of Train A (L₁) = 496 m

Length of Train B (L₂) = 490 m

Speed of Train A (S₁) = 12 m/s

Speed of Train B (S₂) = 22 m/s

Formula Used:

Time = (L₁ + L₂) ÷ (S₁ + S₂) [When moving in opposite directions]

Calculations:

⇒ Total distance to be covered = 496 + 490 = 986 m

⇒ Relative speed = Speed of A + Speed of B

⇒ Relative speed = 12 + 22 = 34 m/s

⇒ Time required = 986 ÷ 34

⇒ Time required = 29 sec

∴ The correct answer is 29 sec.

Additional Information

Relative Speed Concept

When two objects move in the opposite direction, their relative speed is S₁ + S₂. When they move in the same direction, it is S₁ − S₂.

Unit Conversion

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

Crossing a Stationary Object

When a train crosses a pole or a man, the distance covered equals the length of the train itself.

Shortcut Trick

Convert ratio to integers by multiplying with LCM of denominators (9, 2, 5) = 90.

New ratio: (7/9 × 90) : (3/2 × 90) : (2/5 × 90) = 70 : 135 : 36.

Total units = 70 + 135 + 36 = 241 units.

Given 241 units = 241 bananas → 1 unit = 1 banana.

Third monkey share = 36 units = 36 bananas.

∴ The correct answer is 36 bananas.

Alternate Method

Given: Total bananas = 241, Ratio = 7/9 : 3/2 : 2/5

⇒ Simplify the ratio: 7/9 : 3/2 : 2/5

⇒ Multiply each term by LCM(9, 2, 5) = 90

⇒ Ratio = (70/90) : (135/90) : (36/90)

⇒ Ratio in simplest form = 70 : 135 : 36

⇒ Sum of ratio terms = 70 + 135 + 36 = 241

⇒ Third monkey's share = (36 ÷ 241) × 241

⇒ Third monkey's share = 36

∴ The correct answer is 36 bananas.

Additional Information

Ratio Simplification

To convert a fractional ratio a/b : c/d into an integer ratio, multiply both terms by the Lowest Common Multiple (LCM) of the denominators b and d.

Proportional Division

If a total quantity X is divided in the ratio a : b : c, then the share of the third part is calculated as [c / (a + b + c)] × X.

Compound Ratio

The compound ratio of (a : b) and (c : d) is (a × c) : (b × d), used when combining different proportional relationships.

Shortcut Trick

Combined efficiency of P + Q = 1/15 and P = 1/24.

Efficiency of Q = 1/15 − 1/24 = (8 − 5)/120 = 3/120 = 1/40.

Time for Q to fill one-fourth = (1/4) × 40 = 10 hours.

∴ The correct answer is 10 hours.

Alternate Method

Given: P + Q = 15 hours, P = 24 hours.

Formula Used: Total Work = LCM of time; Efficiency = Total Work ÷ Time.

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

⇒ Efficiency of Q = 8 − 5 = 3 units/hour

⇒ One-fourth of cistern = 120 ÷ 4 = 30 units

⇒ Time taken by Q alone = 30 ÷ 3 = 10 hours

∴ The correct answer is 10 hours.

Additional Information

Pipes & Cisterns Efficiency

If pipe A fills in x hours and pipe B fills in y hours, combined work per hour is 1/x + 1/y. If B is an outlet pipe, it is 1/x − 1/y.

Work & Wages

Wages are always distributed in the ratio of the efficiencies of the workers (or work done), not the time taken.

Alternate Day Work

For problems where workers work on alternate days, calculate the combined work in one cycle (e.g., 2 days) and divide total work by cycle work.

Shortcut Trick

Ratio 8 years ago = 4 : 3 | Ratio 8 years later = 5 : 4

Difference in ratio units = 5 − 4 = 1 unit and 4 − 3 = 1 unit

Time difference = 8 + 8 = 16 years

⇒ 1 unit = 16 years

Aman’s age 8 years ago = 4 units = 4 × 16 = 64 years

Aman’s present age = 64 + 8 = 72 years

∴ The correct answer is 72 years.

Alternate Method

Let Aman’s age and Riya’s age 8 years ago be 4x and 3x respectively.

Present ages: Aman = 4x + 8, Riya = 3x + 8

Ages 8 years from now: Aman = 4x + 16, Riya = 3x + 16

According to the question:

⇒ (4x + 16) / (3x + 16) = 5 / 4

⇒ 4(4x + 16) = 5(3x + 16)

⇒ 16x + 64 = 15x + 80

⇒ x = 80 − 64 = 16

⇒ Aman’s present age = 4x + 8 = 4 × 16 + 8

⇒ 64 + 8 = 72 years

∴ The correct answer is 72 years.

Additional Information

Ratio Difference Method

If the difference between ratio parts is same (5−4 = 1 and 4−3 = 1), then the difference corresponds directly to the years passed.

Age Difference Constancy

The numerical difference between the ages of two persons remains constant throughout their lives, regardless of how many years pass.

Cross-Multiplication Method

Used for solving ratio changes: (Initial Ratio) → (Add Years) → (Final Ratio) to find the value of one unit quickly.

Shortcut Trick

The surface area of a sphere is given by 4πr².

Equating the given value: 4πr² = 144π

Dividing both sides by 4π: r² = 36

Taking the square root: r = 6 units.

∴ The correct answer is 6 units.

Alternate Method

Given:

Surface area of the sphere = 144π sq units

Formula Used:

Surface Area of a Sphere = 4πr², where r is the radius.

Calculations:

⇒ 4 × π × r² = 144 × π

⇒ 4r² = 144

⇒ r² = 144 ÷ 4

⇒ r² = 36

⇒ r = √36

⇒ r = 6 units

∴ The correct answer is 6 units.

Additional Information

Volume of a Sphere

The amount of space occupied by a sphere is given by the formula V = (4/3)πr³.

Hemisphere Properties

For a hemisphere, the Curved Surface Area (CSA) is 2πr² and the Total Surface Area (TSA) is 3πr².

Diameter of a Sphere

The diameter is the longest distance through the sphere, calculated as D = 2 × r.

Shortcut Trick

Direct formula: LCM × HCF = Product of two numbers.

LCM × 12 = 2160

LCM = 2160 ÷ 12 = 180

∴ The correct answer is 180.

Alternate Method

Given:

HCF = 12

Product of two numbers = 2160

Formula Used:

HCF × LCM = n₁ × n₂

Calculations:

⇒ 12 × LCM = 2160

⇒ LCM = 2160 ÷ 12

⇒ LCM = 180

∴ The correct answer is 180.

Additional Information

HCF and LCM Relation

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.

HCF of Fractions

HCF of fractions = HCF of Numerators ÷ LCM of Denominators.

LCM of Fractions

LCM of fractions = LCM of Numerators ÷ HCF of Denominators.

Shortcut Trick

Convert Mass to grams: 3.168 kg = 3168 g.

Find Volume: 3168 ÷ 3.5 = 6336/7 cm³.

Solve for Radius (r): (4/3) × (22/7) × r³ = 6336/7 ⇒ r³ = 216 ⇒ r = 6 cm.

Find Surface Area: 4 × (22/7) × 6² = 3168/7 cm².

Calculate Total Cost: (3168/7) × 5.6 = 3168 × 0.8 = ₹2534.40.

∴ The correct answer is ₹2,534.40.

Alternate Method

Given:

Density of the spherical ball = 3.5 g/cm³

Total Mass = 3.168 kg = 3168 g

Rate of painting = ₹5.6 per cm²

Formula Used:

Volume (V) = Mass ÷ Density

Volume of sphere = (4/3) × π × r³

Surface Area of sphere = 4 × π × r²

Calculations:

⇒ Volume of the ball = 3168 ÷ 3.5 = 6336/7 cm³

⇒ (4/3) × (22/7) × r³ = 6336/7

⇒ (88/21) × r³ = 6336/7

⇒ r³ = (6336/7) × (21/88)

⇒ r³ = (6336 ÷ 88) × (21 ÷ 7)

⇒ r³ = 72 × 3 = 216

⇒ r = √216 = 6 cm

⇒ Surface Area of the ball = 4 × (22/7) × 6²

⇒ Surface Area = (4 × 22 × 36) ÷ 7 = 3168/7 cm²

⇒ Total Cost of Painting = Surface Area × Rate

⇒ Cost = (3168/7) × 5.6

⇒ Cost = 3168 × 0.8 = ₹2534.40

∴ The correct answer is ₹2,534.40.

Additional Information

Volume and Surface Area of Sphere

For a sphere of radius r, the volume is (4/3)πr³ and the total surface area is 4πr².

Density Formula

Density is mass per unit volume (ρ = Mass ÷ Volume). Ensure mass and density units are compatible (e.g., grams and g/cm³).

Hemisphere Properties

For a solid hemisphere, Curved Surface Area = 2πr², Total Surface Area = 3πr², and Volume = (2/3)πr³.

Shortcut Trick

Assume a base value (Assumed Mean) of 50.

Calculate deviations from 50: −3, +21, +32, +9, −28, −7, −4, +46, +6.

Sum of deviations = (−3 + 21 + 32 + 9 + 46 + 6) − (28 + 7 + 4) = 111 − 39 = 72.

Average deviation = 72 ÷ 9 = +8.

Actual Mean = Assumed Mean + Average deviation = 50 + 8 = 58.

∴ The correct answer is 58.

Alternate Method

Given: Observations are 47, 71, 82, 59, 22, 43, 46, 96, and 56.

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

⇒ Total sum of observations = 47 + 71 + 82 + 59 + 22 + 43 + 46 + 96 + 56

⇒ Total sum = 522

⇒ Total number of observations (n) = 9

⇒ Arithmetic Mean = 522 ÷ 9

⇒ Arithmetic Mean = 58

∴ The correct answer is 58.

Additional Information

Arithmetic Mean of First n Natural Numbers

The mean of the first n natural numbers is given by the formula: (n + 1) ÷ 2.

Empirical Relationship

The relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Effect of Change in Origin and Scale

If a constant k is added to or multiplied by each observation, the new mean becomes Mean + k or Mean × k respectively.

Shortcut Trick

If Savings = 60%, then Expenditure = 100% − 60% = 40%.

Given, 40% of Income = ₹560.

100% of Income = (560 ÷ 40) × 100 = 14 × 100 = ₹1400.

∴ The correct answer is ₹1400.

Alternate Method

Given:

Savings percentage = 60%

Total Expenditure = ₹560

Formula Used:

Income = Expenditure + Savings

Expenditure % = 100% − Savings %

Calculations:

⇒ Expenditure % = 100% − 60% = 40%

⇒ Let Total Income be 'I'.

⇒ 40% of I = 560

⇒ (40 ÷ 100) × I = 560

⇒ I = (560 × 100) ÷ 40

⇒ I = 14 × 100

⇒ I = 1400

∴ The correct answer is ₹1400.

Additional Information

Percentage Change in Income

If income increases by x% and expenditure increases by y%, the change in savings is calculated using the base equation: Income = Expenditure + Savings.

Base Change Property

When a value is expressed as a percentage of a total, the total can always be found by: (Value ÷ Percentage) × 100.

Shortcut Trick

Average Speed = Total Distance ÷ Total Time

Total Distance = 70 + 80 + 50 = 200 km

Total Time = (70/60) + (80/80) + (50/40) = 7/6 + 1 + 5/4 = (14 + 12 + 15)/12 = 41/12 hours

Average Speed = 200 × (12/41) = 2400/41 ≈ 58.54 km/hr

∴ The correct answer is 58.54 km/hr.

Alternate Method

Given:

Distance₁ = 70 km, Speed₁ = 60 km/hr

Distance₂ = 80 km, Speed₂ = 80 km/hr

Distance₃ = 50 km, Speed₃ = 40 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = Total Distance ÷ Total Time

Calculations:

⇒ Total Distance = 70 + 80 + 50 = 200 km

⇒ Time₁ = 70/60 = 7/6 hours

⇒ Time₂ = 80/80 = 1 hour

⇒ Time₃ = 50/40 = 5/4 hours

⇒ Total Time = 7/6 + 1 + 5/4

⇒ Total Time = (14 + 12 + 15) / 12 = 41/12 hours

⇒ Average Speed = 200 ÷ (41/12) = (200 × 12) / 41

⇒ Average Speed = 2400 / 41 ≈ 58.5365...

∴ The correct answer is 58.54 km/hr.

Additional Information

Average Speed for Equal Distances

If a car covers two equal distances at speeds u and v, the average speed is given by (2 × u × v) ÷ (u + v).

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed is the difference of their speeds. In opposite directions, it is the sum of their speeds.

Shortcut Trick

Ratio of Cost Price (CP) to Marked Price (MP) is CP : MP = (100 − Discount%) : (100 + Profit%).

CP : MP = (100 − 10) : (100 + 10) = 90 : 110 = 9 : 11.

Given 11 units = &8377;880 ⇒ 1 unit = &8377;80.

Therefore, 9 units = 9 × 80 = &8377;720.

∴ The correct answer is &8377;720.

Alternate Method

Given → Marked Price (MP) = &8377;880, Discount = 10%, Gain = 10%. Formulas: SP = MP × (100 − D%) ÷ 100 | CP = SP × 100 ÷ (100 + G%).

⇒ Selling Price (SP) = 880 × (100 − 10) ÷ 100 = 880 × 0.9 = &8377;792.

⇒ Cost Price (CP) = 792 × 100 ÷ (100 + 10) = 792 × 100 ÷ 110.

⇒ CP = 72 × 10 = &8377;720.

∴ The correct answer is &8377;720.

Additional Information

Marked Price and Discount

The Marked Price (MP) is the price printed on the article. Discount is always calculated on the MP to arrive at the Selling Price (SP).

CP-MP Relation Formula

The direct relation between CP and MP is given by CP / MP = (100 − D%) / (100 + P%), where D is discount and P is profit percentage.

Shortcut Trick

Prime numbers (60 to 70): 61, 67 → Average = 128 ÷ 2 = 64

Multiples of 4 (50 to 58): 52, 56 → Average = 108 ÷ 2 = 54

Difference = 64 − 54 = 10

∴ The correct answer is 10.

Alternate Method

Given:

Range 1: Prime numbers between 60 and 70.

Range 2: Multiples of 4 between 50 and 58.

Calculations:

⇒ Prime numbers between 60 and 70 = {61, 67}

⇒ Average of primes = (61 + 67) ÷ 2 = 128 ÷ 2 = 64

⇒ Multiples of 4 between 50 and 58 = {52, 56}

⇒ Average of multiples = (52 + 56) ÷ 2 = 108 ÷ 2 = 54

⇒ Required Difference = 64 − 54 = 10

∴ The correct answer is 10.

Additional Information

Prime Numbers between 1 and 100

There are 25 prime numbers between 1 and 100. These are numbers having exactly two factors: 1 and the number itself.

Average of an Arithmetic Progression

The average of any series in arithmetic progression (like multiples of a number) is simply (First Term + Last Term) ÷ 2.

Composite Numbers

Numbers with more than two factors are composite numbers. Note that 1 is neither prime nor composite.

Shortcut Trick

Split inside root: √(a × b) = √a × √b

= √144×√49 + √64×√25 − √100×√36

= 12×7 + 8×5 − 10×6

= 84 + 40 − 60 = 64

∴ Answer = 64

Alternate Method

Given:

√(144 × 49) + √(64 × 25) − √(100 × 36)

Concept Used:

√(a × b) = √a × √b

Calculation:

⇒ √(144 × 49) = √144 × √49 = 12 × 7 = 84

⇒ √(64 × 25) = √64 × √25 = 8 × 5 = 40

⇒ √(100 × 36) = √100 × √36 = 10 × 6 = 60

⇒ Expression = 84 + 40 − 60

⇒ = 124 − 60

⇒ = 64

∴ The correct answer is 64.

Additional Information

Square Root Property

√(a × b) = √a × √b

Perfect Squares

144 = 12², 49 = 7², 64 = 8², 25 = 5²

Simplification Tip

Break numbers into perfect squares for quick solving

Common Pattern

Use factorization inside root to simplify faster

Shortcut Trick

To get the maximum selling price, the total discount percentage must be the minimum.

For two successive discounts a% and b%, effective discount = a + b − (a × b ÷ 100).

Scheme I: 14 + 14 − 1.96 = 26.04% | Scheme II: 22 + 29 − 6.38 = 44.62%.

Scheme III: 22 + 7 − 1.54 = 27.46% | Scheme IV: 14 + 6 − 0.84 = 19.16%.

Since 19.16% is the lowest discount, Scheme IV results in the highest selling price.

∴ The correct answer is IV.

Alternate Method

Given → Four different successive discount schemes. Formulas: Effective Discount = a + b − (ab ÷ 100) | SP = MP × (100 − Discount%) ÷ 100.

⇒ Calculate Effective Discount (ED) for each scheme: ⇒ Scheme I: 14 + 14 − (14 × 14 ÷ 100) = 28 − 1.96 = 26.04% ⇒ Scheme II: 22 + 29 − (22 × 29 ÷ 100) = 51 − 6.38 = 44.62% ⇒ Scheme III: 22 + 7 − (22 × 7 ÷ 100) = 29 − 1.54 = 27.46% ⇒ Scheme IV: 14 + 6 − (14 × 6 ÷ 100) = 20 − 0.84 = 19.16% ⇒ Comparing all, Scheme IV has the minimum discount of 19.16%.

∴ The correct answer is IV.

Additional Information

Marked Price & Selling Price

The price printed on an article is the Marked Price (MP). Selling Price is obtained after subtracting the discount from the MP.

Successive Discount Concept

When two or more discounts are applied one after another, they are called Successive Discounts. The order of applying these discounts does not change the final selling price.

Shortcut Trick

For mean proportion between x and b to be M, the relationship is x × b = M².

Given M = 63 and b = 147.

x = 63² ÷ 147 = (63 × 63) ÷ 147.

x = (9 × 7 × 63) ÷ (21 × 7) ⇒ x = (9 × 63) ÷ 21 = 9 × 3 = 27.

∴ The correct answer is 27.

Alternate Method

Given: Mean proportion = 63, First number = x, Second number = 147.

Formula Used: Mean Proportion = √(a × b)

⇒ 63 = √(x × 147)

Squaring both sides:

⇒ 63² = x × 147

⇒ 3969 = x × 147

⇒ x = 3969 ÷ 147

⇒ x = 27

∴ The correct answer is 27.

Additional Information

Third Proportion

The third proportion of two numbers a and b is c, such that a : b :: b : c, given by c = b²/a.

Fourth Proportion

The fourth proportion of three numbers a, b, and c is d, such that a : b :: c : d, given by d = (b × c)/a.

Compounded Ratio

The compounded ratio of (a : b) and (c : d) is (a × c) : (b × d).

Shortcut Trick

In right-angled triangle BDC, ∠C = 90° − ∠DBC = 90° − 21° = 69°.

In ΔAEC, ∠AEB is the exterior angle at vertex E.

By the Exterior Angle Theorem, ∠AEB = ∠CAE + ∠ACE.

∠AEB = 27° + 69° = 96°.

∴ The correct answer is 96°.

Alternate Method

Given: BD ⊥ AC, ∠DBC = 21°, ∠CAE = 27°.

Formula Used: Angle Sum Property of a triangle (sum = 180°) and Linear Pair property.

In ΔBDC:

⇒ ∠BDC = 90° (Since BD ⊥ AC)

⇒ ∠BCD = 180° − (∠BDC + ∠DBC)

⇒ ∠BCD = 180° − (90° + 21°) = 69

Shortcut Trick

Let the number be 10 (LCM of denominators 5 and 2).

Three-fifths of 10 = 6 units.

One-half of 10 = 5 units.

Difference (6 − 5) = 1 unit → 4.

Original number (10 units) = 10 × 4 = 40.

∴ The correct answer is 40.

Alternate Method

Let the unknown number be x.

As per the problem statement:

⇒ (3/5) × x = (1/2) × x + 4

⇒ (3x/5) − (x/2) = 4

⇒ (6x − 5x) / 10 = 4

⇒ x / 10 = 4

⇒ x = 40

∴ The correct answer is 40.

Additional Information

Fractions to Percentage

Converting fractions to percentages often simplifies logic: 3/5 is 60% and 1/2 is 50%. The 10% difference equals 4, so 100% equals 40.

Linear Equations

A linear equation in one variable follows the form ax + b = 0. Solving involves isolating the variable x on one side.

LCM Method for Fractions

When dealing with multiple fractions, assuming the number as the LCM of the denominators eliminates fractional calculations and speeds up solving.

Shortcut Trick

Let the total voters be 100%.

Voters who cast their votes = 60%.

Candidate B received 25% (100% − 75%) of the casted votes.

Fraction of B's votes = 60% × 25% = 0.60 × 0.25 = 0.15 (or 15%).

15% → 7500 ⇒ 1% → 500 ⇒ 100% → 50,000.

∴ The correct answer is 50,000 voters.

Alternate Method

Given:

Voter turnout = 60% of total registered voters.

Votes for Candidate A = 75% of polled votes.

Votes for Candidate B = 7500.

Calculations:

Let the total number of registered voters be x.

⇒ Total votes cast = 60% of x = 0.6x

⇒ Percentage of votes for Candidate B = 100% − 75% = 25%

⇒ Votes for B = 25% of (0.6x)

⇒ 7500 = (25/100) × (60/100) × x

⇒ 7500 = (1/4) × (3/5) × x

⇒ 7500 = (3/20) × x

⇒ x = (7500 × 20) ÷ 3

⇒ x = 2500 × 20

⇒ x = 50,000

∴ The correct answer is 50,000 voters.

Additional Information

Percentage Breakdown

In election problems, always distinguish between Total Voters, Votes Cast, and Valid Votes. Use the tree method for clarity.

Majority Rule

The Majority or Winning Margin is the difference between the votes of the winner and the runner-up: Margin = |Votes_A − Votes_B|.

Invalid Votes

If invalid votes are mentioned, subtract them from the total votes cast before distributing percentages to candidates.

Shortcut Trick

Total Salary is always considered as 100%.

If Savings = 30%, then Expenditure = 100% − 30% = 70%.

Expenditure = 70% of ₹1,890 = 0.7 × 1890.

Expenditure = ₹1,323.

∴ The correct answer is ₹1,323.

Alternate Method

Given:

Total Salary = ₹1,890

Savings percentage = 30%

Formula Used:

Expenditure = Total Salary − Savings

Calculations:

⇒ Amount of Savings = 30% of ₹1,890

⇒ Savings = (30 ÷ 100) × 1890

⇒ Savings = 0.3 × 1890 = ₹567

⇒ Expenditure = Total Salary − Savings

⇒ Expenditure = ₹1,890 − ₹567

⇒ Expenditure = ₹1,323

∴ The correct answer is ₹1,323.

Additional Information

Percentage Change

Percentage increase or decrease is calculated as: (Change ÷ Original Value) × 100.

Income and Expenditure Relation

Income = Expenditure + Savings. If any two are known, the third can be found by simple subtraction.

Successive Percentage

If a value is changed by a% and then by b%, the net effective change is (a + b + ab/100)%.

Shortcut Trick

Check the divisibility of numbers between 469 and 474.

Numbers to check: 470, 471, 472, 473.

471: 4 + 7 + 1 = 12 (divisible by 3). 473: 473 ÷ 11 = 43 (divisible by 11).

All numbers are composite.

∴ The correct answer is 0.

Alternate Method

Given: Range of numbers between 469 and 474.

Formula Used: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

⇒ 470: Ends in 0, so divisible by 2, 5, 10. (Composite)

⇒ 471: Sum of digits = 4 + 7 + 1 = 12. 12 is divisible by 3, so 471 is divisible by 3. (Composite)

⇒ 472: Even number, divisible by 2. (Composite)

⇒ 473: Sum of alternate digits = (4 + 3) − 7 = 0. Divisible by 11. (Composite)

⇒ Total prime numbers = 0.

∴ The correct answer is 0.

Additional Information

Divisibility Rule of 3

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

Divisibility Rule of 11

Subtract the sum of digits in even positions from the sum of digits in odd positions. If the result is 0 or divisible by 11, the number is divisible by 11.

Prime Numbers

To check if N is prime, test divisibility by primes up to √N. For 473, √473 ≈ 21.7. Primes to check: 2, 3, 5, 7, 11, 13, 17, 19.

Shortcut Trick

Substitute a standard angle A = 45° to verify the expression value.

⇒ Numerator: 2√(2−1) × √(1−1/2) = 2 × 1 × 1/√2 = √2

⇒ Denominator: √(1+1) × 3√(1−1/2) = √2 × 3 × 1/√2 = 3

⇒ Expression = √2 / 3

Checking options for A = 45°: (1) (2 × 1/√2)/3 = √2 / 3. Matches perfectly.

∴ The correct answer is (2 cos A)/3.

Alternate Method

Given: $\frac{2\sqrt{\csc^{2}A-1} \times \sqrt{1-\cos^{2}A}}{\sqrt{1+\tan^{2}A} \times 3\sqrt{1-\sin^{2}A}}$

Formula Used:

1. √(csc²A − 1) = √cot²A = cot A
2. √(1 − cos²A) = √sin²A = sin A
3. √(1 + tan²A) = √sec²A = sec A
4. √(1 − sin²A) = √cos²A = cos A

Calculations:

⇒ Numerator = 2 × cot A × sin A

⇒ Numerator = 2 × (cos A / sin A) × sin A = 2 cos A

⇒ Denominator = sec A × 3 × cos A

⇒ Denominator = (1 / cos A) × 3 × cos A = 3

⇒ Final Expression = (2 cos A) / 3

∴ The correct answer is (2 cos A)/3.

Additional Information

Pythagorean Identities

sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ | 1 + cot²θ = csc²θ.

Reciprocal Relations

sin θ = 1/csc θ | cos θ = 1/sec θ | tan θ = 1/cot θ.

Quotient Relations

tan θ = sin θ / cos θ | cot θ = cos θ / sin θ.

Shortcut Trick

If a + 1/a = k, then the direct formula for a² + 1/a² is k² − 2.

Given: k = 6.

Calculation: 6² − 2 = 36 − 2 = 34.

∴ The correct answer is 34.

Alternate Method

Given:

a + 1/a = 6

Formula Used:

The Algebraic Identity for the square of a binomial: (x + y)² = x² + y² + 2xy

Calculation:

⇒ Taking the square on both sides of the given equation:

⇒ (a + 1/a)² = 6²

⇒ a² + (1/a)² + 2 × a × (1/a) = 36

⇒ a² + 1/a² + 2 = 36

⇒ a² + 1/a² = 36 − 2

⇒ a² + 1/a² = 34

∴ The correct answer is 34.

Additional Information

Square of a Difference

If a − 1/a = k, then a² + 1/a² = k² + 2.

Cubic Identities

If a + 1/a = k, then a³ + 1/a³ = k³ − 3k. If a − 1/a = k, then a³ − 1/a³ = k³ + 3k.

Higher Power Identities

If a + 1/a = k, then a⁴ + 1/a⁴ = (k² − 2)² − 2.

The logic followed here is:

thus, "520" will replace the question mark in the given series.

Hence, the correct option is "Option 4".

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

Now,

Even digits, which are immediately preceded by an even digit and also immediately followed by an even digit.

So, needed pattern is: Even - Even - Even

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

So, total valid even digit pattern = 1 

Hence, the correct answer is Option 4.

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

Now,

Odd numbers, which are immediately preceded by an odd number and also immediately followed by an odd number.

So, needed pattern is: Odd - Odd - Odd

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

1. 4 9 4 8 [3 3 3] 1 4 7 8 7 1 7 2 9 1 3 5 1 5 9 6 4 8 4 6
2. 4 9 4 8 3 [3 3 1] 4 7 8 7 1 7 2 9 1 3 5 1 5 9 6 4 8 4 6
3. 4 9 4 8 3 3 3 1 4 7 8 [7 1 7] 2 9 1 3 5 1 5 9 6 4 8 4 6
4. 4 9 4 8 3 3 3 1 4 7 8 7 1 7 2 [9 1 3] 5 1 5 9 6 4 8 4 6
5. 4 9 4 8 3 3 3 1 4 7 8 7 1 7 2 9 [1 3 5] 1 5 9 6 4 8 4 6
6. 4 9 4 8 3 3 3 1 4 7 8 7 1 7 2 9 1 [3 5 1] 5 9 6 4 8 4 6
7. 4 9 4 8 3 3 3 1 4 7 8 7 1 7 2 9 1 3 [5 1 5] 9 6 4 8 4 6
8. 4 9 4 8 3 3 3 1 4 7 8 7 1 7 2 9 1 3 5 [1 5 9] 6 4 8 4 6

So, total valid odd number patterns = 8

Hence, the correct answer is Option 1.

BODMAS Table: 

Given equation: 6 ÷ 3 + 16 × 4 − 9 = ? + 5

Now, replacing the symbols as:

 

⇒ 6 × 3 − 16 ÷ 4 + 9 = ? − 5

⇒ 18 − 16 ÷ 4 + 9 = ? − 5

⇒ 18 − 4 + 9 = ? − 5

⇒ 14 + 9 = ? − 5

⇒ 23 = ? − 5

⇒ ? = 23 + 5

⇒ 28

Hence, the correct answer is Option 4.

Given: Six friends — P, Q, R, S, U and V are sitting around a circular table facing the centre.

P is the immediate neighbour of both Q and R.

R is sitting to the immediate left of U.

V is sitting second to the left of P.

So, only Two persons sits between P and S when counted from the right of S.

Hence, the correct answer is "Option 3".

Given Statements:

Some peas are dogs.

No dog is a cub.

The least possible Venn diagram for the given statement is:

Conclusions:

I. Some peas are cubs - Does not follow (No definite relationship is given between peas and cubs. They may or may not overlap).

II. Some dogs are peas - Follows (Since some peas are dogs, it definitely implies that some dogs are peas).

So, only conclusion II follows.

Hence, the correct answer is Option 1.

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

Now,

Odd numbers which are immediately preceded by an odd number and immediately followed by an even number.

So, needed pattern is: Odd - Odd - Even

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

So, total valid patterns = 4

Hence, the correct answer is Option 3.

The logic followed here is to compare the given sentences and identify the common word and its corresponding common code.

Given:

speak the truth → pc rd st

truth always wins → mo st kl

So, truth → st

Hence, the correct answer is Option 1.

Given: Seven boxes W, T, D, B, P, E and S are kept one over the other but not necessarily in the same order.

Only three boxes are kept above B.

Only one box is kept between P and B.

Only three boxes are kept between P and T.

T is kept at some place above B.

D is kept immediately below T.

W is not kept immediately above or below P.

E is kept at one of the places above S.

Now, Boxes between S (7) and D (3): B (4), E (5), P (6) → 3 boxes

Hence, the correct answer is Option 1.