RRB Group D Official Paper (Held On: 08 Dec, 2025 Shift 3) (Previous Year Paper)

The correct answer is Highly diminished, real and inverted.

Key Points

• When an object is placed at infinity in front of a concave mirror, the light rays from the object are considered to be parallel to the principal axis.
• These parallel rays converge at the focus of the concave mirror after reflection, forming the image.
• The image formed is highly diminished because all the rays meet at a single point, making the size of the image negligible compared to the object.
• The image is real because it is formed by the actual convergence of rays after reflection.
• The image is inverted because the reflected rays flip the orientation of the object.

Additional Information

• Concave Mirrors:
• Concave mirrors are spherical mirrors that curve inward, resembling the inside of a sphere.
• They are capable of converging light rays and are commonly used in applications like telescopes, headlights, and shaving mirrors.
• Image Formation Based on Object Position:
• If the object is at infinity, the image is formed at the focus, is highly diminished, real, and inverted.
• If the object is beyond the center of curvature, the image is diminished, real, and inverted.
• If the object is at the center of curvature, the image is the same size, real, and inverted.
• If the object is between the focus and the mirror, the image is enlarged, virtual, and upright.
• Applications of Concave Mirrors:
• Used in telescopes for capturing distant celestial objects.
• Applied in dentistry and surgical procedures for magnified viewing.
• Used in solar concentrators to focus sunlight for energy generation.
• In car headlights, concave mirrors are used to create a parallel beam of light.
• Real and Virtual Images:
• A real image is formed when light rays actually converge, and it can be captured on a screen.
• A virtual image is formed when the rays appear to diverge from a point and cannot be captured on a screen.

Important Points

• Principal Focus: The point where parallel rays converge after reflection in a concave mirror.
• Center of Curvature: The center of the sphere from which the mirror segment is derived.
• Focal Length: The distance between the principal focus and the pole of the mirror.
• Mirror Formula: The mathematical relationship governing mirrors: 1/f = 1/v - 1/u, where f = focal length, v = image distance, and u = object distance.

The correct answer is exchange of metal ions between two metal salts.

Key Points

• In a double displacement reaction, two ionic compounds exchange their ions to form two new compounds. This process involves the exchange of cations (positively charged ions) and anions (negatively charged ions) between the reacting substances.
• The general representation of a double displacement reaction is:
  

AB + CD → AD + CB, where A and C are cations, and B and D are anions.

• This type of reaction typically occurs in aqueous solutions, where the ionic compounds are dissolved in water, allowing the ions to move freely and interact with each other.
• An example of a double displacement reaction is the reaction between sodium chloride (NaCl) and silver nitrate (AgNO₃):
  

NaCl + AgNO₃ → NaNO₃ + AgCl (precipitate).

• Such reactions often result in the formation of a precipitate, a gas, or a neutral molecule such as water, making them easily observable in a laboratory setting.

Additional Information

• Characteristics of Double Displacement Reactions:
• Double displacement reactions typically occur between two ionic compounds in aqueous solutions.
• These reactions often produce a precipitate, a gas, or a liquid (such as water) as a product.
• They do not involve the transfer of electrons, as opposed to redox reactions. Instead, they involve the rearrangement of ions.
• Types of Double Displacement Reactions:
• Precipitation Reactions: These occur when the exchange of ions results in the formation of an insoluble compound (precipitate). For example:
  

BaCl₂ + Na₂SO₄ → BaSO₄ (precipitate) + 2NaCl.

• Neutralization Reactions: These occur between an acid and a base to produce water and a salt. For example:
  

HCl + NaOH → NaCl + H₂O.

• Gas Formation Reactions: In some cases, a gas is released as a result of the reaction. For example:
  

Na₂CO₃ + 2HCl → 2NaCl + H₂O + CO₂ (gas).

• Explanation of Incorrect Options:
• Option 1: "Only one product is formed" – Incorrect. Double displacement reactions result in the formation of two products, not one.
• Option 2: "The reaction mixture is always colourless" – Incorrect. The reaction mixture can vary in colour based on the compounds involved and the products formed, such as precipitates of varying colours.
• Option 4: "The reaction requires heat to initiate" – Incorrect. Double displacement reactions usually occur spontaneously in aqueous solutions without the need for external heat.
• Examples in Daily Life:
• Soap Scum Formation: When soap reacts with hard water containing calcium or magnesium ions, a precipitate is formed.
• Neutralization in Medicine: Antacids neutralize stomach acid through double displacement reactions.

The correct answer is People are using more disposable items instead of long-lasting ones.

Key Points

• Plastic waste has significantly increased due to the growing reliance on disposable items like single-use plastic bags, cutlery, bottles, and packaging. These items are often discarded after one use, contributing to the accumulation of non-biodegradable waste.
• Unlike long-lasting or reusable items, disposable items are designed for convenience and are frequently replaced, leading to higher consumption and waste generation. This habit is a primary driver of plastic pollution in cities.
• Disposable plastics are widely used in the food and beverage industry, e-commerce packaging, and retail sectors, which often prioritize cost-effectiveness and convenience over sustainability.
• While some cities have implemented plastic bans or promoted reusable alternatives, the persistent demand for disposable plastics undermines these efforts, further exacerbating the waste crisis.
• Increasing awareness about the environmental impact of disposable plastics and promoting behavioral changes, such as opting for reusable alternatives, can help reduce plastic waste significantly.

Additional Information

• Plastic waste statistics:
• According to the United Nations Environment Programme (UNEP), over 300 million tons of plastic are produced globally each year, and at least 8 million tons end up in the ocean annually.
• In India, the Central Pollution Control Board (CPCB) reported that plastic waste generation amounted to 3.5 million tons annually in 2020.
• Environmental impact of disposable plastics:
• Plastic waste contributes to soil and water pollution, as it takes hundreds of years to decompose, releasing harmful chemicals in the process.
• Animals often mistake plastic for food, leading to ingestion and severe health issues, and in some cases, fatalities.
• Incineration of plastic waste releases toxic gases, contributing to air pollution and climate change.
• Government initiatives to combat plastic waste:
• The Indian government launched the Plastic Waste Management Rules (2016) to regulate the use and disposal of plastic. In 2022, single-use plastics were banned in India.
• The Swachh Bharat Abhiyan (Clean India Mission) emphasizes waste segregation and recycling, including plastic waste.
• Alternatives to disposable plastics:
• Use of biodegradable materials such as jute bags, bamboo straws, and paper-based packaging is gaining popularity.
• Encouraging the use of metal or glass containers for storage and reusable bottles for water and beverages can significantly reduce plastic waste.

Important Points

• Features of disposable plastics:
• They are lightweight, low-cost, and convenient for single-use applications, making them widely used in packaging, food services, and retail.
• However, their non-biodegradable nature poses a significant threat to the environment.
• Impact of bans on plastic waste:
• Bans on single-use plastics have been implemented in various countries, including India, Canada, and the European Union, to curb plastic pollution.
• While these measures are effective, they require strict enforcement and public cooperation to achieve desired results.

The correct answer is double.

Key Points

• Potential Energy (PE) is given by the formula: PE = m × g × h, where:
• m is the mass of the object (in kilograms).
• g is the acceleration due to gravity (approximately 9.8 m/s²).
• h is the height of the object above a reference point (in meters).
• When the height (h) of an object is doubled, the formula becomes: PE = m × g × (2h).
• This results in the potential energy becoming 2 × (m × g × h), i.e., double the original potential energy.
• Hence, the potential energy of the object is directly proportional to its height. Doubling the height leads to a doubling of the potential energy.
• Thus, the correct answer is that the potential energy becomes double.

Additional Information

• Potential Energy:
• It is a type of mechanical energy associated with the position of an object in a gravitational field.
• Measured in Joules (J), the SI unit of energy.
• Examples include energy stored in a raised weight, water in a dam, or a compressed spring.
• Gravitational Potential Energy:
• This refers specifically to the energy an object possesses due to its height above a reference level in the Earth's gravitational field.
• It is influenced by three factors: mass (m), gravity (g), and height (h).
• Gravitational potential energy increases with an increase in height or mass.
• Applications of Potential Energy:
• In hydropower plants, water stored in dams possesses potential energy that is converted into kinetic energy to generate electricity.
• In transportation systems, potential energy is utilized in mechanisms like elevators and escalators.
• Roller coasters rely on the conversion of potential energy to kinetic energy for motion.
• Important Characteristics:
• Potential energy depends only on the vertical height and not on the path taken to reach that height.
• It is a scalar quantity and has no direction.

Important Points

• Relationship Between Potential Energy and Height:
• Since PE = m × g × h, if height (h) is halved, the potential energy becomes half of its original value.
• If height is tripled, the potential energy becomes three times the original value.
• Other Forms of Energy:
• Kinetic Energy: Energy associated with the motion of an object, given by KE = 1/2 × m × v².
• Elastic Potential Energy: Stored energy in stretched or compressed elastic materials, such as springs.
• Chemical Potential Energy: Energy stored in chemical bonds, such as in fuels or food.
• Practical Implications:
• Understanding potential energy is crucial in fields like engineering, physics, and environmental studies.
• It plays a significant role in designing structures like dams, bridges, and buildings to ensure stability and energy efficiency.

The correct answer is calcium hydroxide.

Key Points

• When calcium oxide (commonly known as quicklime) reacts with water, it forms calcium hydroxide (slaked lime) through an exothermic reaction.
• The chemical equation for this reaction is:
  

CaO + H₂O → Ca(OH)₂ + Heat.

• This reaction releases a significant amount of heat, making it an exothermic reaction.
• The product formed, calcium hydroxide, is a white powder or a suspension in water, often used in construction (e.g., making mortar) and as a disinfectant in water treatment.
• Other options like calcium chloride, calcium carbonate, and calcium sulphate are not formed in this reaction. These compounds are the result of entirely different chemical reactions.

Additional Information

• Calcium Oxide (Quicklime):
• It is a chemical compound with the formula CaO.
• Quicklime is a white, caustic, alkaline crystalline solid at room temperature.
• It is commonly used in the production of cement, steel manufacturing, and as a chemical reagent.
• Calcium Hydroxide (Slaked Lime):
• The chemical formula of calcium hydroxide is Ca(OH)₂.
• It is widely used in whitewashing, water purification, and as a neutralizing agent in acidic soils.
• When dissolved in water, it forms a solution known as lime water, which turns milky in the presence of carbon dioxide.
• Exothermic Reaction:
• Reactions that release heat to the surroundings are termed as exothermic reactions.
• The reaction between calcium oxide and water is highly exothermic, which can cause boiling and splashing if not handled carefully.
• Incorrect Options Explained:
• Calcium Chloride: Formed by the reaction of calcium and hydrochloric acid, not by reacting calcium oxide with water.
• Calcium Carbonate: Formed when calcium hydroxide reacts with carbon dioxide, resulting in CaCO₃.
• Calcium Sulphate: Commonly known as gypsum, it is formed by the reaction of calcium compounds with sulphuric acid or sulphates.

Important Points

• The reaction between calcium oxide and water is an industrially significant process in the production of slaked lime, used in various construction and chemical applications.
• Lime water (a solution of calcium hydroxide in water) is a popular test for carbon dioxide gas as it turns milky in its presence due to the formation of calcium carbonate.
• Calcium hydroxide is used in agriculture to neutralize acidic soils and improve soil fertility.
• Safety measures should be taken while handling quicklime as its reaction with water is vigorous and can cause burns.

The correct answer is A bud forms as an outgrowth due to repeated cell division and detaches when mature..

Key Points

• Budding is an asexual reproduction process observed in organisms like Hydra, yeast, and some other species.
• In this process, a bud forms as an outgrowth on the body of the parent organism due to repeated mitotic cell division.
• The bud remains attached to the parent organism while it grows and matures, drawing nutrients and support from the parent.
• Once the bud is fully developed, it detaches from the parent and becomes an independent organism capable of surviving on its own.
• This process ensures the continuity of the species without the need for gamete fusion, as seen in sexual reproduction. Hydra is a well-known example of organisms reproducing through budding.

Additional Information

• Explanation of Other Options:
• Option 1: "A bud forms only during sexual reproduction and fuses with the parent organism" – This is incorrect. Budding is a form of asexual reproduction, and the bud does not fuse with the parent; it detaches to form a new organism.
• Option 3: "A bud forms from specialized cells inside the organism and remains attached for life" – This is incorrect. The bud forms externally and detaches once it matures.
• Option 4: "A bud forms from a broken body part and grows into a new organism" – This is incorrect. This description refers to regeneration, not budding. For example, starfish regenerate lost arms, but this is not budding.
• Characteristics of Budding:
• Budding is common in Hydra, yeast, and certain plants and fungi.
• In yeast, the process involves the formation of a small bulge on the parent cell, which grows and eventually separates as a new cell.
• In Hydra, budding occurs through mitotic cell division in the body wall, forming a miniature version of the parent.
• This process is energy-efficient as it does not require the production of gametes or mating.
• Comparison with Other Asexual Reproduction Methods:
• Fragmentation: Seen in organisms like spirogyra, where the body breaks into fragments, and each fragment grows into a new organism.
• Binary Fission: Found in bacteria, where the organism divides into two identical daughter cells.
• Sporulation: Seen in fungi and some plants, where spores develop into new organisms under favorable conditions.
• Importance of Budding:
• Ensures rapid reproduction in favorable conditions.
• Helps maintain the population of species in stable environments.
• Simple and efficient mechanism for producing genetically identical offspring.

Important Points

• Hydra:
• Hydra is a freshwater organism belonging to the phylum Cnidaria.
• It reproduces both sexually and asexually, with budding being its common mode of asexual reproduction.
• Hydra’s ability to regenerate is another significant characteristic, which allows it to recover lost body parts.
• Yeast:
• Yeast is a unicellular fungus that reproduces primarily through budding.
• It is used in baking, brewing, and scientific research due to its rapid growth and easy genetic manipulation.
• In yeast, a small bud appears, grows, and eventually separates to form an independent organism.

The correct answer is Pea plant.

Key Points

• Gregor Mendel, known as the "Father of Genetics," conducted his experiments on the pea plant (Pisum sativum).
• Mendel chose pea plants because they have distinct and easily observable traits, such as flower color, seed shape, seed color, pod shape, pod color, flower position, and plant height.
• Pea plants are self-pollinating, which allowed Mendel to control breeding and cross-pollination experiments systematically.
• He performed experiments between 1856 and 1863, analyzing the inheritance patterns of these traits over multiple generations.
• Mendel used pea plants to establish the foundational principles of inheritance, including the laws of segregation and the law of independent assortment.

Additional Information

• Why Mendel chose pea plants:
• Pea plants have short life cycles, enabling Mendel to observe many generations in a short period.
• They exhibit seven distinct traits, each with two clear variations (e.g., tall vs. short plants).
• Pea plants can be artificially cross-pollinated or self-pollinated, allowing Mendel to control the experiment.
• Mendel’s Laws of Inheritance:
• Law of Segregation: Each organism inherits two alleles for each trait, one from each parent, and these alleles segregate during gamete formation.
• Law of Independent Assortment: Alleles for different traits segregate independently of each other during gamete formation.
• Impact of Mendel’s work:
• Mendel’s findings laid the foundation for the modern field of genetics.
• His principles were rediscovered in the early 20th century, decades after his experiments were initially overlooked.
• Other plants Mendel could have experimented with:
• Sunflower: Known for its large flowers and ability to grow quickly but lacks distinct, easily observable traits for genetic studies.
• Tomato: Tomatoes also exhibit genetic diversity, but they were not as suitable for controlled pollination experiments.
• Bean plant: Bean plants have genetic variability but were not as ideal for Mendel’s experiments as pea plants.

Important Points

• Gregor Mendel:
• Mendel was an Austrian monk who conducted pioneering experiments in genetics.
• He published his work in 1866, but it gained recognition only in the early 20th century.
• Applications of Mendel’s discoveries:
• His principles are used in plant breeding and the study of heredity in humans.
• Mendel’s laws are fundamental to understanding genetic disorders and inheritance patterns.

The correct answer is -12.0 m/s.

Key Points

• Initial velocity of the object: The object has an initial velocity of 3.0 m/s in a specific direction. This is the starting motion of the object. (Correct)
• Mass of the object: The mass of the object is given as 10.0 kg. This is used to compute the acceleration when the force is applied. (Correct)
• Force applied: A constant force of 30.0 N is applied in the opposite direction of motion. This force will decelerate the object. (Correct)
• Duration of force application: The force acts on the object for 5.0 seconds. This time duration is crucial in determining the change in velocity. (Correct)
• Final velocity calculation: Using Newton's second law and kinematic equations, the final velocity is calculated as -12.0 m/s, indicating the object is moving in the opposite direction after deceleration. (Correct)

Additional Information

• Newton's Second Law of Motion: The law states that the force acting on an object is equal to the product of its mass and acceleration. Formula: F = ma. Here, force is 30.0 N, and mass is 10.0 kg. This gives an acceleration of a = F/m = 30/10 = 3.0 m/s².
• Acceleration Direction: Since the force is applied in the opposite direction to motion, the acceleration is negative. Thus, a = -3.0 m/s².
• Kinematic Equation: The final velocity can be calculated using the equation: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Substituting the values: v = 3.0 + (-3.0) * 5.0 = 3.0 - 15.0 = -12.0 m/s.
• Negative Velocity: The final velocity being -12.0 m/s indicates that the object has reversed its direction and is now moving opposite to its initial motion.
• Applications of the Concept: This principle is widely used in understanding vehicle braking systems, rocket propulsion, and other real-world scenarios involving motion under opposing forces.

Important Points

• Force Opposing Motion: When a force is applied opposite to the direction of motion, it causes deceleration, which is reflected as a negative acceleration in calculations.
• Sign Convention: In physics problems, the direction of motion is often considered positive, and opposing forces or velocities are assigned negative values.
• Real-Life Example: A car braking to a stop exemplifies deceleration. If the braking force is strong enough, the car can even reverse its direction, similar to the object in this problem.
• Key Formula: Always remember the equation v = u + at for problems involving uniform acceleration or deceleration.
• Practical Implications: Understanding motion under opposing forces is crucial in designing safety systems like airbags, seat belts, and crash avoidance technologies.

The correct answer is diffusion.

Key Points

• Diffusion is the process by which particles of two different types of matter mix randomly due to their inherent kinetic energy and random motion.
• It occurs naturally without the need for external force or pressure, as particles move from regions of higher concentration to regions of lower concentration.
• This phenomenon is common in both gases and liquids, where molecules are free to move and spread.
• Examples: The fragrance of perfume spreading in a room or sugar dissolving in water are examples of diffusion.
• Diffusion is crucial in biological processes, such as gas exchange in the lungs (oxygen diffuses into blood, and carbon dioxide diffuses out).

Additional Information

• Effusion:
• Effusion refers to the movement of gas particles through a small hole without collisions between the particles.
• It is governed by Graham's Law, which states that the rate of effusion is inversely proportional to the square root of the molar mass of the gas.
• Example: Gas leaking from a punctured tire is an example of effusion.
• Intermixing:
• Intermixing is a general term referring to the mixing of two or more substances.
• Unlike diffusion, it does not specifically depend on random molecular motion.
• Intermixing can occur due to stirring, agitation, or external forces.
• Homogenisation:
• Homogenisation is a process used to create a uniform mixture where particles are evenly distributed.
• It typically involves mechanical or chemical methods, such as blending or emulsifying.
• Example: Milk is homogenized to distribute fat particles evenly throughout the liquid.
• Scientific Importance:
• Diffusion is a fundamental physical process that helps explain molecular behavior in gases, liquids, and even solids.
• It plays a vital role in numerous fields, including chemistry, biology, and environmental science.

Important Points

• Factors Influencing Diffusion:
• Temperature: Higher temperatures increase the kinetic energy of particles, speeding up diffusion.
• Concentration Gradient: Diffusion occurs faster when there is a greater difference in concentration between two regions.
• Medium: Diffusion is faster in gases, slower in liquids, and extremely slow in solids due to differences in molecular mobility.
• Real-Life Applications:
• Industrial Processes: Diffusion is used in distillation, purification, and chemical reactions.
• Environmental Science: Diffusion explains the dispersion of pollutants in air and water.
• Medical Science: Oxygen and drug molecules diffuse into cells for vital functions.

The correct answer is Ingestion → Digestion → Absorption → Egestion.

Key Points

• Amoeba, a unicellular organism, follows a specific nutrition process that involves four steps: Ingestion, Digestion, Absorption, and Egestion. This sequence ensures that the amoeba meets its nutritional requirements efficiently.
• Ingestion: The amoeba engulfs food particles through a process called phagocytosis. It extends its pseudopodia (temporary projections of the cell membrane) to surround the food and forms a food vacuole. Hence, this step is correct.
• Digestion: The food inside the vacuole is broken down into simpler substances with the help of enzymes secreted into the vacuole. These enzymes convert complex nutrients into simpler, absorbable forms. This step is correct.
• Absorption: The digested nutrients are absorbed into the cytoplasm of the amoeba and are used for various cellular processes such as growth and energy production. Hence, this step is correct.
• Egestion: The undigested waste material is expelled out of the cell by the reverse process of phagocytosis, where the cell membrane opens to release the waste. This step is correct.
• The sequence Ingestion → Digestion → Absorption → Egestion accurately describes the process of nutrition in amoeba, ensuring its survival and metabolic activities.

Additional Information

• Nutrition in Amoeba: Amoeba exhibits holozoic nutrition, which involves ingestion of solid food particles and their subsequent digestion, absorption, and egestion.
• Pseudopodia: Amoeba uses its pseudopodia to capture food. These are temporary extensions of the cytoplasm that help in locomotion and food capture.
• Food Vacuole: This is a temporary structure formed around the food particle during ingestion. It acts as the site for digestion where enzymes are secreted.
• Enzymes: The enzymes secreted into the food vacuole play a critical role in breaking down complex substances like carbohydrates, proteins, and lipids into simpler substances that can be absorbed.
• Egestion Process: The waste materials left after digestion are expelled through the cell membrane by a process called exocytosis.
• Energy Usage: The absorbed nutrients are used for various cellular activities such as reproduction, locomotion, and repair.

Important Points

• Incorrect Sequences: Other options provided in the question incorrectly describe the order of the nutrition process in amoeba. For example:
• Option 2: "Absorption → Ingestion → Digestion → Egestion" is incorrect because absorption cannot occur before ingestion.
• Option 3: "Diffusion → Ingestion → Digestion → Absorption" is incorrect because diffusion is not a part of the amoeba's nutrition process; it is a mechanism for gas exchange.
• Option 4: "Ingestion → Egestion → Absorption → Digestion" is incorrect as egestion cannot occur before digestion and absorption.
• Locomotion and Feeding: Amoeba's pseudopodia not only help in capturing food but also aid in its movement, making it a versatile survival tool.
• Significance: Understanding the amoeba's nutrition process helps in studying basic life processes and cellular mechanisms in unicellular organisms.

The correct answer is Change in the water content of leaf cells.

Key Points

• Mimosa pudica, commonly known as the "touch-me-not" plant, exhibits a phenomenon known as thigmonasty, where the leaves fold upon touch or mechanical stimulus.
• This movement results from a rapid change in the turgor pressure of cells in specialised structures called the pulvini at the bases of the leaflets and petioles.
• When stimulated, ions like potassium (K⁺) are pumped out of the cells in the pulvini, causing water to exit the cells via osmosis. This leads to a loss of turgidity in the cells, resulting in the folding of the leaves.
• This reaction is a defensive mechanism believed to protect the plant from herbivores or environmental stressors, as the sudden movement might deter predators.
• The change in water content is a hydraulic mechanism and occurs within seconds, making it one of the fastest plant movements observed in nature.

Additional Information

• Mechanism of Leaf Folding in Mimosa pudica:
• The specialized structure called the pulvinus contains motor cells that regulate leaf movement.
• When touched, electrical signals called action potentials are generated, triggering ion exchange and water movement.
• The outer cells lose water, causing them to shrink, while inner cells remain turgid, resulting in leaf folding.
• Role of Thigmonasty in Plant Survival:
• Thigmonasty acts as a defense mechanism by making the plant less appealing to herbivores.
• The folding reduces the surface area exposed to sunlight, potentially helping the plant conserve moisture under extreme heat.
• It may also help dislodge harmful insects or dirt particles from the leaves.
• Key Characteristics of Mimosa pudica:
• Mimosa pudica is a leguminous plant and belongs to the family Fabaceae.
• It is a perennial herb native to South and Central America but is now found in tropical and subtropical regions worldwide.
• The plant is known for its sensitive nature, with leaves responding not only to touch but also to temperature changes, vibrations, and light intensity.
• Comparison of Other Stimuli:
• Change in pH: While pH changes can affect cellular processes, they are not directly involved in the rapid folding of Mimosa leaves.
• Change in Light Intensity: Some plants exhibit nyctinasty (movements in response to light), but Mimosa's folding mechanism is unrelated to light changes.
• Change in Temperature: Although temperature can affect plant physiology, it does not trigger the rapid folding of Mimosa pudica leaves.

Important Points

• Electrochemical Signals: Mimosa pudica relies on electrochemical signaling for rapid responses, similar to the nervous system in animals. These signals travel at speeds of up to 10 cm/s.
• Hydraulic Mechanism: The folding of leaves is a result of hydraulic pressure changes and not due to muscular or mechanical forces.
• Scientific Study: The behavior of Mimosa pudica has been studied extensively in plant physiology to understand tropic and nastic movements.

The correct answer is It is energetically unfavourable to lose or gain 4 electrons..

Key Points

• Carbon's electronic configuration: Carbon has an atomic number of 6, and its electronic configuration is 1s² 2s² 2p². It has 4 valence electrons in its outermost shell.
• Energetic considerations for ionic bonding: To form an ionic bond, carbon would need to either gain 4 electrons to achieve a stable octet (resulting in a -4 charge) or lose 4 electrons (resulting in a +4 charge). Both processes require a tremendous amount of energy, making them highly energetically unfavourable.
• Covalent bonding as a solution: Instead of forming ionic bonds, carbon achieves stability by sharing its 4 electrons with other atoms through covalent bonds. This sharing allows carbon to complete its octet with minimal energy expenditure.
• Small atomic size: Carbon's small atomic size makes it suitable for forming stable covalent bonds by overlapping its orbitals with those of other atoms, enabling strong bond formation.
• Versatility of covalent bonding: Carbon forms covalent bonds with a wide variety of elements, including hydrogen, oxygen, nitrogen, and other carbon atoms. This property allows it to create an extensive range of compounds, including organic molecules essential for life.

Additional Information

• Carbon's tetravalency: Carbon is tetravalent, meaning it can form four covalent bonds. This property allows it to create a diverse array of compounds, ranging from simple molecules like methane (CH₄) to complex macromolecules like DNA.
• Allotropes of carbon: Carbon exists in several allotropes, including diamond, graphite, and fullerenes. These allotropes demonstrate the versatility of carbon's covalent bonding.
• Hybridization in carbon: Carbon exhibits different types of hybridization (sp, sp², sp³), allowing it to form single, double, and triple bonds. For example, sp³ hybridization occurs in methane, while sp² hybridization is seen in ethene.
• Importance in organic chemistry: Carbon's ability to form stable covalent bonds is the foundation of organic chemistry, which studies compounds primarily composed of carbon and hydrogen.
• Comparison with other elements: Unlike elements such as sodium or chlorine, which readily lose or gain one electron to form ionic bonds, carbon's intermediate position in the periodic table makes covalent bonding the most practical and energetically efficient option.

The correct answer is completely dissipated as heat.

Key Points

• In a purely resistive circuit, the electrical energy supplied by the battery is fully converted into heat energy. This is a fundamental property of resistive components, governed by Joule’s Law of Heating.
• According to Joule's Law, the heat energy generated in a resistive element is proportional to the square of the current, the resistance, and the time for which the current flows. The formula is: H = I²RT, where H is heat, I is current, R is resistance, and T is time.
• Resistors do not store energy in any form. They dissipate energy continuously as heat when current flows through them.
• Unlike inductors or capacitors, which can store energy as magnetic or electric fields respectively, resistors solely serve to oppose the flow of current and convert electrical energy into heat.
• This phenomenon is utilized in various applications, such as electric heaters, toasters, and incandescent light bulbs, where resistive elements are designed to generate heat intentionally.

Additional Information

• Energy Conversion in Resistive Circuits:
• In a resistive circuit, electrical energy is transformed directly into thermal energy due to the opposition to current flow caused by the resistor.
• The power dissipated as heat in the resistor is given by P = I²R = V²/R, where P is power, I is current, V is voltage, and R is resistance.
• Comparison with Other Components:
• Inductors: Store energy in the form of a magnetic field. The energy stored is given by E = ½LI², where L is inductance and I is current.
• Capacitors: Store energy in the form of an electric field. The energy stored is given by E = ½CV², where C is capacitance and V is voltage.
• Resistors: Do not store energy but dissipate it entirely as heat.
• Applications of Resistive Heating:
• Electric Heating Devices: Appliances like electric heaters, hair dryers, and ovens use resistive elements to convert electrical energy into heat.
• Incandescent Bulbs: Filaments in these bulbs are resistive and glow due to the heat generated, producing light.
• Fuses: Resistive heating is utilized in fuses to break the circuit when excessive current flows.
• Important Characteristics of a Resistive Circuit:
• Ohmic Behavior: Resistors obey Ohm's Law, where voltage (V) across the resistor is directly proportional to the current (I) flowing through it, i.e., V = IR.
• No Energy Storage: Unlike capacitors or inductors, resistors do not have the ability to store energy.
• Uniform Dissipation: The energy is uniformly dissipated as heat along the resistive element.

Important Points

• Option 1: Stored as magnetic energy – This is incorrect because only inductors can store energy as a magnetic field.
• Option 2: Stored as potential energy – This is incorrect because capacitors, not resistors, store energy as potential energy in the form of an electric field.
• Option 3: Completely dissipated as heat – This is correct as resistors convert all electrical energy into heat.
• Option 4: Converted only into light – This is incorrect. While resistors in devices like incandescent bulbs emit light, it is a by-product of the heat generated.

The correct answer is Medium.

Key Points

• Sound waves require a medium for propagation, as they travel by causing vibrations in the particles of a material medium such as solids, liquids, or gases.
• In a solid medium, particles are closely packed, allowing sound to travel faster compared to liquids and gases.
• Sound cannot travel through a vacuum, as there are no particles present to vibrate and transfer the sound energy. (This makes Option 3 incorrect).
• Unlike light waves, which can travel through a vacuum, sound waves are mechanical waves and depend on the interaction of particles in a medium. (This excludes Options 1, 2).
• The speed of sound varies depending on the medium. For example, sound travels at approximately 343 m/s in air, 1,480 m/s in water, and up to 5,000 m/s in steel.

Additional Information

• Characteristics of Sound Waves:
• Sound waves are longitudinal waves, meaning the particles of the medium vibrate parallel to the direction of wave propagation.
• They are also mechanical waves, requiring a medium such as air, water, or solids to transmit energy.
• Sound waves consist of compressions (regions of high pressure) and rarefactions (regions of low pressure).
• Role of Medium:
• The type of medium affects the speed, intensity, and quality of sound.
• Sound travels fastest in solids, slower in liquids, and slowest in gases, due to the density and elasticity of the medium.
• For example, sound travels faster in water (1,480 m/s) than in air (343 m/s).
• Sound in a Vacuum:
• In a vacuum, there are no particles to vibrate and carry sound waves, so sound cannot propagate.
• This principle is why astronauts in space require radio communication, as space is essentially a vacuum.
• Examples of Sound Propagation:
• In air: Sound from a speaker or a person talking travels through the air to reach your ears.
• In water: Marine animals like dolphins use sound to communicate over long distances underwater.
• In solids: Vibrations from a tuning fork can be felt through a table or metal rod.

Important Points

• Option 1: Magnet - A magnet is unrelated to sound propagation, as it affects magnetic fields, not mechanical waves.
• Option 2: Light Source - A light source emits electromagnetic waves, which can travel through a vacuum, unlike sound waves.
• Option 3: Vacuum - A vacuum lacks particles, making it impossible for sound to propagate through it.
• Option 4: Medium - A medium (solid, liquid, or gas) is essential for the propagation of sound waves, as it provides the particles needed for the transfer of energy.
• Option 5: None - This is incorrect, as a medium is mandatory for sound to travel.

The correct answer is Meiosis.

Key Points

• Meiosis is a type of cell division that occurs in germ cells (reproductive cells).
• It reduces the chromosome number by half, producing four haploid cells (gametes) from a single diploid parent cell.
• This process ensures that the offspring have the same chromosome number as the parent organism when gametes fuse during fertilization.
• Meiosis consists of two sequential stages: Meiosis I and Meiosis II, each with specific phases like prophase, metaphase, anaphase, and telophase.
• During Meiosis I, homologous chromosomes separate, and during Meiosis II, sister chromatids separate, resulting in the formation of four genetically distinct gametes.
• This process is crucial for maintaining genetic diversity through mechanisms like crossing over and independent assortment.

Additional Information

• Budding:
• This is an asexual form of reproduction seen in organisms like yeast and hydra.
• A new individual develops as an outgrowth (bud) on the parent organism and eventually detaches to live independently.
• It does not involve the reduction of chromosome number or the formation of gametes.
• Mitosis:
• Mitosis is a type of cell division in which a single parent cell divides to produce two genetically identical daughter cells.
• It occurs in somatic (non-reproductive) cells and is responsible for growth, tissue repair, and asexual reproduction.
• Unlike meiosis, mitosis maintains the original chromosome number (diploid) in daughter cells.
• Amitosis:
• Amitosis is a direct type of cell division without the formation of spindle fibers or distinct chromosomes.
• It occurs in some unicellular organisms like bacteria and certain protozoa.
• This process is much simpler and lacks the complexity of meiosis or mitosis.
• Role of Meiosis in Genetic Diversity:
• Genetic recombination occurs during crossing over in Prophase I of Meiosis I, which exchanges genetic material between homologous chromosomes.
• Independent assortment of chromosomes during Metaphase I leads to a variety of genetic combinations.
• These mechanisms contribute to evolutionary processes by introducing variability in populations.

Important Points

• Meiosis and Fertilization:
• Meiosis ensures that gametes (sperm and egg) have half the chromosome number (haploid).
• Fertilization restores the diploid chromosome number in the zygote, ensuring species consistency.
• Organisms Exhibiting Meiosis:
• Meiosis occurs in sexually reproducing organisms, including animals, plants, and fungi.
• In plants, meiosis leads to the formation of spores, which eventually develop into gametophytes.
• Medical Relevance:
• Errors in meiosis can lead to chromosomal abnormalities such as Down syndrome, Klinefelter syndrome, or Turner syndrome.
• Understanding meiosis is crucial for fertility treatments and genetic counseling.

The correct answer is remains the same.

Key Points

• When an object falls freely near the Earth, it is influenced only by the force of gravity. The motion remains in a straight line towards the center of the Earth, and the direction does not change. Hence, the direction of motion remains the same.
• Gravitational force acts vertically downward, pulling the object towards the Earth's surface. There is no force acting to change the direction of motion during free fall.
• The motion of a freely falling object is considered uniformly accelerated motion due to the constant acceleration caused by gravity (9.8 m/s² near the Earth's surface).
• In the absence of air resistance, the object moves along a vertical line, with no horizontal forces or components acting on it. Thus, the trajectory remains vertical throughout the fall.
• Options like "becomes horizontal," "changes continuously," or "becomes opposite to gravity" are incorrect as they contradict the principle of free-fall motion governed by gravity.

Additional Information

• Free Fall:
• Free fall is defined as the motion of a body where gravity is the only force acting on it.
• The acceleration due to gravity is denoted by g and has a standard value of approximately 9.8 m/s² on the Earth's surface.
• During free fall, the object's velocity increases linearly with time, while its position follows a quadratic relationship with time.
• The formula for the distance traveled during free fall is s = 1/2 g t², where s is the distance, g is the acceleration due to gravity, and t is time.
• Examples include a stone dropped from a height, a ball falling from a building, or a skydiver (without air resistance).
• Influence of Air Resistance:
• In real-world scenarios, air resistance can affect the motion of falling objects, causing deviations from ideal free fall.
• Without air resistance, all objects fall at the same rate regardless of their mass, as demonstrated by Galileo's famous experiment at the Leaning Tower of Pisa.
• Air resistance depends on factors such as the object's shape, surface area, and velocity.
• Misconceptions:
• It is a common misconception that heavier objects fall faster than lighter ones. In the absence of air resistance, objects of different masses fall at the same rate due to gravity.
• Another misconception is that the direction of motion might change during free fall. However, in the absence of external forces like wind or air resistance, the direction remains vertical and constant.
• Applications:
• Free fall concepts are used in calculating time of fall, velocity, and impact force in physics and engineering problems.
• It is also crucial in understanding planetary motion, satellite launches, and celestial mechanics.

The correct answer is bends towards the normal.

Key Points

• When a light ray passes from a rarer medium (like air) to a denser medium (like water or glass), it bends towards the normal. This phenomenon occurs due to the change in the speed of light as it moves between media of different optical densities.
• The bending of light at the interface of two media is governed by Snell's Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media.
• The refractive index of the denser medium is greater than that of the rarer medium, causing the light ray to slow down and bend towards the normal.
• The angle of refraction is always smaller than the angle of incidence when light travels from a rarer to a denser medium.
• Examples of this phenomenon include the bending of light when it passes from air into water, making objects submerged in water appear displaced or distorted.

Additional Information

• Snell's Law:
• Snell's Law is mathematically expressed as: n₁ sin θ₁ = n₂ sin θ₂, where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively.
• It explains the relationship between the angles and the refractive indices of the two media.
• Refractive Index:
• The refractive index is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum.
• For example, the refractive index of air is approximately 1.0003, while that of water is about 1.33, and glass ranges from 1.5 to 1.9.
• Applications of Refraction:
• Refraction is utilized in the design of optical instruments such as lenses, microscopes, and telescopes.
• It is also responsible for natural phenomena like the formation of rainbows and the apparent bending of objects in water.
• Critical Angle and Total Internal Reflection:
• When light travels from a denser medium to a rarer medium, at a certain angle called the critical angle, the refracted ray travels along the boundary.
• If the angle of incidence exceeds the critical angle, the light undergoes total internal reflection, a principle used in optical fibers and periscopes.

Important Points

• Option 2 (bends at a 90° angle): This is incorrect because light does not bend at a 90° angle unless it is incident at the critical angle and refracts along the boundary.
• Option 3 (bends away from the normal): This occurs when light travels from a denser medium to a rarer medium, not the other way around.
• Option 4 (does not bend at all): This is true only if the light ray is incident normally (perpendicular) to the surface, in which case it passes straight through without refraction.
• Understanding the concept of refraction is crucial in fields like optics, physics, and engineering, where precision in light manipulation is required.

The correct answer is Charcoal.

Key Points

• Charcoal burns without a flame because it is primarily made up of carbon. During combustion, it undergoes a process called smoldering combustion, which does not produce a visible flame but generates heat and light. Hence, the correct answer is Charcoal.
• Flames are typically the result of the combustion of volatile gases. Substances like petrol, wax, and LPG release significant amounts of volatile gases when heated, leading to the formation of a flame. However, charcoal does not release such gases in large quantities.
• Charcoal undergoes a slow oxidation process when it burns, producing carbon dioxide (CO₂) and heat without the presence of a flame. This is why it is commonly used in applications like barbeques and heating where a steady heat source is needed.
• Unlike petrol, wax, or LPG, charcoal does not contain hydrocarbons that vaporize and ignite, which is necessary for the creation of a flame. Instead, it burns steadily in its solid form.
• Substances like petrol, wax, and LPG produce flames because they involve the combustion of liquid or gaseous hydrocarbons, which are inherently different from the solid carbon structure of charcoal.

Additional Information

• Charcoal:
• Charcoal is produced by heating wood or organic matter in the absence of oxygen, a process called pyrolysis.
• It is primarily composed of carbon, with small amounts of ash and other impurities.
• Charcoal is widely used for cooking and heating because it provides a long-lasting and steady heat source without flames.
• It is also used in industries as a reducing agent in metallurgical processes.
• Petrol:
• Petrol, also known as gasoline, is a hydrocarbon-based liquid fuel derived from crude oil.
• When petrol burns, it vaporizes and ignites, producing a bright flame and significant heat.
• It is primarily used as a fuel for internal combustion engines in vehicles.
• Its high volatility makes it highly flammable, and it burns with a visible flame.
• Wax:
• Wax is a hydrocarbon compound used in candles and other applications. It melts when heated and releases vapors that ignite to produce a flame.
• The combustion of wax involves the release of energy in the form of light and heat.
• Wax candles are commonly used for illumination, decoration, and religious purposes.
• LPG (Liquefied Petroleum Gas):
• LPG is a mixture of propane and butane, which are hydrocarbons. It is stored as a liquid under pressure and vaporizes when released.
• When ignited, LPG burns with a flame, making it suitable for cooking, heating, and other applications.
• It is widely used as a clean and efficient energy source in households and industries.

Important Points

• Burning Process: Substances burn in different ways depending on their physical and chemical properties. For instance:
• Solid Fuels: Like charcoal, they burn without a visible flame due to the lack of volatile compounds.
• Liquid Fuels: Like petrol, they vaporize and burn with a flame due to the combustion of hydrocarbon vapors.
• Gaseous Fuels: Like LPG, they burn with a flame as they are already in the gaseous state and easily mix with oxygen for combustion.
• Flame and Combustion: A flame is produced when a combustible material releases vapors that mix with oxygen and ignite. The absence of such vapors in substances like charcoal results in flameless combustion.
• Applications: Charcoal is preferred in situations where controlled, flameless heat is required, while substances like petrol, wax, and LPG are used where a visible flame is needed.

The correct answer is Number of protons in an atom.

Key Points

• The atomic number of an element represents the number of protons present in the nucleus of an atom.
• The atomic number is a unique identifier for each element in the periodic table, meaning no two elements share the same atomic number.
• The number of protons in an atom determines its chemical properties and its position in the periodic table.
• For a neutral atom, the number of protons is equal to the number of electrons, ensuring electrical neutrality.
• The atomic number is denoted by the symbol Z in nuclear and atomic physics.

Additional Information

• Option 1: Number of Neutrons in an Atom
• The number of neutrons in an atom is not represented by its atomic number. Instead, neutrons are calculated by subtracting the atomic number from the mass number (Mass Number = Protons + Neutrons).
• The number of neutrons can vary among isotopes of the same element, leading to different atomic masses.
• Option 2: Mass Number of the Atom
• The mass number is the sum of protons and neutrons in the nucleus of an atom.
• Unlike the atomic number, the mass number is not unique to an element because it can vary for isotopes of the same element.
• Option 3: Total Number of Electrons and Protons
• The atomic number does not represent the combined total of electrons and protons. However, in a neutral atom, the number of electrons is equal to the number of protons.
• In ions, the number of electrons differs from the number of protons due to gain or loss of electrons.
• Option 4: Number of Protons in an Atom
• This is the correct answer as the atomic number directly corresponds to the number of protons in the nucleus of an atom.
• For example, the element Hydrogen has an atomic number of 1, meaning it has 1 proton, while Carbon has an atomic number of 6, meaning it has 6 protons.
• Importance of Atomic Number
• The atomic number determines the element's identity and its position in the periodic table.
• It is crucial in defining the element's chemical behavior and reactivity.

Important Points

• Periodic Table: The periodic table is organized based on the atomic number of elements, with elements arranged in increasing order of atomic number.
• Isotopes: Atoms of the same element with the same atomic number but different mass numbers are called isotopes. For example, Carbon-12 and Carbon-14 are isotopes of Carbon.
• Discovery: The concept of the atomic number was proposed by Henry Moseley in 1913 through his work on X-ray spectra.
• Neutral Atoms: In a neutral atom, the atomic number also indicates the number of electrons, which balance the positive charge of the protons.

The correct answer is Nucleus.

Key Points

• Nucleus is the control center of the cell and plays a critical role in cellular reproduction. It houses the cell's genetic material (DNA), which contains instructions for cell division.
• The process of cellular reproduction, including mitosis and meiosis, is regulated by the nucleus. It ensures the accurate replication and distribution of genetic material during cell division.
• The nucleus contains chromosomes composed of DNA and proteins, which are essential for genetic inheritance and cellular functionality.
• It also contains the nucleolus, which is involved in the synthesis of ribosomes, aiding in protein production necessary for cell growth and division.
• Any damage or mutation in the nucleus can lead to errors in cellular reproduction, which might result in diseases such as cancer.

Additional Information

• Lysosomes:
• Lysosomes are known as the "suicidal bags" of the cell due to their ability to break down cellular waste and damaged organelles.
• They play no direct role in cellular reproduction but are involved in autophagy (self-digestion) and maintaining cellular health.
• Endoplasmic Reticulum (ER):
• The ER is responsible for protein and lipid synthesis. It aids in cellular growth but does not directly participate in cellular reproduction.
• The smooth ER synthesizes lipids, while the rough ER assists in protein production through ribosomes.
• Plastids:
• Plastids, including chloroplasts, are found in plant cells and are involved in photosynthesis and storage.
• They do not directly contribute to cellular reproduction but play a role in energy production, supporting cell growth.
• Cellular Reproduction:
• Cellular reproduction is divided into two main types: mitosis (for somatic cells) and meiosis (for gametes).
• Both processes are tightly regulated by the nucleus, ensuring proper genetic material distribution.

Important Points

• Chromosomes:
• These are thread-like structures in the nucleus that contain DNA. Human cells typically have 46 chromosomes, arranged in 23 pairs.
• During cell division, chromosomes ensure genetic material is accurately copied and distributed.
• Cell Cycle:
• The cell cycle consists of Interphase (growth and DNA replication) and Mitotic Phase (division).
• The nucleus plays a vital role in regulating the cell cycle stages, especially during DNA replication in the S-phase of Interphase.
• Mutations:
• Errors in the nucleus during replication can lead to mutations, which may cause genetic disorders or cancer.
• Cellular mechanisms, such as DNA repair processes, work to prevent these errors.

The correct answer is Both Statements 1 and 2 are correct..

Key Points

• Statement 1: A fuse is connected in series in an electric circuit.
• A fuse is always connected in series with the electrical circuit it protects.
• This ensures that the fuse can effectively interrupt the flow of current to the entire circuit when there is an overload or short circuit.
• By being in series, any excessive current passes through the fuse first, protecting the devices downstream. Hence, Statement 1 is correct.
• Statement 2: A fuse wire melts when the current exceeds a safe limit.
• A fuse is designed with a wire of low melting point and specific current-carrying capacity.
• When the current in the circuit exceeds the safe limit, the heat generated by the excess current causes the fuse wire to melt.
• This action breaks the circuit and prevents damage to electrical appliances and fires caused by overheating. Hence, Statement 2 is correct.
• Both statements are true as they describe the purpose and working principle of a fuse in an electric circuit.

Additional Information

• Fuse:
• A fuse is a safety device used in electrical circuits to protect appliances from overloads and short circuits.
• It is made of a metal or alloy with a low melting point and high conductivity, typically tin-lead alloy or copper.
• Functions of a Fuse:
• It protects electrical devices by interrupting excessive current flow.
• Prevents overheating and potential fire hazards in the circuit.
• Ensures the safety of users by disconnecting the faulty circuit.
• Types of Fuses:
• Cartridge Fuse: Used in appliances and automotive circuits.
• Rewireable Fuse: Common in domestic wiring systems.
• Miniature Circuit Breaker (MCB): Modern alternative to traditional fuses, automatically trips during faults.
• Applications of Fuses:
• Used in domestic wiring systems to protect home appliances.
• Widely utilized in industrial machinery to prevent electrical hazards.
• Essential in automotive circuits to safeguard vehicle electronics.

Important Points

• Melting Point of Fuse Wire:
• The fuse wire is selected based on its low melting point so that it melts quickly during excessive current flow.
• Proper Rating:
• Fuses must have the correct current rating to provide effective protection; an undersized fuse may blow unnecessarily, while an oversized fuse may fail to protect the circuit.
• Maintenance:
• Regular inspection of fuse connections is necessary to ensure they are functioning properly and are not damaged.

The correct answer is A straight line passing through the origin with positive slope.

Key Points

• A particle starting from rest and moving with uniform acceleration experiences a constant rate of change in velocity over time.
• The mathematical relation between velocity (v), acceleration (a), and time (t) is given by the equation v = u + at, where u is the initial velocity. Since the particle starts from rest, u = 0. Thus, v = at.
• This equation shows that velocity (v) is directly proportional to time (t), resulting in a linear relationship with a positive slope determined by the acceleration (a).
• The graph of velocity versus time for a uniformly accelerated motion is a straight line passing through the origin, as the velocity is zero at time t = 0.
• Among the given options, only a straight line with a positive slope satisfies the condition of uniform acceleration. Hence, the correct graph is a straight line passing through the origin with a positive slope.

Additional Information

• Option 1: A parabola
• A parabolic velocity-time graph is characteristic of non-uniform acceleration, where the rate of change of velocity is not constant. This does not apply to the given scenario of uniform acceleration.
• Option 2: A straight line passing through the origin with negative slope
• A straight line with a negative slope represents uniform deceleration, where the velocity decreases linearly with time. This is not applicable as the particle is accelerating uniformly.
• Option 3: A horizontal line
• A horizontal velocity-time graph indicates a constant velocity, implying zero acceleration. This contradicts the condition of uniform acceleration given in the question.
• Uniform Acceleration:
• It refers to motion where the velocity of an object changes by an equal amount in equal intervals of time. Examples include freely falling bodies under gravity.
• Velocity-Time Graph:
• The slope of a velocity-time graph represents the acceleration. A straight line with a positive slope indicates constant acceleration, while a horizontal line indicates zero acceleration.
• For uniformly accelerated motion starting from rest, the velocity-time graph is always a straight line passing through the origin.

Important Points

• Key Equations of Motion:
• v = u + at: Relates velocity, initial velocity, acceleration, and time.
• s = ut + 1/2 at²: Describes the displacement of an object under uniform acceleration.
• v² = u² + 2as: Relates final velocity, initial velocity, acceleration, and displacement.
• Real-Life Applications of Uniform Acceleration:
• Free fall: Objects falling under gravity experience uniform acceleration (9.8 m/s² on Earth).
• Vehicle acceleration: When a car accelerates uniformly from rest, its velocity-time graph forms a straight line.
• Graphical Analysis:
• The area under the velocity-time graph represents the displacement of the object.
• The slope of the graph provides the acceleration of the motion.

The correct answer is 16u.

Key Points

• The molecular mass of HNO₃: The molecular mass of nitric acid (HNO₃) is given as 63u, where "u" represents atomic mass units.
• Composition of HNO₃: Nitric acid is composed of 1 hydrogen atom (H), 1 nitrogen atom (N), and 3 oxygen atoms (O).
• Atomic masses of hydrogen and nitrogen: The atomic mass of hydrogen (H) is 1u, and the atomic mass of nitrogen (N) is 14u.
• Mass contribution of oxygen: The total molecular mass of HNO₃ is calculated as follows:
  

Molecular mass = Mass of H + Mass of N + (3 × Mass of O)

Substituting the values: 63 = 1 + 14 + (3 × Mass of O).

• Solving for oxygen's atomic mass: Simplifying the equation:
  

63 = 15 + 3 × Mass of O

48 = 3 × Mass of O

Mass of O = 48 ÷ 3 = 16u.

• Conclusion: The atomic mass of oxygen is 16u.

Additional Information

• Atomic Mass Unit (u):
• An atomic mass unit (u) is a standard unit of mass that quantifies the mass of an atom or molecule. It is equivalent to 1/12th the mass of a carbon-12 atom.
• The atomic masses of elements are expressed in "u" and are based on their isotopic abundances.
• Atomic Mass of Common Elements:
• Hydrogen (H): 1u.
• Carbon (C): 12u.
• Nitrogen (N): 14u.
• Oxygen (O): 16u.
• Molecular Mass:
• The molecular mass of a compound is the sum of the atomic masses of all atoms in its molecular formula.
• For example, the molecular mass of water (H₂O) is calculated as:
  

Molecular mass = (2 × Mass of H) + (1 × Mass of O) = (2 × 1u) + (1 × 16u) = 18u.

• Applications of Atomic and Molecular Mass:
• Used in stoichiometric calculations to determine the proportions of reactants and products in chemical reactions.
• Helps in calculating molar masses, which are crucial for laboratory preparations and industrial processes.
• Essential for understanding molecular structures and their behavior in different states of matter.
• Oxygen in Nature:
• Oxygen is a vital element, making up approximately 21% of Earth's atmosphere.
• It is essential for respiration in living organisms and is a key component of water (H₂O).
• Oxygen is also involved in combustion and various industrial processes, such as steelmaking.

The correct answer is Lead.

Key Points

• The reactivity series is a list of metals arranged in order of their reactivity from most reactive to least reactive. It is a fundamental concept in chemistry used to predict the behavior of metals in reactions.
• Lead is placed just above Copper in the reactivity series. This is because lead is slightly more reactive than copper but less reactive than metals like iron, zinc, or aluminum.
• The placement of lead above copper in the reactivity series means that lead can displace copper from its compounds in certain chemical reactions, such as in displacement reactions.
• Copper, being lower in the series, is less reactive and does not displace hydrogen from acids or react with water under normal conditions. Lead, however, can react with certain acids, although it reacts more slowly compared to more reactive metals like zinc or aluminum.
• The reactivity series is critical in understanding metal extraction and corrosion. For example, lead is used in applications like batteries because of its moderate reactivity and resistance to corrosion, whereas copper is widely used in electrical wiring due to its low reactivity and excellent conductivity.

Additional Information

• Reactivity Series:
• The reactivity series includes metals like potassium, sodium, and calcium at the top (most reactive), and metals like gold and platinum at the bottom (least reactive).
• Metals higher in the reactivity series can displace those lower in the series from their compounds during chemical reactions.
• The series is determined based on reactions with water, acids, and displacement reactions.
• Characteristics of Lead:
• Atomic Number: 82.
• Symbol: Pb (from its Latin name "Plumbum").
• It is a soft, malleable metal with a bluish-white appearance when freshly cut but tarnishes to a dull gray upon exposure to air.
• Lead is used in batteries, radiation shielding, and weights. It is also resistant to corrosion due to the formation of an oxide layer.
• Characteristics of Copper:
• Atomic Number: 29.
• Symbol: Cu (from its Latin name "Cuprum").
• Copper is a reddish-brown metal known for its excellent thermal and electrical conductivity.
• It is used in electrical wiring, plumbing, and coinage.
• Copper is less reactive, does not corrode easily, and is often alloyed with other metals like zinc to form brass.
• Applications of the Reactivity Series:
• The reactivity series helps in predicting the outcome of displacement reactions.
• It is used in the extraction of metals from their ores. For example, highly reactive metals like aluminum are extracted via electrolysis, while less reactive metals like iron are extracted by reduction with carbon.
• The series also plays a role in understanding and preventing corrosion. Metals lower in the series, like copper, are less prone to corrosion compared to those higher, like iron.

Important Points

• Displacement Reactions: A more reactive metal displaces a less reactive metal from its compound. For example, lead can displace copper from copper sulfate solution.
• Corrosion Resistance: Metals lower in the series, like gold and copper, are less reactive and more resistant to corrosion, making them ideal for jewelry and electrical applications.
• Industrial Usage: Lead, due to its moderate reactivity, is extensively used in lead-acid batteries, while copper's low reactivity and high conductivity make it ideal for electrical wiring.
• Environmental Impact: Both lead and copper must be handled responsibly to prevent environmental contamination, as lead is toxic and copper can accumulate in aquatic ecosystems.

The correct answer is Nissl granules.

Key Points

• Nissl granules are specialized structures found in the cytoplasm of neurons, particularly in the soma (cell body) and dendrites.
• These granules are composed of rough endoplasmic reticulum (RER) and ribosomes, which are crucial for protein synthesis.
• Neurons require extensive protein synthesis to produce neurotransmitters, enzymes, and structural proteins essential for maintaining their function and communication.
• Proteins synthesized by Nissl granules are transported throughout the neuron, including the axon and dendrites, to support cellular activities.
• Nissl granules are absent in the axon and axon hillock, which is why protein synthesis predominantly occurs in the soma and dendrites.
• Hence, Nissl granules are vital for maintaining the structural and functional integrity of neurons.

Additional Information

• Axon:
• The axon is a long, slender projection of a neuron that conducts electrical impulses away from the cell body.
• It does not contain Nissl granules, so protein synthesis does not occur here.
• The axon is primarily involved in transmitting signals to other neurons, muscles, or glands.
• Proteins required in the axon are synthesized in the soma and transported via axoplasmic transport.
• Dendrites:
• Dendrites are branched extensions of the neuron that receive signals from other neurons.
• They contain Nissl granules, allowing limited protein synthesis to occur locally.
• Dendrites play a crucial role in integrating synaptic inputs and transmitting them to the soma.
• Synaptic Bulb:
• The synaptic bulb, also known as the synaptic terminal, is the end of the axon where neurotransmitters are released.
• It does not contain Nissl granules or engage in protein synthesis.
• Proteins and neurotransmitters used in the synaptic bulb are transported from the soma.
• Importance of Protein Synthesis in Neurons:
• Protein synthesis is essential for producing neurotransmitters, maintaining neuronal structure, and repairing damage.
• Neurons have a high metabolic demand, making efficient protein synthesis crucial for their survival and functionality.
• Disruption in protein synthesis can lead to neurological disorders such as Alzheimer's disease or Parkinson's disease.

Important Points

• Location of Nissl Granules:
• Found in the soma and dendrites but absent in the axon and axon hillock.
• This localization ensures that protein synthesis primarily occurs in the regions where it is most required.
• Functions of Nissl Granules:
• Involved in the synthesis of proteins, including enzymes and structural proteins.
• Play a role in neuronal repair and regeneration.
• Essential for maintaining neuronal communication and health.
• Role in Neurological Research:
• Nissl granules are studied extensively in neurobiology to understand neuronal health and diseases.
• Their presence and distribution are used as markers to identify neurons in histological studies.

Shortcut Trick

Total Voters = 1800

Valid Votes = 1800 × 0.80 × 0.80 = 1152

Ratio of X : Y = 7 : 2 (Total parts = 9)

Difference = (7 − 2)/9 × 1152 = 5/9 × 1152 = 5 × 128 = 640

∴ The correct answer is 640 votes.

Alternate Method

Given:

Total registered voters = 1800

Voters who did not vote = 20%

Invalid votes = 20% of votes cast

Calculations:

⇒ Votes cast = 1800 − (20% of 1800) = 1800 − 360 = 1440

⇒ Valid votes = 1440 − (20% of 1440) = 1440 − 288 = 1152

⇒ Total ratio parts = 7 + 2 = 9 units

⇒ 9 units = 1152

⇒ 1 unit = 1152 ÷ 9 = 128

⇒ Difference between X and Y = (7 − 2) units = 5 units

⇒ 5 units = 5 × 128 = 640

∴ The correct answer is 640 votes.

Additional Information

Successive Percentage Decrease

When a value decreases by a% and then b%, the final value is Initial × (100−a)/100 × (100−b)/100.

Ratio Distribution

To find the difference between two shares in ratio a:b, use Difference = (a−b)/(a+b) × Total Value.

Valid Votes Calculation

Valid Votes = Total Voters × (Voting %) × (100% − Invalid %). Always calculate invalidity on the casted votes.

Shortcut Trick

List the first 12 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.

Group them for faster addition: (2+3+5+7+13+17) + (11+19+31) + (23+29+37).

Calculation: 47 + 61 + 89 = 197.

∴ The correct answer is 197.

Alternate Method

Given: First 12 prime numbers.

Formula Used: Sum of numbers = n₁ + n₂ + n₃ + ... + n₁₂

⇒ Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and itself.

⇒ Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37

⇒ Sum = (2 + 3 + 5 + 7) + (11 + 13 + 17 + 19) + (23 + 29 + 31 + 37)

⇒ Sum = 17 + 60 + 120

⇒ Sum = 197

∴ The correct answer is 197.

Additional Information

Definition of Prime Numbers

A prime number is a natural number greater than 1 that has exactly two factors: 1 and the number itself. 2 is the only even prime number.

Sum of First 'n' Primes

There is no simple arithmetic formula for the sum of the first n prime numbers; they must be listed and added individually.

Prime Numbers up to 100

There are exactly 25 prime numbers between 1 and 100. The sum of prime numbers from 1 to 50 is 328.

Given:

$\frac{8^{21} \div 4^{28} \times 2^5}{(4^{14})^{1/7} \times 8^2}$

Concept Used:

$\text{Laws of exponents}$

Calculation:

$\Rightarrow \frac{(2^3)^{21} \div (2^2)^{28} \times 2^5}{((2^2)^{14})^{1/7} \times (2^3)^2}$

$\Rightarrow \frac{2^{63} \div 2^{56} \times 2^5}{(2^{28})^{1/7} \times 2^6}$

$\Rightarrow \frac{2^{63-56} \times 2^5}{2^{28/7} \times 2^6}$

$\Rightarrow \frac{2^7 \times 2^5}{2^4 \times 2^6}$

$\Rightarrow \frac{2^{12}}{2^{10}}$

$\Rightarrow 2^{12-10}$

$\Rightarrow 2^2$

$\Rightarrow 4$

∴ The correct answer is $4$

Shortcut Trick

The age difference between two people remains constant over time.

Difference = 30 years. After 5 years, Father : Son = 2 : 1.

Ratio difference (2 − 1) = 1 unit → 30 years.

Son's age after 5 years = 1 unit = 30 years.

Son's present age = 30 − 5 = 25 years.

∴ The correct answer is 25 years.

Alternate Method

Given: Father is 30 years older than Son. After 5 years, Father = 2 × Son.

Let Son's present age = x years.

Then Father's present age = (x + 30) years.

After 5 years:

⇒ Son = x + 5

⇒ Father = (x + 30) + 5 = x + 35

According to the condition:

⇒ x + 35 = 2 × (x + 5)

⇒ x + 35 = 2x + 10

⇒ 2x − x = 35 − 10

⇒ x = 25

∴ The correct answer is 25 years.

Additional Information

Constant Age Difference

The numerical difference between the ages of two persons never changes regardless of the number of years passed.

Ratio Method for Ages

If the ratio of ages at two different times is given, equate the parts representing the constant difference to find the value of 1 unit.

Verification

If son is 25, father is 55. In 5 years, son is 30 and father is 60. Since 60 = 2 × 30, the answer is verified.

Shortcut Trick

Direct Calculation: x × 0.13 × 0.16 = 59.28

x = 59.28 ÷ (0.13 × 0.16)

x = 59.28 ÷ 0.0208

x = 2850

∴ The correct answer is 2850.

Alternate Method

Given:

13% of 16% of x = 59.28

Formula Used:

Percentage Calculation: (a/100) × (b/100) × x = Value

Calculations:

⇒ (13/100) × (16/100) × x = 59.28

⇒ (13 × 16 × x) / 10000 = 59.28

⇒ 208 × x = 59.28 × 10000

⇒ 208 × x = 592800

⇒ x = 592800 ÷ 208

⇒ x = 2850

∴ The correct answer is 2850.

Additional Information

Successive Percentage

When multiple percentages are applied one after another, it is a chain multiplication of fractions: x × (a/100) × (b/100).

Percentage to Fraction

To simplify calculations, convert common percentages to fractions. For example, 25% = 1/4 and 20% = 1/5.

Base Change Rule

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

Shortcut Trick

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

Free Items = 29, Total Items = 2 + 29 = 31

Discount % = (29 ÷ 31) × 100 ≈ 93.55%

∴ The correct answer is 93.55%.

Alternate Method

Given: Items bought = 2, Items free = 29.

Let the Marked Price (MP) of each item be ₹1.

⇒ Total Marked Price of 31 items = ₹31

⇒ Selling Price (SP) (Price paid for 2 items) = ₹2

⇒ Discount Amount = MP − SP = 31 − 2 = ₹29

⇒ Discount % = (Discount ÷ MP) × 100

⇒ Discount % = (29 ÷ 31) × 100

⇒ Discount % = 93.548...

∴ The correct answer is 93.55%.

Additional Information

Buy X Get Y Free Concept

In such offers, the discount is calculated on the total number of items (X+Y) because the shopkeeper gives away (X+Y) items but charges only for X items.

Successive Discounts

If two discounts d₁% and d₂% are given, the effective discount is [d₁ + d₂ − (d₁×d₂)/100]%.

Marked Price (MP) vs Selling Price (SP)

Discount is always calculated on the Marked Price. SP = MP × (100 − Discount%)/100.

Shortcut Trick

Cost Price of 15 notebooks = ₹1,200 ⇒ CP of 1 notebook = 1200 ÷ 15 = ₹80.

Total Loss = CP − SP = 1,200 − 1,040 = ₹160.

Number of notebooks = Total Loss ÷ CP of one notebook = 160 ÷ 80 = 2.

∴ The correct answer is 2.

Alternate Method

Given:

Total Cost Price (CP) of 15 notebooks = ₹1,200

Total Selling Price (SP) of 15 notebooks = ₹1,040

Formula Used:

Loss = CP − SP

Loss = n × CP per unit (where n is the number of notebooks)

Calculations:

⇒ Cost Price (CP) of 1 notebook = 1200 ÷ 15 = ₹80

⇒ Total Loss = 1,200 − 1,040 = ₹160

⇒ Let the loss be equal to the CP of x notebooks.

⇒ 160 = x × 80

⇒ x = 160 ÷ 80 = 2

∴ The correct answer is 2.

Additional Information

Loss and Profit Percentage

Loss% = (Loss ÷ CP) × 100 and Profit% = (Profit ÷ CP) × 100. Both are always calculated on the Cost Price unless specified.

Unitary Method in Profit & Loss

To find the price of a single item when the total is given, divide the Total Price by the Total Quantity.

Selling Price Calculation

When Loss occurs: SP = CP − Loss. When Profit occurs: SP = CP + Profit.

Shortcut Trick

Total Distance = Train Length + Platform Length = 1 + 3 = 4 units of Train Length.

Speed (m/s) = (4 × 275) ÷ 44 = 1100 ÷ 44 = 25 m/s.

Speed (km/hr) = 25 × (18/5) = 5 × 18 = 90 km/hr.

∴ The correct answer is 90 km/hr.

Alternate Method

Given:

Length of Train (L_T) = 275 m

Length of Platform (L_P) = 3 × 275 = 825 m

Time (t) = 44 seconds

Formula Used:

Speed = Total Distance ÷ Time

Total Distance = Length of Train + Length of Platform

Calculations:

⇒ Total Distance = 275 + 825 = 1100 m

⇒ Speed in m/s = 1100 ÷ 44

⇒ Speed in m/s = 25 m/s

⇒ Speed in km/hr = 25 × (18 ÷ 5)

⇒ Speed in km/hr = 5 × 18 = 90 km/hr

∴ The correct answer is 90 km/hr.

Additional Information

Unit Conversion

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

Relative Speed

When two objects move in the same direction, relative speed = S₁ − S₂. In opposite directions, relative speed = S₁ + S₂.

Train Crossing Stationary Objects

When a train crosses a pole/man, distance = length of train. When it crosses a bridge/platform, distance = length of train + length of object.

Shortcut Trick

Use the trigonometric identity: tan θ = sin θ ÷ cos θ.

⇒ tan 30° × cos 30° = (sin 30° ÷ cos 30°) × cos 30° = sin 30°.

We know that sin 30° = 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given: expression tan 30° × cos 30°.

Formula Used: Standard values of trigonometric ratios at 30°.

⇒ Value of tan 30° = 1/√3

⇒ Value of cos 30° = √3/2

⇒ Calculation: (1/√3) × (√3/2)

⇒ Result: 1/2

∴ The correct answer is 1/2.

Additional Information

Trigonometric Ratios of 30°

sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, cot 30° = √3, sec 30° = 2/√3, cosec 30° = 2.

Fundamental Identities

tan θ = sin θ / cos θ and sin²θ + cos²θ = 1.

Complementary Angles

sin 30° = cos 60° and tan 30° = cot 60°.

Shortcut Trick

Simplify A : B = 4 : 6 = 2 : 3.

Multiply the first ratio by the first term of the second ratio (7) and the second ratio by the second term of the first ratio (3).

A : B = 2×7 : 3×7 = 14 : 21

B : C = 7×3 : 10×3 = 21 : 30

∴ The correct answer is 14 : 21 : 30.

Alternate Method

Given:

A : B = 4 : 6

B : C = 7 : 10

To combine the ratios, the value of B must be the same in both.

⇒ LCM of 6 and 7 = 42

⇒ Multiply (A : B) by 7 → (4 × 7) : (6 × 7) = 28 : 42

⇒ Multiply (B : C) by 6 → (7 × 6) : (10 × 6) = 42 : 60

⇒ A : B : C = 28 : 42 : 60

⇒ Divide by 2 → 14 : 21 : 30

∴ The correct answer is 14 : 21 : 30.

Additional Information

Compound Ratio

The compound ratio of a:b and c:d is (ac) : (bd). It is obtained by multiplying the corresponding terms.

Duplicate and Sub-duplicate Ratio

The duplicate ratio of a : b is a² : b², while the sub-duplicate ratio is √a : √b.

Proportion Rule

If a, b, c, d are in proportion, then a × d = b × c (Product of extremes = Product of means).

Given:

x + 1/x = 13

Formula Used:

(x + 1/x)² = x² + 1/x² + 2

x² + 1/x² = (x + 1/x)² - 2

Calculation:

⇒ (13)² = 169

⇒ x² + 1/x² = 169 - 2

⇒ x² + 1/x² = 167

∴ The correct answer is 167.

Shortcut Trick

Relative Speed (Opposite) = 130 + 60 = 190 km/hr.

Convert to m/s = 190 × (5/18) = 475/9 m/s.

Total Distance (L₁ + L₂) = Relative Speed × Time = (475/9) × 12 = 1900/3 ≈ 633.33 m.

L₂ = 633.33 − 360 = 273.33 m.

∴ The correct answer is 273.33 m.

Alternate Method

Given: Speed of Train 1 (S₁) = 130 km/hr, Speed of Train 2 (S₂) = 60 km/hr, Length of Train 1 (L₁) = 360 m, Time (t) = 12 s.

Formula Used: Relative Speed = (L₁ + L₂) ÷ Time

⇒ Relative Speed = 130 + 60 = 190 km/hr

⇒ Relative Speed in m/s = 190 × (5/18) = 52.777... m/s

⇒ Distance = Speed × Time

⇒ 360 + L₂ = (190 × 5/18) × 12

⇒ 360 + L₂ = 190 × (5/3) × 2 = 190 × 10/3

⇒ 360 + L₂ = 1900/3

⇒ L₂ = 633.333... − 360

⇒ L₂ = 273.33 m

∴ The correct answer is 273.33 m.

Additional Information

Relative Speed Concept

When two objects move in opposite directions, their speeds are added (S₁ + S₂). When they move in the same direction, their speeds are subtracted (S₁ − S₂).

Unit Conversion

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

Train Crossing Case

When two trains of lengths L₁ and L₂ cross each other, the total distance traveled is always L₁ + L₂, regardless of the direction.

Shortcut Trick

Let the cost of a TV = 100. Then the cost of a Washing Machine (WM) = 50.

Initial Total Cost (6 WM + 2 TV) = (6 × 50) + (2 × 100) = 300 + 200 = 500.

Net change in total cost = [6 × (50 × 57%) + 2 × (100 × −74%)] = [171 − 148] = +23.

Percentage change = (23 ÷ 500) × 100 = 4.6%.

∴ The correct answer is Increase by 4.6%.

Alternate Method

Given:

⇒ Let Cost Price (CP) of a TV = ₹100.

⇒ CP of a Washing Machine (WM) = 50% less than TV = 100 − 50 = ₹50.

Calculations:

⇒ Total Initial Cost of 6 WM and 2 TV = (6 × 50) + (2 × 100) = 300 + 200 = ₹500.

⇒ New CP of WM = 50 + (57% of 50) = 50 + 28.5 = ₹78.5.

⇒ New CP of TV = 100 − (74% of 100) = 100 − 74 = ₹26.

⇒ Total New Cost = (6 × 78.5) + (2 × 26) = 471 + 52 = ₹523.

⇒ Change in Cost = 523 − 500 = ₹23.

⇒ Percentage Change = (23 ÷ 500) × 100 = 4.6%.

∴ The correct answer is Increase by 4.6%.

Additional Information

Percentage Change

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

Weighted Average Method

For multiple items, Net % Change = (n₁c₁ + n₂c₂ + ...) ÷ (n₁ + n₂ + ...) where n is quantity and c is percentage change.

Successive Percentage

If a value changes by a% then b%, the net change is [a + b + (ab/100)]%.

Shortcut Trick

For a right-angled triangle with legs 9 m and 12 m, the hypotenuse is 15 m (using the standard 3-4-5 triplet scaled by 3).

Area of right-angled ΔABC = (1/2) × 9 × 12 = 54 m².

The other triangle ΔACD has sides 13 m, 14 m, and 15 m. This is a standard 13-14-15 triangle whose area is always exactly 84 m².

Total Area of quadrilateral ABCD = 54 + 84 = 138 m².

∴ The correct answer is 138 m².

Alternate Method

Given:

Sides of quadrilateral ABCD: AB = 9 m, BC = 12 m, CD = 14 m, DA = 13 m

Angle ∠ABC = 90°

Formula Used:

Area of right-angled triangle = (1/2) × base × height

Heron's Formula for area of a triangle = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2 is the semi-perimeter.

Calculations:

In right-angled ΔABC, applying Pythagoras' theorem:

⇒ AC² = AB² + BC²

⇒ AC² = 9² + 12²

⇒ AC² = 81 + 144 = 225

⇒ AC = √225 = 15 m

Now, calculate the area of ΔABC:

⇒ Area(ΔABC) = (1/2) × AB × BC

⇒ Area(ΔABC) = (1/2) × 9 × 12 = 54 m²

For ΔACD, the sides are AC = 15 m, CD = 14 m, and DA = 13 m:

⇒ Semi-perimeter (s) = (15 + 14 + 13) ÷ 2 = 42 ÷ 2 = 21 m

Using Heron's Formula for Area(ΔACD):

⇒ Area(ΔACD) = √[21 × (21 − 15) × (21 − 14) × (21 − 13)]

⇒ Area(ΔACD) = √[21 × 6 × 7 × 8]

⇒ Area(ΔACD) = √[7056] = 84 m²

Total Area of quadrilateral ABCD:

⇒ Area(ABCD) = Area(ΔABC) + Area(ΔACD)

⇒ Area(ABCD) = 54 + 84 = 138 m²

Option Analysis:

⇒ Option (1) - 128 m²: Incorrect, as it does not equal the sum of the areas of the two constituent triangles.

⇒ Option (2) - 135 m²: Incorrect, which may arise from calculation errors in applying Heron's formula.

⇒ Option (3) - 148 m²: Incorrect, as the sum of the areas of ΔABC and ΔACD is exactly 138 m².

⇒ Option (4) - 138 m²: Correct, as the total area of the quadrilateral is the sum of the areas of ΔABC (54 m²) and ΔACD (84 m²).

∴ The correct answer is 138 m².

Additional Information

Pythagorean Triplets

A set of three positive integers a, b, and c such that a² + b² = c². Common triplets include (3, 4, 5), (5, 12, 13), (8, 15, 17), and their multiples.

Heron's Formula

Used to find the area of any triangle when the lengths of all three sides are known: Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter.

Standard 13-14-15 Triangle

A triangle with side lengths 13, 14, and 15 is a classic geometry problem favorite. Its semi-perimeter is always 21, and its area is always exactly 84.

Shortcut Trick

In an equilateral triangle, the sum of perpendicular distances from any point inside the triangle to its sides is always equal to the altitude (height) of the triangle.

This is known as Viviani's Theorem.

Height (h) = (√3 ÷ 2) × side

Sum of distances = h = (√3 ÷ 2) × 18 = 9√3 cm.

∴ The correct answer is 9√3 cm.

Alternate Method

Given: Side of the equilateral triangle (a) = 18 cm.

Formula Used: Height (h) = (√3 ÷ 2) × a; Distance of centroid from side = h ÷ 3.

⇒ Height (h) of triangle ABC = (√3 ÷ 2) × 18 = 9√3 cm.

⇒ In an equilateral triangle, the centroid (G) is also the incenter.

⇒ The perpendicular distance from the centroid to any side is the inradius (r).

⇒ r = h ÷ 3 = 9√3 ÷ 3 = 3√3 cm.

⇒ Sum of distances (p₁ + p₂ + p₃) = 3 × r = 3 × 3√3 = 9√3 cm.

∴ The correct answer is 9√3 cm.

Additional Information

Viviani's Theorem

For any point inside an equilateral triangle, the sum of its perpendicular distances to the sides is constant and equal to the altitude (h) of the triangle.

Inradius and Circumradius

In an equilateral triangle of side 'a', the inradius (r) is a ÷ (2√3) and the circumradius (R) is a ÷ √3. Note that R = 2r.

Area of Equilateral Triangle

The area of an equilateral triangle is given by the formula (√3 ÷ 4) × a², where 'a' is the side length.

Shortcut Trick

Sum of parallel sides = (3x + 1) + (2x + 2) = 5x + 3; Height = 3y = 3(x − 1).

Area = 1/2 × (Sum) × Height ⇒ 54 = 1/2 × (5x + 3) × 3(x − 1).

Test options for (5x + 3): If Sum = 18 units, then 5x + 3 = 18 ⇒ 5x = 15 ⇒ x = 3.

If x = 3, then Height = 3(3 − 1) = 6 units.

Area = 1/2 × 18 × 6 = 54 sq. units. This matches the given area.

∴ The correct answer is 18 units.

Alternate Method

Given:

Area of trapezium = 54 sq. units

Parallel sides (a, b) = (3x + 1) units and (2x + 2) units

Distance (h) = 3y units and x − y = 1 ⇒ y = x − 1

Formula Used:

Area of a trapezium = 1/2 × (sum of parallel sides) × height

Calculations:

⇒ Sum of parallel sides = (3x + 1) + (2x + 2) = 5x + 3

⇒ Height (h) = 3(x − 1)

⇒ 54 = 1/2 × (5x + 3) × 3(x − 1)

⇒ 108 = 3 × (5x + 3)(x − 1)

⇒ 36 = 5x² − 5x + 3x − 3

⇒ 5x² − 2x − 39 = 0

⇒ 5x² − 15x + 13x − 39 = 0

⇒ 5x(x − 3) + 13(x − 3) = 0

⇒ (x − 3)(5x + 13) = 0

⇒ x = 3 (Since x must be positive to keep lengths valid)

⇒ Sum of parallel sides = 5(3) + 3 = 18 units.

∴ The correct answer is 18 units.

Additional Information

Area and Median of Trapezium

The median (or midsegment) of a trapezium is parallel to the bases and its length is the average of the bases: Median = (a + b) / 2. Thus, Area = Median × Height.

Quadratic Roots Shortcut

For ax² + bx + c = 0, the roots are given by [−b ± √(b² − 4ac)] / 2a. In geometry, discard negative or non-physical roots.

Isosceles Trapezium

A trapezium where the non-parallel sides are equal is an isosceles trapezium, and its base angles are equal.

Shortcut Trick

Apply the relation: MP/CP = (100 + Profit%)/(100 − Discount%)

MP/CP = (100 + 62)/(100 − 40) = 162/60 = 2.7/1

Marked-up Percentage X = (2.7 − 1) × 100 = 170%

∴ The correct answer is 170.

Alternate Method

Given: Discount = 40%, Profit = 62%

Formula: Marked Price (MP) = CP × (1 + X/100) and Selling Price (SP) = MP × (1 − D/100)

⇒ Let Cost Price (CP) = 100

⇒ Since Profit = 62%, Selling Price (SP) = 100 + 62 = 162

⇒ Discount = 40%, so SP = 60% of MP

⇒ 162 = 0.60 × MP

⇒ MP = 162 / 0.60 = 270

⇒ Marked-up value (X) = MP − CP = 270 − 100 = 170

∴ The correct answer is 170.

Additional Information

Mark-up and Profit Relation

The effective percentage change after a mark-up (X%) and a discount (D%) results in Profit (P%), given by: P = X − D − (XD/100).

Successive Discounts

If two successive discounts d₁% and d₂% are given, the single equivalent discount is (d₁ + d₂ − d₁d₂/100)%.

No Profit No Loss

When SP = CP, it is called the break-even point. This occurs when the mark-up percentage is equal to the discount percentage applied on the marked price.

Shortcut Trick

To find the largest number that divides a set of numbers leaving the same remainder, find the HCF of their differences.

Differences: 703 − 555 = 148, 999 − 703 = 296, 1147 − 999 = 148.

HCF of (148, 296, 148) = 148.

∴ The correct answer is 148.

Alternate Method

Given:

Four numbers: 555, 703, 999, and 1147.

Formula Used:

The largest number dividing a, b, c, d leaving same remainder = HCF of (|a − b|, |b − c|, |c − d|).

Calculations:

⇒ First Difference = 703 − 555 = 148

⇒ Second Difference = 999 − 703 = 296

⇒ Third Difference = 1147 − 999 = 148

⇒ Required number = HCF of (148, 296, 148)

⇒ 148 = 148 × 1

⇒ 296 = 148 × 2

⇒ Therefore, HCF = 148

∴ The correct answer is 148.

Additional Information

HCF with Same Remainder Case

When the remainder is same but not given, the required greatest divisor is the HCF of the absolute differences between the numbers.

HCF with Different Remainders Case

If the remainders r₁, r₂, r₃ are different and given for numbers x, y, z, then the required greatest divisor is HCF of (x − r₁, y − r₂, z − r₃).

Shortcut Trick

Combined efficiency of A, B, and C = (1/2 + 1/6 + 1/12) of total work.

Total work done by them in 1 day = (6 + 2 + 1)/12 = 9/12 = 3/4 units.

Time taken for 1 unit of work = 4/3 days.

Time taken for thrice the work = 3 × (4/3) = 4 days.

∴ The correct answer is 4 days.

Alternate Method

Given: Time taken by A = 2 days, B = 6 days, C = 12 days.

Formula Used: Total Work = LCM of individual days; Efficiency = Total Work ÷ Days.

⇒ Total efficiency of A + B + C = 6 + 2 + 1 = 9 units/day.

⇒ Original work = 12 units.

⇒ Thrice the work = 3 × 12 = 36 units.

⇒ Time taken = New Total Work ÷ Total Efficiency

⇒ Time taken = 36 ÷ 9 = 4 days.

∴ The correct answer is 4 days.

Additional Information

Work and Efficiency Relation

Time and efficiency are inversely proportional. If a person finishes work in 'n' days, their 1-day efficiency is 1/n.

Combined Work Formula

For three people A, B, and C, time taken together is 1/[(1/A) + (1/B) + (1/C)] or (ABC)/(AB + BC + CA) days.

Multiple Work Factor

If the work is multiplied by 'k', the time taken by the same team also gets multiplied by 'k', keeping efficiency constant.

Shortcut Trick

B = sum of first 3 even natural numbers = 2 + 4 + 6 = 12.

For third proportional C of 16 and 12: 16 : 12 :: 12 : C.

C = (12 × 12) ÷ 16 = 144 ÷ 16 = 9.

∴ The correct answer is 9.

Alternate Method

Given: First value = 16, Second value = B, Third proportional = C.

Formula Used: If c is the third proportional to a and b, then a : b = b : c ⇒ c = b² ÷ a.

Calculations:

⇒ First three even natural numbers are 2, 4, and 6.

⇒ B = 2 + 4 + 6 = 12.

⇒ 16 : 12 = 12 : C

⇒ C = (12 × 12) ÷ 16

⇒ C = 144 ÷ 16

⇒ C = 9

∴ The correct answer is 9.

Additional Information

Mean Proportional

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

Fourth Proportional

If a, b, and c are three numbers, their fourth proportional d is calculated as a : b = c : d, which gives d = (b × c) ÷ a.

Sum of First n Even Numbers

The sum of the first n even natural numbers is given by the formula n(n + 1). For n=3, Sum = 3(3 + 1) = 12.

Shortcut Trick

Identify the modal class from the given mode 133 2/11, which lies in the interval 130 − 135.

Apply the simplified mode relation: Mode − L = [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h

3 2/11 = [ (14 − 7) ÷ (28 − 7 − X) ] × 5

35/11 = [ 7 ÷ (21 − X) ] × 5 ⇒ 35/11 = 35 ÷ (21 − X)

Comparing denominators: 11 = 21 − X ⇒ X = 10

∴ The correct answer is 10.

Alternate Method

Given:

Mode = 133 2/11 cm = 1465/11 cm

Modal Class = 130 − 135 (since the mode lies in this range)

Lower limit (L) = 130

Frequency of modal class (f₁) = 14

Preceding frequency (f₀) = 7

Succeeding frequency (f₂) = X

Class width (h) = 5

Formula Used:

Mode = L + [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h

Calculations:

⇒ 1465/11 = 130 + [ (14 − 7) ÷ (2 × 14 − 7 − X) ] × 5

⇒ 1465/11 − 130 = [ 7 ÷ (28 − 7 − X) ] × 5

⇒ (1465 − 1430) / 11 = 35 ÷ (21 − X)

⇒ 35/11 = 35 ÷ (21 − X)

⇒ 21 − X = 11

⇒ X = 21 − 11

⇒ X = 10

∴ The correct answer is 10.

Additional Information

Definition of Mode

The Mode is the value that appears most frequently in a data set. For grouped data, it is estimated using the modal class (the class with the highest frequency).

Empirical Relationship

For a moderately skewed distribution, the relationship between measures of central tendency is: Mode = 3 × Median − 2 × Mean.

Modal Class Property

The modal class is the interval with the highest frequency. In this problem, after finding X = 10, the frequency 14 remains the maximum, confirming 130-135 is the correct modal class.

Shortcut Trick

Ratio of two parts = 5 : 3

Difference in ratio units = 5 − 3 = 2 units

Actual difference given = 10

2 units = 10 ⇒ 1 unit = 5

Parts are 5 × 5 = 25 and 3 × 5 = 15

∴ The correct answer is 25 and 15.

Alternate Method

Given: Ratio = 5 : 3, Difference = 10

Method: Linear Equation

⇒ Let the two parts be 5x and 3x.

⇒ 5x − 3x = 10

⇒ 2x = 10

⇒ x = 5

⇒ First part = 5 × 5 = 25

⇒ Second part = 3 × 5 = 15

∴ The correct answer is 25 and 15.

Additional Information

Ratio and Proportion

A ratio compares two quantities by division. If a : b = c : d, then a, b, c, d are in proportion where ad = bc.

Unitary Method in Ratios

To find individual values, divide the total quantity or the difference by the sum or difference of the ratio parts respectively.

Shortcut Trick

Mode is defined as the observation that appears most frequently in the data set.

By inspection, the number 20 appears 4 times, which is more than any other number in the list.

Numbers like 14, 15, 16, and 18 appear only 2 times, while 17, 19, and 21 appear only once.

∴ The correct answer is 20.

Alternate Method

Given data: 20, 21, 15, 16, 20, 17, 18, 16, 15, 14, 19, 20, 18, 14, 20

Formula: Mode = Value with the Maximum Frequency

Frequency of each term:

⇒ 14 occurs 2 times

⇒ 15 occurs 2 times

⇒ 16 occurs 2 times

⇒ 17 occurs 1 time

⇒ 18 occurs 2 times

⇒ 19 occurs 1 time

⇒ 20 occurs 4 times

⇒ 21 occurs 1 time

⇒ Maximum frequency is 4, which belongs to the number 20.

∴ The correct answer is 20.

Additional Information

Mean (Arithmetic Average)

It is the sum of all observations divided by the total number of observations: Mean = Σx_i ÷ n.

Median

The middle value of a data set when it is arranged in ascending or descending order. For odd n, it is the ((n+1)/2)^th term.

Empirical Relationship

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Bimodal and Multimodal Data

If two or more values have the same maximum frequency, the data set is called bimodal (two modes) or multimodal (multiple modes).

Shortcut Trick

Difference in Annual Interest Rate = 7% − 3.5% = 3.5%

Total difference in Percentage Interest for 3 years = 3.5% × 3 = 10.5%

Positive difference in interest amount = 10.5% of ₹4,80,000

⇒ 4,80,000 × (10.5 ÷ 100) = 4,800 × 10.5 = 50,400

∴ The correct answer is ₹50,400.

Alternate Method

Given:

Principal (P) for both banks = ₹4,80,000

Rate of interest for Bank A (R₁) = 3.5% p.a.

Rate of interest for Bank B (R₂) = 7% p.a.

Time period (T) = 3 years

Formula Used:

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

Calculations:

⇒ SI paid to Bank A = (4,80,000 × 3.5 × 3) ÷ 100

⇒ SI paid to Bank A = 4,800 × 10.5 = ₹50,400

⇒ SI paid to Bank B = (4,80,000 × 7 × 3) ÷ 100

⇒ SI paid to Bank B = 4,800 × 21 = ₹1,00,800

⇒ Required Difference = ₹1,00,800 − ₹50,400

⇒ Required Difference = ₹50,400

∴ The correct answer is ₹50,400.

Additional Information

Simple Interest Properties

Simple Interest remains constant every year if the principal and rate are unchanged. Total SI = Principal × Net Rate %.

Interest Difference Concept

When Principal and Time are same, Difference in Interest = P × (R₂ − R₁) × T ÷ 100.

SI vs CI for 2 years

The difference between Compound Interest and Simple Interest for 2 years is given by P(R ÷ 100)².

Shortcut Trick

Time for 100% tank: A = 30 min, B = 20 min, A + B − C = 15 min.

Total Capacity (LCM of 30, 20, 15): 60 units.

Efficiencies: A = 2, B = 3, A + B − C = 4 ⇒ 2 + 3 − C = 4 ⇒ C = 1.

Emptying Time for 2/3 tank: (2/3 × 60) ÷ 1 = 40 minutes.

∴ The correct answer is 40 minutes.

Alternate Method

Given:

Pipe A fills 50% in 15 min ⇒ 100% in 30 min.

Pipe B fills 50% in 10 min ⇒ 100% in 20 min.

Combined time (A + B + C) = 15 min (C is emptying).

Calculations:

⇒ Efficiency of A + B = 2 + 3 = 5 units/min

⇒ Efficiency of (A + B − C) = 4 units/min

⇒ Efficiency of C = 5 − 4 = 1 unit/min

⇒ Capacity to be emptied = 2/3 × 60 = 40 units

⇒ Time taken by C = 40 ÷ 1 = 40 minutes

∴ The correct answer is 40 minutes.

Additional Information

Efficiency of Emptying Pipe

An emptying pipe always carries a negative efficiency in calculations when combined with filling pipes.

Time and Work Relation

Total Work = Time × Efficiency. When work is constant, time is inversely proportional to efficiency.

Pipes and Cisterns Logic

If a pipe fills 1/n part in 1 minute, the total time to fill the tank is n minutes.

The position and positional values or English Alphabets are:

Given:

QUT−XBA

<br>

YCB−FJI

The pattern followed here is:

Each letter-cluster pair follows the pattern +7, +7, +7.

Let's check each option:

1. IML–OTR

This letter-cluster pair does not follow the pattern +7, +7, +7.

2. ILJ–OSR

This letter-cluster pair does not follow the pattern +7, +7, +7.

3. ILJ–PTS

This letter-cluster pair does not follow the pattern +7, +7, +7.

4. IML–PTS

This letter-cluster pair follows the pattern +7, +7, +7.

Thus, IML−PTS follows the same pattern.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

Given: YPMG is related to VKJB in a certain way based on the English alphabetical order. In the same way, MVAM is related to JQXH. 

The logic followed here is:

Similarly,

Thus, DGRX is related to ABOS.

Hence, the correct answer is "Option 2".

The position and positional values or English Alphabets are:

Given: MP33 KN41 IL49 GJ57 ?

The logic followed here is:

Thus, EH65 will replace the question mark (?) in the following series to make it logically complete.

Hence, the correct answer is "Option 4".

Given: In a certain code language, ‘iron hip cheek’ is coded as ‘nn nu qw’ and ‘mud iron globe’ is coded as ‘nu yx mt’.

Thus, ‘iron’ is coded as "nu" in the given language.

Hence, the correct answer is "Option 4".

The position and positional values or English Alphabets are:

The pattern followed here is:

Each letter-cluster pair follows the pattern +7, -2.

Let's check each option:

1. PS–WR

This letter-cluster pair does not follow the pattern +7, -2.

2. SV–ZT

This letter-cluster pair follows the pattern +7, -2.

3. MP–TN

This letter-cluster pair follows the pattern +7, -2.

<br>

4. JM–QK

This letter-cluster pair follows the pattern +7, -2.

Thus, PS-WR is the odd one out.

 

Hence, the correct answer is "Option 1".

Given: Seven boxes, A, B, C, D, I, J and K, are kept one over the other, but not necessarily in the same order.

1. Only A is kept above I. 2. Only K is kept below J.
2. D is not kept immediately above C.
3. Only two boxes are kept between I and C. 

B, the only person left will occupy the vacant position.

Thus, two boxes are kept between B and K.

Hence, the correct answer is "Option 4".

Statements:

Some mangoes are bananas.

No banana is a kiwi.

The least possible Venn diagram for the given statement is:

<br> <br>

Conclusions:

(I): All bananas are mangoes. - False (Because, Some mangoes are bananas.)

(II): No mango is a kiwi. - False (There is no definite relation between mango and kiwi, it can be possible but it is not definite)

Thus, Neither follows.

Hence, the correct answer is "Option 3".

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

Now,

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

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

Thus, there are 3 even numbers, each of which is immediately preceded by an odd number and also immediately followed by an odd number.

Hence, the correct answer is "Option 1".

Given: A, B, C, E, F, P and Q are sitting around a circular table facing the centre.

1. A sits fourth to the left of C.
2. P sits fourth to the right of B.
3. Only two people sit between C and B when counted from the left of C.
4. E is not an immediate neighbour of P or B.
5. F is not an immediate neighbour of P.

Q, the only person left will occupy the vacant position.

Thus, F sits third to the right of Q.

Hence, the correct answer is "Option 2".

Given: 73 63 53 43 33 ?

The logic followed here is:

Thus, 23 should come in place of the question mark (?) in the given series.

Hence, the correct answer is "Option 1".

BODMAS Rule:

Given equation:  223 ÷ 4 + 164 × 2 − 37 = ?

Now, if ‘+’ and ‘−’ are interchanged and ‘×’ and ‘÷’ are interchanged, then:

⇒ 223 × 4 − 164 ÷ 2 + 37 = ?

⇒ 223 × 4 − 82 + 37 = ?

⇒ 892 + 37 − 82 = ?

⇒ 929 − 82 = ?

⇒ 847 = ?

Hence, the correct answer is "Option 1".

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

Now,

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

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

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

Hence, the correct answer is "Option 1".

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

Now,

Odd digits, each of which is immediately preceded by an odd digit and also immediately followed by an even digit.

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

Thus, there are 3 odd digits, each of which is immediately preceded by an odd digit and also immediately followed by an even digit.

Hence, the correct answer is "Option 4".

Given: P, Q, R, S, T and U are sitting around a circular table facing the centre.

1. R sits on the immediate left of U. 2. P sits second to the left of R.
2. S sits second to the right of T.

Q, the only person left will occupy the vacant position.

Thus, 1 people sit between R and Q when counted from the right of R.

Hence, the correct answer is "Option 2".

Statements:

All Zucchinis are Leeks.

All Leeks are Radishes.

Some Radishes are Celeries.

The least possible Venn diagram for the given statement is:

<br> <br>

Conclusions:

(I) All Leeks are Celeries. - False (There is no definite relation between leeks and celeries, it can be possible but it is not definite)

(II) Some Radishes are Zucchinis. - True (Because, All Zucchinis are Leeks and All Leeks are Radishes, which means some radishes being leeks are Radishes.)

Thus, Only conclusion (II) follows.

Hence, the correct answer is "Option 3".

Given: A, B, C, D, W, X and Y are sitting around a circular table facing the centre of the table.

1. A sits third to the left of X.
2. Only one person sits between X and C when counted from the left of C.
3. Only two people sit between B and W when counted from the right of W.
4. D is an immediate neighbour of B.

Thus, A sits third to the right of Y.

Hence, the correct answer is "Option 2".

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

Now,

Numbers, each of which is immediately preceded by a perfect square and also immediately followed by a perfect square.

So, the needed pattern is: Perfect Square — Number — Perfect Square

Thus, there are four numbers, each of which is immediately preceded by a perfect square and also immediately followed by a perfect square.

Hence, the correct answer is "Option 2".

Given: 20 25 15 20 10 ?

The logic followed here is:

Thus, 15 should come in place of the question mark (?) in the given series.

Hence, the correct answer is "Option 1".

The position and positional values or English Alphabets are:

Given: SGHE is related to XCMA in a certain way based on the English alphabetical order. In the same way, CYRW is related to HUWS. 

The logic followed here is:

Similarly,

Thus, MQBO is related to RMGK.

Hence, the correct answer is "Option 2".

The position and positional values or English Alphabets are:

The pattern followed here is:

Each letter-cluster pair follows the pattern +2, +1.

Let's check each option:

1. FC–HE

This letter-cluster pair does not follow the pattern +2, +1.

2. HE–JF

This letter-cluster pair follows the pattern +2, +1.

3. KH–MI

This letter-cluster pair follows the pattern +2, +1.

4. NK–PL

This letter-cluster pair follows the pattern +2, +1.

Thus, FC-HE is the odd one out.

Hence, the correct answer is "Option 1".