RRB Group D Official Paper (Held On: 06 Feb, 2026 Shift 2) (Previous Year Paper)

The correct answer is Accommodation.

Key Points

• The ability of the eye lens to adjust its focal length to focus on both near and distant objects is called Accommodation.
• This process is facilitated by the ciliary muscles, which can modify the curvature of the crystalline lens.
• When the ciliary muscles are relaxed, the lens becomes thin and its focal length increases, allowing the eye to see distant objects clearly.
• When the ciliary muscles contract, the lens becomes thicker and its focal length decreases, enabling the eye to focus on nearby objects.
• The power of accommodation for a normal human eye is limited; for a healthy adult, the closest distance an object can be seen clearly without strain is about 25 cm.
• Hence, the correct answer is Accommodation.

Additional Information

• Near Point and Far Point:
• The near point of the eye (least distance of distinct vision) is 25 cm for a normal young adult.
• The far point of a normal eye is infinity, meaning it can see objects at a very great distance clearly.
• Ciliary Muscles and Suspensory Ligaments:
• The lens is held in place by suspensory ligaments attached to the ciliary body.
• The tension in these ligaments changes based on the contraction or relaxation of the ciliary muscles, thereby altering the lens shape.
• Presbyopia:
• With age, the power of accommodation usually decreases as the ciliary muscles weaken and the crystalline lens loses its flexibility.
• This condition, where people find it difficult to see nearby objects comfortably and distinctly without corrective eye-wear, is called Presbyopia.
• Other Eye Functions:
• Adaptation: This refers to the ability of the pupil and iris to adjust to varying levels of light intensity.
• Reflection: This is the bouncing back of light when it hits a surface; it is not the primary mechanism of focusing within the biological lens of the eye.

Important Points

• Image Formation: The eye lens forms a real and inverted image of the object on the retina, which is a light-sensitive screen at the back of the eye.
• Refractive Power: The cornea actually performs about 70-80% of the total refraction (focusing) of the eye. The lens provides the final, adjustable 20-30% of the power required for fine-tuning.
• Optical Power Unit: The power of a lens is measured in Dioptres (D). The accommodation power of a healthy young human eye is approximately 4 Dioptres.
• Refractive Defects: When the eye cannot properly accommodate or focus light on the retina, it leads to defects like Myopia (nearsightedness), Hypermetropia (farsightedness), and Astigmatism.

The correct answer is 4.0 m/s.

Key Points

• The slope of a distance-time graph is a physical representation of the speed of an object. In physics, the slope is defined as the vertical change (rise) divided by the horizontal change (run).
• In this specific problem, the particle covers a distance of 12 m in every 3 s. This indicates that the particle is moving at a constant rate, which is a characteristic of uniform motion.
• To calculate the slope, we use the formula: Slope = Change in Distance / Change in Time. This is equivalent to the formula for speed: v = d / t.
• By substituting the given numerical values into the formula: Slope = 12 m / 3 s. Performing the division gives a result of 4.0 m/s.
• Since the slope is 4.0 m/s, it tells us that for every 1 second that passes, the particle moves exactly 4 meters. The linear nature of the graph (a straight line) confirms that the speed remains constant throughout the motion.

Additional Information

• Distance-Time Graph Fundamentals:
• On a distance-time graph, the time is always plotted on the x-axis (the independent variable), and the distance is plotted on the y-axis (the dependent variable).
• The unit for the slope in this graph is meters per second (m/s), which is the standard SI unit for speed.
• Types of Motion on Graphs:
• Uniform Motion: Represented by a straight line passing through the origin. The constant slope indicates a constant speed.
• Non-Uniform Motion: Represented by a curved line (parabola). A curve indicates that the object is accelerating or decelerating, meaning the speed is changing over time.
• Object at Rest: Represented by a horizontal line parallel to the time axis. In this case, the slope is 0, indicating a speed of 0 m/s.
• Speed vs. Velocity:
• While the slope of a distance-time graph gives the speed (a scalar quantity), the slope of a displacement-time graph provides the velocity (a vector quantity).
• Speed only considers magnitude, whereas velocity considers both magnitude and direction.
• Graphical Interpretation:
• A steeper line on the graph indicates a higher slope and thus a greater speed. Conversely, a line with a low angle of inclination indicates a slower speed.
• If the line is vertical, it would imply infinite speed (covering distance in zero time), which is physically impossible for a particle with mass.

Important Points

• The calculation of the slope 12 / 3 = 4 is a direct application of the rate of change concept in kinematics.
• In calculus terms, the speed is the first derivative of distance with respect to time (v = dd/dt). For linear graphs, this derivative is simply the constant slope of the line.
• Understanding distance-time graphs is crucial for analyzing real-world data, such as the average speed of vehicles, satellite orbits, or athletic performance.

The correct answer is It is a concave lens of focal length 25 cm..

Key Points

• The power of a lens is defined as the reciprocal of its focal length (f) in meters. It represents the degree of convergence or divergence of light rays falling on it.
• The formula to calculate the power of a lens is P = 1/f, where P is the power in Dioptres (D) and f is the focal length in meters (m).
• Given the power P = -4 D. To find the focal length, we use f = 1/P. Substituting the value, we get f = 1/(-4) = -0.25 meters.
• To convert the focal length from meters to centimeters, we multiply by 100. Thus, -0.25 m × 100 = -25 cm.
• According to the Cartesian sign convention, a negative sign in power and focal length indicates that the lens is a diverging lens or a concave lens. Hence, the statement that it is a concave lens of focal length 25 cm is correct.
• Conversely, a convex lens (converging lens) always has a positive focal length and therefore a positive power. Hence, options suggesting a convex lens are incorrect.

Additional Information

• SI Unit of Power: The standard unit of lens power is the Dioptre, denoted by the symbol D. One Dioptre is defined as the power of a lens whose focal length is exactly 1 meter (1 D = 1 per meter).
• Concave Lens Characteristics: A concave lens is thinner at the center than at the edges. It diverges parallel rays of light. The images formed by a concave lens are always virtual, erect, and diminished, regardless of the position of the object.
• Convex Lens Characteristics: A convex lens is thicker at the center than at the edges. It converges parallel rays of light to a point called the focus. It can form both real and virtual images depending on the object's distance from the lens.
• Calculation for other options:
• If the focal length were 2.5 m, the power would be 1/2.5 = 0.4 D.
• If the focal length were 0.4 m, the power would be 1/0.4 = 2.5 D.

Important Points

• Correction of Vision: In ophthalmology, power values are crucial. A person suffering from Myopia (nearsightedness) is prescribed a concave lens with negative power to move the image back onto the retina.
• A person suffering from Hypermetropia (farsightedness) is prescribed a convex lens with positive power to increase the convergence of light rays.
• Combination of Lenses: When two or more lenses are placed in contact, the net power (P) is the algebraic sum of the individual powers: P = P1 + P2 + P3.... This principle is used in designing camera lenses and microscopes to minimize aberrations.
• The focal length of a lens depends on the refractive index of the material and the radii of curvature of its two surfaces, as described by the Lens Maker's Formula.

The correct answer is Decrease in temperature.

Key Points

• Evaporation is a surface phenomenon where liquid molecules gain enough kinetic energy to escape into the air as vapor at any temperature below the boiling point.
• An Increase in surface area significantly enhances the rate of evaporation because more molecules are exposed to the atmosphere at the liquid-air interface. For example, spreading out wet clothes allows them to dry faster. Since this increases the rate, statement 1 is not the correct factor for this question.
• An Increase in wind speed helps in moving the water vapor particles away from the surface of the liquid, thereby reducing the concentration of vapor in the immediate surroundings and allowing more molecules to escape. Since this increases the rate, statement 2 is not the correct factor for this question.
• A Decrease in temperature results in lower kinetic energy for the liquid particles, making it harder for them to overcome the intermolecular forces and escape the surface. This directly leads to a slower evaporation process. Hence, statement 3 is correct as it does NOT increase the rate of evaporation.
• A Decrease in humidity increases the rate of evaporation because dry air has a higher capacity to absorb moisture than moist air. When the moisture content in the air is low, the gradient for evaporation is steeper. Since this increases the rate, statement 4 is not the correct factor for this question.

Additional Information

• Temperature: The rate of evaporation is directly proportional to the absolute temperature. Higher temperatures provide the latent heat of vaporization required for phase change.
• Humidity: This is the measure of water vapor present in the air at a given temperature. On a rainy day, when humidity is near 100%, the rate of evaporation is at its lowest, which is why clothes take longer to dry.
• Latent Heat: During evaporation, the liquid absorbs energy from the surroundings to change into a gas. This is why evaporation always leads to a cooling effect on the remaining liquid or the surface.
• Nature of Liquid: Different liquids have different intermolecular forces. Volatile liquids like acetone or petrol evaporate much faster than water at 25 degrees Celsius because they have weaker bonds.

Important Points

• Bulk vs. Surface Phenomenon: Unlike boiling, which is a bulk phenomenon where the entire mass of the liquid reaches the boiling point, evaporation occurs only at the surface.
• Cooling Applications: The principle of evaporation is used in desert coolers and earthen pots (Matkas). In a desert cooler, the evaporation of water absorbs heat from the air, providing a cool breeze during hot and dry summer months.
• Atmospheric Pressure: While not mentioned in the options, a decrease in atmospheric pressure also increases the rate of evaporation, as there is less pressure pushing down on the surface to prevent molecules from escaping.

The correct answer is The speed of the object is changing with time.

Key Points

• In a distance–time graph, the time is typically plotted on the x-axis and the distance covered by the object is plotted on the y-axis.
• The slope (or gradient) of a distance–time graph represents the speed of the object. If the graph is a straight line, the slope is constant, indicating that the object is moving with a constant speed. Hence, statement 1 is incorrect.
• When the graph is a curved line, it implies that the slope of the graph is changing at different points in time. Since the slope represents speed, a changing slope means the speed of the object is changing. Hence, statement 4 is correct.
• A change in speed over time is defined as acceleration (or deceleration). If the graph is curved, it indicates that the motion is accelerated. Hence, statement 2 is incorrect.
• If an object remains stationary, its distance does not change as time passes. This would result in a horizontal straight line parallel to the time axis. Hence, statement 3 is incorrect.

Additional Information

• Uniform Motion:
• An object is said to be in uniform motion when it covers equal distances in equal intervals of time.
• The distance–time graph for uniform motion is always a straight line passing through the origin (if starting from zero distance).
• In this case, the acceleration of the object is zero.
• Non-Uniform Motion:
• An object is in non-uniform motion when it covers unequal distances in equal intervals of time.
• The distance–time graph for this motion is a curve, such as a parabola, which indicates that the speed is either increasing or decreasing.
• Non-uniform motion is also referred to as accelerated motion.
• Instantaneous Speed:
• On a curved distance–time graph, the speed at any particular instant can be found by drawing a tangent to the curve at that point and calculating its slope.
• The steeper the curve, the higher the instantaneous speed of the object at that time.

Important Points

• The SI unit of speed is meters per second (m/s), while time is measured in seconds (s) and distance in meters (m).
• A distance–time graph can never go downwards because distance is a scalar quantity and can only increase or stay constant with time; it can never be negative.
• If the curve bends upwards (getting steeper), the object is speeding up (positive acceleration).
• If the curve starts to level off (getting flatter), the object is slowing down (negative acceleration or deceleration).

The correct answer is 2 – 1 – 4 – 3.

Key Points

• Fragmentation is a form of asexual reproduction where an organism physically splits into distinct segments, each of which can grow into a new individual.
• Statement 2: The process begins when a multicellular organism, such as a filamentous alga, reaches maturity and breaks down its body or filament into smaller pieces or fragments. Hence, statement 2 is the first step in the sequence.
• Statement 1: Once the body has fragmented, the cells within these fragments initiate the growth of lost body parts. This is achieved through the process of mitosis (equational cell division), where cells replicate to form the necessary biological structures. Hence, statement 1 is the second step in the sequence.
• Statement 4: As cellular division and differentiation continue, each fragment develops into a complete organism that is morphologically and genetically similar to the parent. Hence, statement 4 is the third step in the sequence.
• Statement 3: The cycle concludes when the mature organism lives independently, capable of performing all life functions and eventually undergoing fragmentation itself to repeat the reproductive cycle. Hence, statement 3 is the final step in the sequence.
• Therefore, the logical progression of fragmentation-based reproduction follows the sequence of breakdown, mitotic growth, full development, and independent maturity. Hence, the correct answer is 2 – 1 – 4 – 3.

Additional Information

• Fragmentation in Biology:
• This method of reproduction is common in organisms with simple body designs. Unlike complex animals, these organisms do not have highly specialized organ systems for reproduction.
• Example - Spirogyra: This is a green alga that grows as long filaments in freshwater. Upon maturity, the filament naturally breaks into small pieces, and each piece grows into a new Spirogyra plant.
• Example - Planaria: While often cited under regeneration, simple animals like Planaria (flatworms) can be cut into pieces, and each piece will grow into a complete organism.
• Asexual Reproduction Characteristics:
• It involves only a single parent, meaning there is no fusion of gametes or fertilization.
• The offspring produced are clones, maintaining the exact genetic makeup of the parent.
• This process allows for rapid population growth in favorable environmental conditions.

Important Points

• Fragmentation vs. Regeneration:
• While fragmentation is a deliberate reproductive strategy used by organisms like algae and fungi, regeneration is often an accidental process or a response to injury (e.g., a lizard regrowing its tail).
• In regeneration, specialized regenerative cells proliferate and undergo morphogenesis to form various cell types and tissues.
• Role of Mitosis:
• Mitosis is crucial because it ensures that every new cell in the developing fragment receives a complete set of chromosomes, preserving the species-specific characteristics.
• This process occurs in somatic cells and is the foundation for growth, repair, and asexual reproduction in eukaryotes.

The correct answer is Round seeds.

Key Points

• Gregor Mendel, often referred to as the Father of Genetics, conducted his primary experiments using Pisum sativum (the garden pea plant) in the 19th century.
• In this specific experiment, seed shape is the trait under observation. Mendel identified round seeds as the dominant trait and wrinkled seeds as the recessive trait.
• A pure-breeding plant is homozygous, meaning it carries two identical alleles for a gene. The pure-breeding round plant has the genotype RR, while the pure-breeding wrinkled plant has the genotype rr.
• When these two are crossed, the F1 generation (first filial generation) receives one allele from each parent, resulting in a heterozygous genotype of Rr.
• According to Mendel’s Law of Dominance, in a heterozygous individual, the dominant allele (R) completely masks the expression of the recessive allele (r).
• Consequently, every single seed produced in the F1 generation will exhibit the round phenotype, even though they carry the hidden gene for wrinkled seeds. Hence, the correct answer is Round seeds.

Additional Information

• Mendelian Inheritance Patterns:
• Monohybrid Cross: The cross mentioned is a monohybrid cross because it focuses on a single trait (seed shape). Mendel performed these crosses for seven different traits including plant height, flower color, and pod shape.
• F2 Generation: When F1 plants (Rr) are self-pollinated, they produce the F2 generation. In this stage, the recessive trait (wrinkled) reappears in a specific phenotypic ratio of 3:1 (3 round to 1 wrinkled).
• Key Genetic Terminology:
• Genotype: This refers to the genetic makeup of the organism (e.g., RR, Rr, or rr).
• Phenotype: This refers to the observable physical characteristics of the organism (e.g., round or wrinkled).
• Law of Segregation: This law states that during the formation of gametes, the alleles for each gene segregate from each other so that each gamete carries only one allele for each gene.
• Experimental Data:
• In his 1865 paper, Mendel reported that out of 7,324 seeds in the F2 generation, 5,474 were round and 1,850 were wrinkled, which closely approximates the 3:1 ratio.
• The genotypic ratio of the F2 generation is 1:2:1, representing one homozygous dominant (RR), two heterozygous (Rr), and one homozygous recessive (rr).

Important Points

• The Seven Traits Studied by Mendel:
• Seed Shape: Round (Dominant) vs. Wrinkled (Recessive).
• Seed Color: Yellow (Dominant) vs. Green (Recessive).
• Flower Color: Violet (Dominant) vs. White (Recessive).
• Pod Shape: Inflated (Dominant) vs. Constricted (Recessive).
• Pod Color: Green (Dominant) vs. Yellow (Recessive).
• Flower Position: Axial (Dominant) vs. Terminal (Recessive).
• Stem Height: Tall (Dominant) vs. Dwarf (Recessive).
• Punnett Square Analysis:
• The Punnett Square for the F1 cross involves a 2x2 grid. If we place the R and R gametes from the round parent on one side and r and r from the wrinkled parent on the other, all four squares result in the Rr genotype.

The correct answer is Copper(II) oxide (CuO) – black in colour.

Key Points

• When copper metal is heated in the presence of oxygen (air), a chemical reaction occurs where the copper atoms combine with oxygen atoms to form a new compound.
• The product formed is Copper(II) oxide, which has the chemical formula CuO. In this compound, the oxidation state of copper is +2.
• The surface of the reddish-brown copper metal becomes coated with a black-coloured layer during this process. This change in color is a clear indicator of a chemical change.
• The balanced chemical equation for this reaction is: 2Cu + O2 → 2CuO (upon heating).
• This reaction is classified as an oxidation reaction because copper gains oxygen to form its oxide. It is also a combination reaction as two reactants combine to form a single product.

Additional Information

• Copper(I) oxide (Cu2O): This is another oxide of copper where the oxidation state of copper is +1. It is typically red or brownish-red in colour. While it can be formed under specific conditions, the primary product of heating copper in excess air is the black Copper(II) oxide.
• Reactivity of Copper: Copper is a transition metal located in Group 11 of the Periodic Table. It is relatively low in the reactivity series, meaning it does not react with oxygen at room temperature, but requires high temperatures to undergo significant oxidation.
• Basic Nature: Copper(II) oxide is a basic oxide. When it reacts with dilute acids like hydrochloric acid (HCl), it forms copper salts (like copper chloride) and water.
• Difference from Corrosion: The black coating formed by heating is different from the greenish patina seen on copper structures (like the Statue of Liberty) over long periods. That green layer is basic copper carbonate, formed by the slow reaction of copper with moist carbon dioxide and oxygen in the atmosphere at ambient temperatures.
• Industrial Uses: Copper(II) oxide is used as a pigment in ceramics to produce blue, red, and green (and sometimes gray, pink, or black) glazes. It is also used as a catalyst in various chemical reactions and in the production of wood preservatives.

Important Points

• Reduction of CuO: If hydrogen gas is passed over the heated black Copper(II) oxide, the black coating turns brown again as the oxygen is removed, regenerating metallic copper. This is a redox reaction: CuO + H2 → Cu + H2O.
• Physical Properties: Copper(II) oxide is a solid at room temperature and is insoluble in water. It has a monoclinic crystal structure and is often found in nature as the mineral tenorite.
• Safety: While copper is an essential trace element for humans, Copper(II) oxide can be toxic if inhaled or ingested in large quantities, as it acts as an irritant to the respiratory tract and gastrointestinal system.

The correct answer is Potential Energy.

Key Points

• Potential Energy is defined as the energy stored in an object or a system due to its position, shape, or configuration. It is "potential" because it has the capacity to do work when the state of the object changes.
• There are various forms of potential energy, the most common being Gravitational Potential Energy. This depends on an object's mass, its height relative to a reference point, and the acceleration due to gravity. The mathematical expression is P.E. = mgh.
• Another significant form is Elastic Potential Energy, which is the energy stored in objects that can be compressed or stretched, such as a coiled spring, a stretched rubber band, or a drawn archer's bow. This energy is a result of the change in the configuration or shape of the object.
• Potential energy can also be Chemical (stored in bonds), Electrical, or Magnetic, all depending on the relative arrangement of parts within a system.
• Because it represents the energy of an object based on its state of rest and arrangement rather than its motion, it perfectly matches the definition provided in the question. Hence, the correct answer is Potential Energy.

Additional Information

• Kinetic Energy:
• This is the energy possessed by an object by virtue of its motion.
• Unlike potential energy, which is based on position, kinetic energy depends on the mass and the square of the velocity of the object (K.E. = 1/2 mv²).
• Examples include a speeding car, a flowing river, or a falling stone. Hence, this option is incorrect as it relates to motion, not position.
• Heat Energy:
• Also known as thermal energy, it is the internal energy of a system resulting from the microscopic motion (vibration and collision) of atoms and molecules.
• It flows from a body at a higher temperature to one at a lower temperature.
• It is typically measured in Joules or Calories. Hence, this option is incorrect as it refers to internal molecular kinetic energy.
• Muscular Energy:
• This is the energy derived from the metabolic processes within the muscles of living organisms.
• It allows animals and humans to perform physical work, such as lifting weights, walking, or running.
• It is a transformation of chemical energy obtained from food into mechanical energy. Hence, this option is incorrect as it is a biological energy source for activity.
• Mechanical Energy:
• The sum total of potential energy and kinetic energy in a system is called Mechanical Energy.
• According to the Law of Conservation of Energy, in an isolated system, energy can neither be created nor destroyed; it only transforms from one form (like potential) to another (like kinetic).

Important Points

• The S.I. Unit of energy (both Kinetic and Potential) is the Joule (J), named after the physicist James Prescott Joule.
• Energy is a scalar quantity, meaning it has magnitude but no direction.
• The Commercial Unit of energy is the kilowatt-hour (kWh), which is commonly used for electricity billing and is equal to 3.6 million Joules.
• A stretched bow possesses Elastic Potential Energy because of its shape/configuration; as soon as the arrow is released, this energy converts into the Kinetic Energy of the arrow.

The correct answer is The little bee.

Key Points

• Apis florea is popularly known as the little honey bee. It is one of the smallest species of honey bees found in the wild.
• This species is a member of the subgenus Micrapis and is primarily distributed across South and Southeast Asia, extending to parts of the Middle East.
• The little bee is characterized by its nesting habit of building a single, open-air comb. This comb is usually attached to the branches of shrubs, bushes, or small trees, rather than in dark cavities. Hence, the correct answer is "The little bee".
• Due to its small body size, which is approximately 7 to 10 mm in length, it is easily distinguished from larger honey bee species.
• The honey yield from Apis florea is relatively low, typically ranging from 500 grams to 1 kilogram per colony per year, but the honey is highly valued for its medicinal properties in traditional systems.

Additional Information

• The Giant Bee (Apis dorsata):
• Commonly known as the Rock Bee or Giant Honey Bee, it is the largest honey bee species found in India.
• It builds huge, single combs in open spaces, often hanging from high cliffs, tall trees, or ceiling corners of multi-story buildings.
• It is known for its ferocious nature and high honey production capacity, yielding up to 30 to 40 kg of honey per colony.
• The Asian Bee (Apis cerana):
• Also known as the Indian Hive Bee or Asiatic Honey Bee.
• Unlike Apis florea, this species is a cavity-nesting bee, making it suitable for domestication in wooden hives.
• It is widely used in apiculture across Asia, though it produces less honey compared to the European bee (Apis mellifera).
• The Dwarf Bee:
• While Apis florea is sometimes referred to as a dwarf honey bee, the term specifically distinguishes it alongside Apis andreniformis (the black dwarf honey bee).
• Apis andreniformis is generally darker and found in more tropical, humid regions of Southeast Asia.

Important Points

• Behavioral Traits: The little bee is unique because its dance language (used to communicate the location of food) is performed on the horizontal platform at the top of its comb, whereas other species perform it on vertical surfaces.
• Defense Mechanisms: These bees exhibit a collective behavior known as shimmering, where workers vibrate their wings in waves to intimidate predators like wasps.
• Migration: Apis florea is highly migratory and often shifts its colony location based on seasonal changes and the availability of floral nectar.
• Pollination: They play a critical role in the ecosystem as pollinators for various tropical crops and wild flora, contributing to biodiversity conservation.

The correct answer is Higher pitch.

Key Points

• The pitch of a sound is directly determined by its frequency of vibration. When the frequency of vibration of a sound source increases, the resulting sound will have a higher pitch.
• Frequency is defined as the number of oscillations or vibrations per unit of time (usually one second). It is measured in Hertz (Hz).
• A high-frequency vibration produces a sound that is shrill and has a high pitch, such as the sound of a whistle or a bird chirping.
• Conversely, a low-frequency vibration results in a sound with a lower pitch, such as the beating of a drum or the roar of a lion.
• It is important to note that pitch is a subjective perception of the frequency of sound waves by the human ear, whereas frequency is a physical, measurable quantity.

Additional Information

• Amplitude and Loudness:
• The loudness of a sound depends on the amplitude of the vibration. Loudness is proportional to the square of the amplitude of the vibration producing the sound.
• If the amplitude is large, the sound produced is loud; if the amplitude is small, the sound is feeble. Therefore, increasing frequency does not necessarily increase loudness.
• Human Audibility Range:
• The human ear can typically hear sounds with frequencies ranging from 20 Hz to 20,000 Hz (20 kHz).
• Sounds with a frequency below 20 Hz are termed infrasonic, while those above 20,000 Hz are termed ultrasonic.
• Characteristics of Sound Waves:
• Wavelength: The distance between two consecutive compressions or rarefactions in a sound wave.
• Time Period: The time taken by the wave to complete one full oscillation. It is the reciprocal of frequency (T = 1/f).
• Speed of Sound: The distance a point on the wave travels per unit time. In dry air at 20 degrees Celsius, the speed of sound is approximately 343 meters per second.
• Unit of Loudness:
• The loudness of sound is measured in decibels (dB). Normal conversation is typically around 60 dB, while sounds above 80 dB can become physically painful or harmful to the human ear over time.
• Quality (Timbre):
• This is the characteristic of sound that allows us to distinguish between two sounds of the same pitch and loudness, such as different musical instruments or human voices.

Important Points

• In general, the voice of a woman has a higher frequency and is shriller than that of a man, meaning it has a higher pitch.
• The voice of a man typically has a lower frequency of vibration, resulting in a lower pitch or deeper voice.
• This difference is primarily due to the length and thickness of the vocal cords; men usually have longer vocal cords (about 20 mm) compared to women (about 15 mm).

The correct answer is The sum of protons and neutrons.

Key Points

• The mass number (represented by the symbol A) of an element is defined as the total count of protons and neutrons present in the nucleus of an atom.
• Protons and neutrons are collectively referred to as nucleons because they occupy the central core of the atom.
• The mathematical representation of the mass number is A = Z + n, where Z represents the atomic number (the number of protons) and n represents the number of neutrons.
• Since electrons have a negligible mass (approximately 1/1840 of the mass of a proton), they are not included in the calculation of the mass number.
• For example, an atom of Aluminum has 13 protons and 14 neutrons; therefore, its mass number is 27.

Additional Information

• Atomic Number (Z):
• This refers strictly to the number of protons in the nucleus of an atom. It defines the identity of the element and its position in the Modern Periodic Table.
• In a neutral atom, the number of protons is exactly equal to the number of electrons.
• Isotopes:
• These are atoms of the same element that possess the same atomic number but different mass numbers.
• This variation occurs because isotopes have the same number of protons but a different number of neutrons.
• Example: Carbon has isotopes like Carbon-12 (6 protons, 6 neutrons) and Carbon-14 (6 protons, 8 neutrons), the latter of which is used in radiocarbon dating.
• Isobars:
• These are atoms of different chemical elements that have the same mass number but different atomic numbers.
• Example: Argon-40 and Calcium-40 both have a mass number of 40, despite having 18 and 20 protons respectively.
• Relative Atomic Mass:
• Unlike the mass number (which is always a whole number), relative atomic mass is often a decimal because it is a weighted average of all the naturally occurring isotopes of that element.

Important Points

• The discovery of the neutron by James Chadwick in 1932 was crucial for the accurate determination of mass numbers.
• The mass number is typically written as a superscript to the left of the chemical symbol (e.g., 12C), while the atomic number is written as a subscript (e.g., 6C).
• The nucleus contains nearly 99.9% of the total mass of the atom, concentrated in an extremely small volume compared to the total size of the atom, a fact first proposed by Ernest Rutherford in 1911.
• The stability of a nucleus is often determined by the neutron-to-proton ratio (n/p ratio); if this ratio is too high or too low, the atom may become radioactive.

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

Key Points

• Buoyant force, also known as upthrust, is the upward force exerted by a fluid (liquid or gas) that opposes the weight of an object immersed in it.
• All objects, regardless of their material, mass, or shape, experience a force of buoyancy when they are partially or fully immersed in a fluid. This force acts in the upward direction, contrary to the force of gravity. Hence, statement (i) is correct.
• The magnitude of the buoyant force is determined by the formula Fb = ρVg, where ρ (rho) represents the density of the fluid. Because the force is directly proportional to the density of the medium, an object will experience a greater upward force in a denser fluid (like saltwater) compared to a less dense fluid (like freshwater). Hence, statement (ii) is correct.
• According to Archimedes' Principle, the magnitude of the buoyant force is exactly equal to the weight of the fluid displaced by the body.
• The point at which the buoyant force acts on the body is called the centre of buoyancy, which coincides with the centre of gravity of the displaced fluid.
• Hence, the correct answer is Both (i) and (ii).

Additional Information

• Archimedes' Principle:
• This fundamental law of physics was discovered by the Greek mathematician Archimedes in the 3rd century BC.
• It explains why some objects float while others sink. If the buoyant force is equal to or greater than the weight of the object, the object will float; otherwise, it will sink.
• Density and Buoyancy:
• The density of a substance is its mass per unit volume. The SI unit for density is kg/m³.
• Pure water has a density of approximately 1000 kg/m³ at 4°C. Mercury, being much denser at 13,600 kg/m³, exerts a significantly higher buoyant force on objects.
• Practical Applications:
• Ship Design: Large ships made of steel float because they are designed to displace a volume of water whose weight is equal to the weight of the ship.
• Lactometers: These are instruments used to determine the purity of a sample of milk based on the principle of buoyancy.
• Hydrometers: These are used for determining the density of liquids by measuring how deep they sink in a fluid.
• Hot Air Balloons: They rise because the hot air inside the balloon is less dense than the surrounding cool air, creating a net upward buoyant force.

Important Points

• Factors affecting Buoyant Force:
• Volume of the immersed object: The larger the volume of the object submerged, the greater the buoyant force it experiences.
• Density of the fluid: As established, higher fluid density leads to a stronger upthrust.
• Acceleration due to gravity (g): In a zero-gravity environment (like free fall or outer space), the buoyant force becomes zero. On Earth, g is approximately 9.8 m/s².
• Apparent Weight:
• When an object is immersed in a fluid, it appears to lose weight. This apparent weight is equal to the actual weight of the object minus the buoyant force.
• This phenomenon is why it is easier to lift a heavy stone while it is still under water than to lift it once it reaches the surface.

The correct answer is Proper diet and prenatal care.

Key Points

• Proper diet and prenatal care are the most critical factors for ensuring the health of both the mother and the developing fetus during the child-bearing period. A balanced intake of nutrients supports fetal growth and prevents maternal depletion. Hence, statement 3 is correct.
• Prenatal care refers to the regular medical checkups a woman receives during pregnancy. These visits allow healthcare providers to monitor the health of the mother and baby, perform necessary screenings, and provide guidance on lifestyle and nutrition.
• High caffeine intake (Option 1) has been linked to increased risks of miscarriage and low birth weight. While moderate consumption is often considered acceptable, excessive intake is detrimental. Hence, statement 1 is incorrect.
• Skipping meals (Option 2) can lead to hypoglycemia and nutritional deficiencies, which negatively impact the mother's energy levels and the baby's development. Hence, statement 2 is incorrect.
• Smoking daily (Option 4) is extremely harmful as it restricts oxygen supply to the baby and is a leading cause of premature birth, birth defects, and Sudden Infant Death Syndrome (SIDS). Hence, statement 4 is incorrect.

Additional Information

• Nutritional Requirements: During pregnancy, the demand for specific nutrients like Folic Acid, Iron, Calcium, and Iodine increases significantly. Folic Acid is essential in the first trimester to prevent Neural Tube Defects (NTDs).
• Maternal Mortality Ratio (MMR): This is a key health indicator representing the number of maternal deaths per 1,00,000 live births. Improving institutional deliveries and prenatal care are primary strategies to reduce this ratio globally.
• Government Initiatives in India:
• Janani Suraksha Yojana (JSY): Launched in 2005, it is a safe motherhood intervention under the National Health Mission providing cash assistance for institutional delivery.
• Pradhan Mantri Matru Vandana Yojana (PMMVY): A maternity benefit program providing Rs. 5,000 to pregnant and lactating mothers for the first living child to compensate for wage loss and ensure nutrition.
• Pradhan Mantri Surakshit Matritva Abhiyan (PMSMA): Aimed at providing assured, comprehensive, and quality antenatal care on the 9th of every month.
• WHO Guidelines: The World Health Organization recommends a minimum of eight antenatal contacts to reduce perinatal mortality and improve the women's experience of care.

Important Points

• Anemia Prevention: Iron and Folic Acid (IFA) supplementation is a cornerstone of women's health in India, as anemia is a major cause of maternal complications and postpartum hemorrhage.
• Immunization: Pregnant women must receive Tetanus Toxoid (TT) or Td (Tetanus and adult diphtheria) vaccines to prevent neonatal tetanus.
• Weight Management: Monitoring Gestational Weight Gain (GWG) is crucial, as excessive gain can lead to gestational diabetes and hypertension, while insufficient gain leads to underweight newborns.
• Environmental Factors: Avoidance of second-hand smoke, alcohol, and exposure to heavy metals is just as vital as direct healthy habits for reproductive health.

The correct answer is Double displacement reaction.

Key Points

• A double displacement reaction, also known as a metathesis reaction, is a chemical process where the positive ions (cations) and negative ions (anions) of two ionic compounds switch places to form two entirely new compounds.
• The general skeleton of this chemical equation can be represented as AB + CD → AD + CB. In this formula, A and C are positively charged cations, while B and D are negatively charged anions.
• These reactions primarily occur in aqueous solutions (solutions where the solvent is water), allowing the ions to dissociate and move freely to interact with one another.
• A classic example of this reaction is the mixing of silver nitrate (AgNO3) and sodium chloride (NaCl), which yields a white precipitate of silver chloride (AgCl) and sodium nitrate (NaNO3) in solution.
• For a double displacement reaction to occur, one of the products formed is usually a precipitate (an insoluble solid), a gas, or a molecular compound like water.

Additional Information

• Combination Reaction:
• This is a reaction where two or more reactants (elements or compounds) combine to form a single product.
• It follows the general form A + B → AB. An example is the combustion of magnesium (Mg) in oxygen (O2) to produce magnesium oxide (MgO).
• Decomposition Reaction:
• In this reaction, a single reactant breaks down into two or more simpler products. This process usually requires energy in the form of heat (thermal decomposition), light (photolysis), or electricity (electrolysis).
• An example is the heating of calcium carbonate (CaCO3), which decomposes into calcium oxide (CaO) and carbon dioxide (CO2).
• Displacement Reaction:
• Also called a single displacement reaction, it involves a more reactive element pushing out or "displacing" a less reactive element from its compound.
• The reactivity series of metals is used to predict these reactions. For instance, Zinc (Zn) displaces copper (Cu) from a copper sulfate (CuSO4) solution because zinc is more reactive.

Important Points

• Precipitation Reaction: This is a specific type of double displacement where one of the formed products is insoluble in the solvent and settles down as a solid. This solid is referred to as the precipitate.
• Neutralization Reaction: This is another vital subtype of double displacement where an acid reacts with a base to form salt and water. For example, hydrochloric acid (HCl) reacts with sodium hydroxide (NaOH) to produce sodium chloride (NaCl) and H2O.
• In a double displacement reaction, the oxidation states of the elements involved typically remain constant, distinguishing them from Redox (reduction-oxidation) reactions where electrons are transferred.
• The driving force for these reactions is often the removal of ions from the solution through the formation of a gas, a solid, or a weak electrolyte.

The correct answer is Nostrils → Throat → Lungs → Alveoli.

Key Points

• The respiratory system in human beings is responsible for the exchange of gases, primarily Oxygen (O2) and Carbon Dioxide (CO2), between the body and the environment.
• Air enters the body through the Nostrils, where it is filtered by fine hairs and mucus to remove dust and foreign particles.
• From the nostrils, the air passes through the Nasal Passage into the Throat (which includes the Pharynx and Larynx). The Pharynx serves as a common passage for both food and air.
• The air then travels down the Trachea (windpipe) and enters the Lungs through the Bronchi and Bronchioles.
• The final destination of the inhaled air is the Alveoli, which are tiny balloon-like structures where the actual gas exchange takes place with the blood capillaries.
• Option 1 suggests air goes through the Food pipe and Heart. The food pipe (Oesophagus) belongs to the digestive system, and the heart is part of the circulatory system. Hence, statement 1 is incorrect.
• Option 2 correctly describes the flow from Nostrils to the Throat, then into the Lungs, and finally reaching the Alveoli. Hence, statement 2 is correct.
• Option 3 mentions the Stomach, which is a major organ of the Digestive System and not involved in the air pathway. Hence, statement 3 is incorrect.
• Option 4 mentions the Heart as a pathway for air, which is false, as the heart pumps blood, not air. Hence, statement 4 is incorrect.

Additional Information

• Pharynx and Larynx: The Pharynx leads to the Larynx (voice box). A muscular flap called the Epiglottis covers the opening of the Glottis during swallowing to prevent food from entering the windpipe.
• Trachea (Windpipe): It is a tube supported by C-shaped cartilaginous rings. These rings ensure that the air passage does not collapse even when there is little air in it.
• The Bronchial Tree: Inside the lungs, the Trachea divides into Primary Bronchi (right and left), which further branch into Secondary and Tertiary Bronchi, and eventually into thin Bronchioles.
• Mechanism of Alveoli: The Alveoli provide a very large surface area (approximately 80 to 100 square meters in an adult) for the diffusion of gases. The walls of the alveoli are extremely thin and surrounded by a vast network of blood vessels.
• Diaphragm: It is a large, dome-shaped muscle located at the base of the chest cavity. Its contraction increases the chest cavity volume, creating a vacuum that pulls air into the lungs (Inhalation).

Important Points

• Breathing vs. Respiration: Breathing is the physical process of Inhalation and Exhalation, whereas Respiration is a biochemical process where Glucose is broken down in cells to release Energy (ATP).
• Transport of Gases: Haemoglobin, the red pigment in Red Blood Cells (RBCs), has a high affinity for Oxygen and carries it from the lungs to the tissues. Carbon Dioxide is more soluble in water and is mostly transported in the dissolved form in the blood plasma.
• Diffusion: The exchange of gases at the alveolar site occurs through Simple Diffusion based on the pressure gradient or concentration gradient of the gases.
• Residual Volume: Even after forceful expiration, some amount of air always remains in the lungs, known as the Residual Volume, to ensure there is sufficient time for Oxygen to be absorbed.

The correct answer is Current flowing through the conductor.

Key Points

• The Right Hand Thumb Rule is a fundamental convention used in physics to visualize the direction of the magnetic field produced by an electric current.
• According to this rule, if a straight current-carrying conductor is held in the right hand with the thumb pointing in the direction of the current, then the fingers will curl around the conductor in the direction of the magnetic field lines.
• In the context of the question, if the curling of the fingers represents the magnetic field, it naturally follows that the thumb is aligned along the direction of the current flowing through the conductor. Hence, the correct answer is option 2.
• This rule indicates that the magnetic field lines around a straight wire are in the form of concentric circles whose centers lie on the wire.
• The magnitude of the magnetic field produced at any given point is directly proportional to the current passing through the wire and inversely proportional to the distance from the wire.

Additional Information

• Maxwell’s Corkscrew Rule: This is another name for the same principle. It states that if a right-handed corkscrew is rotated such that it advances in the direction of the current, the direction in which the handle rotates gives the direction of the magnetic field.
• Fleming's Left Hand Rule: This rule is used to find the direction of the magnetic force acting on a moving charge or a current-carrying wire. The thumb represents the Force (Motion), the forefinger represents the Magnetic Field, and the middle finger represents the Current. This is the working principle for Electric Motors.
• Fleming's Right Hand Rule: This rule is specifically applied to Electric Generators to determine the direction of induced current. Here, the thumb shows the direction of motion of the conductor, the index finger shows the field, and the middle finger shows the induced current.
• Application in Solenoids: When applying the Right Hand Thumb Rule to a solenoid (a coil of wire), the roles are often swapped for convenience; if the fingers curl in the direction of the current through the loops, the thumb points towards the Magnetic North Pole of the solenoid.
• Oersted's Experiment: In 1820, Hans Christian Oersted discovered that a compass needle deflects when placed near a wire carrying a current, providing the first evidence of electromagnetism.

Important Points

• Directional Sensitivity: If the direction of current is reversed, the direction of the magnetic field lines also reverses (e.g., from clockwise to counter-clockwise).
• Biot-Savart Law: While the Right Hand Thumb Rule gives the direction, the Biot-Savart Law provides the mathematical formula to calculate the intensity of the magnetic field generated by a current element.
• Force on a Conductor: It is important to distinguish this rule from the Lorentz Force. The Right Hand Thumb Rule describes the creation of a field, whereas Fleming's rules describe the interaction between a field and a current.

The correct answer is Mercury.

Key Points

• Metals are typically characterized by their high thermal conductivity and electrical conductivity due to the presence of free electrons that can move easily through the atomic lattice.
• Mercury (Hg) is a notable exception to this trend as it is a poor conductor of heat compared to most other metals. Hence, option 1 is correct.
• Mercury is the only metal that exists in a liquid state at room temperature (approximately 25°C), which significantly influences its physical and thermal properties.
• Iron (Fe) is a good conductor of heat and is widely used in industrial machinery and construction due to its durability and thermal properties. Hence, option 2 is incorrect.
• Aluminium (Al) is an excellent conductor of heat, which is why it is extensively used in kitchen utensils and heat sinks for electronic components. Hence, option 3 is incorrect.
• Copper (Cu) is one of the best thermal conductors among common metals, second only to silver, making it the primary choice for heat exchangers and electrical wiring. Hence, option 4 is incorrect.

Additional Information

• Thermal Conductivity: This is a physical property of a material that defines its ability to transfer heat. In metals, heat is transferred via lattice vibrations (phonons) and the kinetic energy of free-moving electrons.
• Mercury (Hg): With an atomic number of 80, it is used in thermometers, barometers, and fluorescent lamps. Despite being a poor heat conductor, its uniform coefficient of thermal expansion makes it ideal for measuring temperature.
• Lead (Pb): Along with Mercury, Lead is also considered a poor conductor of heat among the metal group. It is commonly used for radiation shielding and in lead-acid batteries.
• Best Conductors: Silver (Ag) is the most efficient conductor of both heat and electricity, followed by Copper (Cu) and Gold (Au).
• Metals in Periodic Table: Most metals are located on the left side and in the center of the Periodic Table, categorized as Alkali metals, Alkaline earth metals, Transition metals, and Post-transition metals.

Important Points

• Malleability and Ductility: Metals are generally malleable (can be beaten into thin sheets) and ductile (can be drawn into thin wires). Gold is the most malleable metal known to man.
• Sonority: Metals are sonorous, meaning they produce a distinct ringing sound when struck. This property is utilized in the manufacturing of musical instruments and bells.
• Metallic Lustre: Metals possess a shiny surface in their pure state, a quality known as lustre. This is due to the reflection of light by the delocalized electrons on the surface.
• Hardness: While most metals are physically hard, elements like Sodium (Na) and Potassium (K) are soft enough to be sliced with a knife.
• Melting and Boiling Points: Metals typically have high melting and boiling points. Tungsten (W) has the highest melting point, making it suitable for light bulb filaments.

The correct answer is Formation of complex sugars from simple sugars.

Key Points

• The Golgi apparatus, also known as the Golgi complex or Golgi body, plays a crucial role in the biosynthesis of various cellular components, specifically in the formation of complex sugars from simple sugars.
• In many cases, the Golgi apparatus is involved in the modification of proteins and lipids by adding sugar moieties, a process known as glycosylation. Hence, statement 2 is correct.
• The synthesis of nucleic acids (DNA and RNA) occurs in the nucleus, mitochondria, and chloroplasts, while ribosomes are the site for protein synthesis, not nucleic acid synthesis. Hence, statement 1 is incorrect.
• The absorption of water through osmosis is a physical process regulated by the plasma membrane and tonoplast (vacuolar membrane), and is not a synthetic role attributed to the Golgi apparatus. Hence, statement 3 is incorrect.
• The conversion of proteins into amino acids is a catabolic or degradative process (proteolysis) that typically occurs in the lysosomes or through proteasomes, rather than being a function of synthesis in the Golgi. Hence, statement 4 is incorrect.
• The Golgi apparatus acts as a dispatch station, where products of the Endoplasmic Reticulum (ER) are modified, packaged, and sorted for delivery to their destinations.

Additional Information

• Discovery and Structure:
• The Golgi apparatus was first identified by the Italian biologist Camillo Golgi in 1898 using a silver staining technique.
• It consists of a series of flattened, membrane-bound sacs called cisternae, which are typically stacked together in 4 to 8 layers.
• The structure has a distinct polarity, featuring a cis face (entry face near the ER) and a trans face (exit face near the plasma membrane).
• Primary Functions:
• Storage and Packaging: It stores proteins and lipids synthesized in the Endoplasmic Reticulum and packages them into vesicles for secretion.
• Modification: It modifies glycoproteins and glycolipids by trimming or adding specific sugar molecules.
• Lysosome Formation: The Golgi apparatus is directly involved in the production of lysosomes, which contain hydrolytic enzymes for intracellular digestion.
• Golgi in Plant Cells:
• In plant cells, the Golgi bodies are often scattered and are referred to as dictyosomes.
• They play a vital role in synthesizing complex polysaccharides like pectin and hemicellulose, which are essential components of the cell wall.
• Vesicular Transport:
• Transport vesicles from the ER fuse with the cis-Golgi, and after processing, secretory vesicles bud off from the trans-Golgi to move toward the cell surface or other organelles.

The correct answer is Produces identical daughter cells.

Key Points

• Mitosis is a process of cell division where a single cell divides into two genetically identical daughter cells. The primary purpose of mitosis is for growth and to replace worn-out cells.
• In Mitosis, the chromosomal number remains the same as that of the parent cell, which is why it is often referred to as equational division. This ensures that each daughter cell receives a complete and identical set of DNA. Hence, the statement "Produces identical daughter cells" is true for mitosis.
• Meiosis, on the other hand, is a specialized form of cell division that reduces the chromosome number by half, resulting in the production of haploid cells (gametes). These daughter cells are genetically distinct from the parent and each other. Hence, the statement "Results in haploid cells" is true for meiosis, not mitosis.
• Genetic variation is a hallmark of Meiosis due to processes like crossing over and independent assortment during Prophase I. Mitosis does not typically promote variation as it aims to produce clones. Hence, the statement "Promotes genetic variation" is true for meiosis, not mitosis.
• Mitosis occurs in somatic cells (all body cells except reproductive ones), whereas Meiosis occurs specifically in reproductive cells (germ cells) to produce sperm and eggs. Hence, the statement "Occurs in reproductive cells" is true for meiosis, not mitosis.

Additional Information

• Stages of Mitosis:
• Prophase: The chromatin condenses into visible chromosomes, and the nuclear envelope begins to break down.
• Metaphase: Chromosomes align at the metaphase plate (the cell's equator) to ensure even distribution.
• Anaphase: Sister chromatids are pulled apart toward opposite poles of the cell.
• Telophase: Two new nuclear envelopes form around the separated sets of chromosomes, followed by cytokinesis (division of the cytoplasm).
• Key Differences in Meiosis:
• Meiosis involves two successive rounds of division: Meiosis I and Meiosis II.
• During Meiosis I, homologous chromosomes pair up and exchange genetic material, a process known as recombination or crossing over.
• The end result of meiosis is four daughter cells, each containing a haploid (n) number of chromosomes, compared to the diploid (2n) nature of the parent cell.
• Biological Significance:
• Mitosis is essential for asexual reproduction in single-celled organisms and for tissue repair and development in multicellular organisms.
• Meiosis is critical for sexual reproduction, as it maintains the constant chromosome number across generations and increases biodiversity within a population.

Important Points

• Somatic Cells vs. Germ Cells: Somatic cells make up the bulk of an organism's body (e.g., skin, muscle, bone) and divide via mitosis. Germ cells are located in the gonads (testes and ovaries) and undergo meiosis to produce gametes.
• DNA Replication: In both mitosis and meiosis, DNA replication occurs only once during the S-phase of the interphase preceding the division.
• Ploidy: Mitosis maintains ploidy (e.g., 2n to 2n), while meiosis reduces ploidy (e.g., 2n to n).

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

 

Key Points

• A heterogeneous mixture is a type of mixture in which the components are not uniformly distributed throughout the volume and exist in distinct phases.
• Sand and water is a heterogeneous mixture because the sand particles do not dissolve in water. Instead, they remain suspended for a while and eventually settle at the bottom due to gravity, a process known as sedimentation. Hence, option 4 is the correct answer.
• Sugar solution is a homogeneous mixture (specifically a true solution) because the sugar molecules are distributed uniformly among the water molecules, and the mixture has a consistent composition throughout. Hence, option 1 is incorrect.
• Salt solution is also a homogeneous mixture. Once Sodium Chloride (NaCl) is dissolved in water, the individual ions are spread evenly, making it impossible to distinguish the salt from the water by simple observation. Hence, option 2 is incorrect.
• Air is considered a homogeneous mixture of various gases, primarily Nitrogen (78%) and Oxygen (21%), along with small amounts of Argon, Carbon dioxide, and water vapor. Under normal conditions, these gases are mixed thoroughly at the molecular level. Hence, option 3 is incorrect.

Additional Information

• Classification of Matter: Matter is generally classified into pure substances (elements and compounds) and mixtures. Mixtures are further divided into homogeneous and heterogeneous categories based on the uniformity of their composition.
• Homogeneous Mixtures:
• These possess a single phase.
• The particle size is typically less than 1 nanometer.
• They do not scatter a beam of light (no Tyndall effect).
• Examples include alloys (like Brass), vinegar, and tap water.
• Heterogeneous Mixtures:
• These consist of two or more phases.
• They include suspensions (large particles that settle) and colloids (smaller particles that stay dispersed).
• Suspensions have particle sizes larger than 1000 nanometers.
• Examples include oil and water, mixture of iron filings and sulfur, and muddy water.
• Physical Separation: Components of a heterogeneous mixture can usually be separated using physical methods such as filtration, decantation, centrifugation, or manual picking. For sand and water, filtration is a highly effective method to recover the solid sand.

Important Points

• Colloids: While they may appear homogeneous to the naked eye (like milk or blood), they are technically heterogeneous when viewed under an ultra-microscope because the particles are not truly dissolved.
• Solubility: The nature of the mixture (homogeneous vs. heterogeneous) often depends on the solubility of the solute in the solvent. Sand (silicon dioxide) is insoluble in water, leading to a heterogeneous result, whereas sugar is highly soluble.
• Proportions: Unlike compounds, which have a fixed chemical ratio, mixtures can be prepared in any proportion. You can have a mixture of 1 gram of sand in 1 liter of water or 100 grams; both remain heterogeneous.

The correct answer is Fleming’s left-hand rule.

Key Points

• Fleming’s left-hand rule is specifically used to determine the direction of the mechanical force (or motion) acting on a current-carrying conductor when it is placed in an external magnetic field.
• According to this rule, if we stretch the thumb, the forefinger, and the middle finger of the left hand such that they are mutually perpendicular to each other, each finger represents a specific vector.
• The Forefinger points in the direction of the Magnetic Field (from North to South pole).
• The Middle finger points in the direction of the Electric Current (flowing through the conductor).
• The Thumb then points in the direction of the Force or the Motion of the conductor. Hence, the correct answer is Fleming’s left-hand rule.
• This principle is the fundamental working basis of an Electric Motor, where electrical energy is converted into mechanical energy.

Additional Information

• Ohm’s Law:
• Formulated by Georg Simon Ohm in 1827, it states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, provided physical conditions remain constant.
• The formula is V = IR, where V is voltage, I is current, and R is resistance. It does not determine the direction of force in magnetic fields.
• Maxwell’s Corkscrew Rule:
• Also known as the Right-Hand Screw Rule, it is used to find the direction of the magnetic field produced by a current-carrying conductor.
• If a right-handed corkscrew is rotated so that it moves forward in the direction of the current, then the direction of rotation of the handle indicates the direction of the magnetic field lines.
• Right-Hand Thumb Rule:
• This rule is used to determine the direction of magnetic field lines around a straight current-carrying wire.
• If the thumb of the right hand points in the direction of the current, the curled fingers show the direction of the magnetic field.
• Fleming’s Right-Hand Rule:
• This rule is used to find the direction of the induced current when a conductor moves inside a magnetic field (phenomenon of Electromagnetic Induction).
• It is primarily applied in Electric Generators. In this rule, the thumb represents motion, the forefinger represents the field, and the middle finger represents the induced current.

Important Points

• Lorentz Force: The force experienced by a charge moving in a magnetic field is given by F = q(v × B). For a conductor of length L, the force is F = BIL sinθ.
• The force is maximum when the conductor is placed perpendicular (at 90 degrees) to the magnetic field.
• The force is zero when the conductor is placed parallel or anti-parallel (at 0 or 180 degrees) to the direction of the magnetic field.
• John Ambrose Fleming, a British physicist, devised these rules in the late 19th century to provide an easy way for students to remember the orientation of vectors in electromagnetism.

The correct answer is Johad.

Key Points

• A Johad is a traditional rainwater harvesting structure used in India, particularly in the state of Rajasthan. It is a crescent-shaped small earthen check dam built across a contour to capture and conserve rainwater runoff.
• The primary purpose of a Johad is to facilitate groundwater recharge. By holding rainwater, it allows the water to slowly percolate into the aquifers, thereby raising the water table in arid and semi-arid regions. Hence, the correct answer is Johad.
• This traditional technique was revived significantly in the Alwar district of Rajasthan during the 1980s and 1990s. The revival led to the restoration of several dead rivers, such as the Arvari River.
• The Tarun Bharat Sangh (TBS), an NGO led by Rajendra Singh (often called the "Waterman of India"), played a pivotal role in promoting Johads. Since 1985, thousands of these structures have been built across Rajasthan, transforming the local ecology and agriculture.
• Johads are community-driven projects that require low maintenance and use locally available materials like soil and stones, making them a sustainable solution for water scarcity.

Additional Information

• Hand pump:
• A hand pump is a manual device used to extract water from a well or an underground aquifer. It is a modern mechanical extraction method rather than a water harvesting technique.
• While widely used in rural India for providing drinking water, it relies on existing groundwater levels rather than contributing to their replenishment.
• Borewell:
• A borewell is a deep, narrow hole drilled into the earth to reach deep-seated groundwater. It typically uses electric or diesel pumps for water extraction.
• Unlike Johads, borewells are a modern intervention that can lead to the depletion of groundwater if overused without proper recharge mechanisms.
• Check dam:
• A check dam is a small, sometimes temporary, dam constructed across a swale, drainage ditch, or waterway to counteract erosion and reduce water velocity.
• While a Johad is technically a type of earthen check dam, the term "check dam" is a general civil engineering term, whereas "Johad" refers specifically to the traditional and cultural practice unique to Rajasthan.
• Other Traditional Methods in Rajasthan:
• Khadin: A long earthen embankment built across the lower hill slopes to collect runoff water for moisture-retention in agricultural fields.
• Tanka: An underground storage tank used to collect rainwater from rooftops or prepared catchment areas, primarily used for drinking purposes in the Thar Desert.
• Baori / Stepwell: Magnificent architectural structures built to reach the water table, serving as community reservoirs for centuries.

Important Points

• Rajendra Singh: Known as the Waterman of India, he won the Ramon Magsaysay Award in 2001 and the Stockholm Water Prize in 2015 for his work in water conservation and the revival of Johads.
• Environmental Impact: The use of Johads has been proven to increase forest cover, improve crop yields, and mitigate the effects of drought in the Arid Zone.
• Geographical Context: These structures are most effective in the Aravalli Range and the Western Rajasthan landscape, where the soil type allows for efficient seepage and recharge.

The correct answer is CaO + H2O → Ca(OH)2.

Key Points

• A combination reaction (also known as a synthesis reaction) is a reaction where two or more elements or compounds combine to form a single compound. The general form is A + B → AB.
• In the reaction CaO + H2O → Ca(OH)2, Calcium Oxide (quicklime) reacts vigorously with water to produce Calcium Hydroxide (slaked lime). Since two reactants combine to form one single product, it is a classic example of a combination reaction. Hence, statement 3 is correct.
• The reaction 2KClO3 → 2KCl + 3O2 involves a single reactant breaking down into two or more simpler products. This is categorized as a decomposition reaction. Hence, statement 1 is incorrect.
• The reaction Fe + CuSO4 → FeSO4 + Cu involves a more reactive element (Iron) displacing a less reactive element (Copper) from its salt solution. This is known as a single displacement reaction. Hence, statement 2 is incorrect.
• The reaction AgNO3 + NaCl → AgCl + NaNO3 involves an exchange of ions between two compounds to form two new compounds. This is a double displacement reaction, specifically a precipitation reaction as AgCl forms a white solid. Hence, statement 4 is incorrect.

Additional Information

• Combination Reactions:
• These reactions are often exothermic in nature because bonds are formed, which releases energy.
• Examples include the burning of coal (C + O2 → CO2) and the formation of water from hydrogen and oxygen (2H2 + O2 → 2H2O).
• In the CaO reaction mentioned above, a large amount of heat energy is released, making the container hot.
• Decomposition Reactions:
• These are the opposite of combination reactions. They require energy in the form of heat, light, or electricity to break bonds.
• Thermal decomposition occurs when heat is used (e.g., heating Calcium Carbonate). Electrolytic decomposition occurs via electricity (e.g., electrolysis of water).
• Displacement Reactions:
• These depend on the reactivity series of metals. A metal placed higher in the series can displace a metal placed lower.
• For example, Zinc can displace copper from copper sulphate because Zinc is more reactive than Copper.
• Double Displacement Reactions:
• Usually occur in aqueous solutions where ions "switch partners."
• Commonly result in the formation of an insoluble solid called a precipitate or the formation of stable molecules like water in neutralization reactions.

Important Points

• Calcium Oxide (CaO): Commonly known as quicklime or burnt lime. It is widely used in the manufacture of cement and glass.
• Calcium Hydroxide (Ca(OH)2): Known as slaked lime. A dilute solution of slaked lime is called lime water.
• White Washing: The slaked lime produced in the combination reaction is used for whitewashing walls. It reacts slowly with Carbon Dioxide (CO2) in the air to form a thin, shiny layer of Calcium Carbonate (CaCO3) after two to three days.
• Chemical Equations: To satisfy the Law of Conservation of Mass, chemical equations must be balanced, meaning the number of atoms of each element must be equal on both the reactant and product sides.

The correct answer is Presence of both oxygen and water (moisture).

Key Points

• Rusting of iron is a specific type of corrosion that occurs when Iron (Fe) comes into contact with Oxygen (O2) and Water (H2O) or moisture in the environment.
• The chemical process of rusting is an oxidation reaction. In this reaction, iron reacts with oxygen in the presence of water to form Hydrated Iron (III) Oxide, which is chemically represented as Fe2O3.nH2O.
• For rusting to take place, the presence of both oxygen and water is absolutely essential. If either of these components is missing, the rusting process cannot proceed. Hence, statement 3 is correct.
• If iron is placed in distilled water that has been boiled to remove dissolved gases and then sealed, it will not rust because of the lack of Oxygen. Hence, statement 2 is incorrect.
• Similarly, if iron is kept in dry air (for example, by using a drying agent like Anhydrous Calcium Chloride), it will not rust because of the absence of Moisture. Hence, statement 4 is incorrect.
• While Carbon dioxide (CO2) is present in the atmosphere, it is not a primary or essential requirement for the basic rusting process. Hence, statement 1 is incorrect.

Additional Information

• Prevention of Rusting:
• Galvanization: This is the process of depositing a thin layer of Zinc (Zn) on iron objects. Since zinc is more reactive than iron, it oxidizes first, protecting the underlying iron from corrosion.
• Alloying: Iron can be mixed with other elements like Chromium, Nickel, and Carbon to create Stainless Steel, which does not rust easily.
• Painting and Greasing: Applying a coat of paint, oil, or grease creates a barrier that prevents iron from coming into direct contact with air and moisture.
• Nature of Rust: Rust is a reddish-brown, flaky substance that is brittle. Unlike the patina on copper, rust does not protect the underlying metal; instead, it flakes off, exposing more iron to the environment, which eventually leads to the total destruction of the object.
• Acceleration Factors: The rate of rusting increases significantly in the presence of salts (electrolytes). This is why iron objects and ships rust much faster in marine environments or coastal areas where the air contains high levels of salt.
• Chemical Change: Rusting is considered a chemical change because a new substance (iron oxide) is formed, and the process cannot be easily reversed by physical means.

Important Points

• The approximate chemical equation for rusting is: 4Fe + 3O2 + 2xH2O → 2Fe2O3.xH2O.
• Humidity: Higher humidity levels in the atmosphere correlate directly with faster rusting rates because there is more water vapor available to facilitate the reaction.
• Economic Impact: Rusting causes massive economic losses globally every year, necessitating the constant maintenance and replacement of bridges, railway tracks, cars, and industrial machinery.

Shortcut Trick

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

Substituting a = x and b = 2:

(x − 2)² = (x)² − 2 × (x) × (2) + (2)²

= x² − 4x + 4

∴ The correct answer is x² − 4x + 4.

Alternate Method

Given: (x − 2)²

Formula Used: (a − b)² = a² − 2ab + b²

⇒ (x − 2)² = (x − 2) × (x − 2)

⇒ x(x − 2) − 2(x − 2)

⇒ x² − 2x − 2x + 4

⇒ x² − 4x + 4

∴ The correct answer is x² − 4x + 4.

Additional Information

Square of a Sum

The algebraic identity for the square of a sum is (a + b)² = a² + 2ab + b².

Difference of Squares

The identity for the difference of two squares is a² − b² = (a − b)(a + b).

Square of a Trinomial

For three terms, the identity is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.

Shortcut Trick

LCM of 30 and 40 minutes = 120 minutes = 2 hours.

Total duration from 8 am to 2 pm = 6 hours.

Number of times = (Total Duration ÷ LCM interval) + 1

Number of times = (6 ÷ 2) + 1 = 3 + 1 = 4 times.

∴ The correct answer is 4.

Alternate Method

Given:

Bell 1 interval = 30 minutes

Bell 2 interval = 40 minutes

Starting time = 8:00 am

Closing time = 2:00 pm

Formula Used:

To find the simultaneous occurrence of cyclic events, calculate the Least Common Multiple (LCM) of their time intervals.

⇒ LCM of 30 and 40:

⇒ 30 = 2 × 3 × 5

⇒ 40 = 2 × 2 × 2 × 5

⇒ LCM = 2³ × 3 × 5 = 120 minutes

⇒ Interval in hours = 120 ÷ 60 = 2 hours

⇒ Total duration = 2:00 pm − 8:00 am = 6 hours

⇒ Number of intervals = 6 ÷ 2 = 3

⇒ Total times the bells ring together = (Number of intervals + 1) for the initial 8 am ring.

⇒ Total = 3 + 1 = 4

∴ The correct answer is 4.

Additional Information

LCM for Cyclic Events

When multiple objects start at the same time and repeat at fixed intervals, they will coincide again at a time which is the LCM of their individual intervals.

Total Occurrences Formula

Total Occurrences = (Total Time Period ÷ LCM of intervals) + 1. The +1 accounts for the first time they ring together at the starting point.

Unit Consistency

Always ensure the time intervals and the total duration are in the same units (either all in minutes or all in hours) before performing division.

Shortcut Trick

Net filling rate = (1/6) − (1/24) = 3/24 = 1/8 tank per minute.

Time to fill the full tank = 8 minutes.

Time to fill one-half of the tank = 8 × (1/2) = 4 minutes.

∴ The correct answer is 4.

Alternate Method

Given:

Time taken by Pipe A to fill = 6 minutes

Time taken by Pipe B to empty = 24 minutes

Formula used: Efficiency = Total Work ÷ Time

⇒ Total capacity of the tank = LCM(6, 24) = 24 units

⇒ Efficiency of Pipe A = 24 ÷ 6 = +4 units/min

⇒ Efficiency of Pipe B = 24 ÷ 24 = −1 units/min

⇒ Net efficiency of (A + B) = 4 − 1 = 3 units/min

⇒ One-half of the tank capacity = 24 ÷ 2 = 12 units

⇒ Time required to fill 12 units = 12 ÷ 3 = 4 minutes

∴ The correct answer is 4.

Additional Information

Negative Work in Pipes

When a pipe empties a tank, its efficiency or rate of work is taken as negative (e.g., −1/T).

Pipes & Cisterns Basic Formula

If pipe A fills in x hours and pipe B empties in y hours, net work in 1 hour = (1/x) − (1/y).

LCM Method for Multiple Pipes

Total capacity is assumed as the LCM of individual times. Efficiency = Capacity ÷ Time.

Shortcut Trick

Ratio of A : B (Present) = 5 : 1 (Difference = 4 units)

Ratio of A : B (5 yrs ago) = 6 : 1 (Difference = 5 units)

Equalize the difference by multiplying ratios by 5 and 4 respectively:

Present ratio = 25 : 5 | Past ratio = 24 : 4

Difference in B's ratio parts = 5 − 4 = 1 unit → 5 years

Present age of B = 5 units × 5 = 25 years

∴ The correct answer is 25 years.

Alternate Method

Given:

Present ratio of A and B = 5 : 1

Ratio 5 years ago = 6 : 1

Let the present age of A be 5x and B be x.

According to the question, five years ago:

⇒ (5x − 5) ÷ (x − 5) = 6 ÷ 1

⇒ 1 × (5x − 5) = 6 × (x − 5)

⇒ 5x − 5 = 6x − 30

⇒ 6x − 5x = 30 − 5

⇒ x = 25

Present age of B = x = 25 years.

∴ The correct answer is 25 years.

Additional Information

Constant Difference Rule

The difference between the ages of two individuals remains constant regardless of the number of years passed or elapsed.

Future and Past Age

To find age 'n' years ago, subtract n from present age (x − n). To find age 'n' years hence, add n to present age (x + n).

Ratio Balancing

When solving age ratios, multiplying the ratios to make the numerical difference between parts equal to the time gap simplifies the calculation.

Shortcut Trick

A number is divisible by 45 if it is divisible by both 5 and 9.

Divisibility by 5: Last digit must be 0 or 5. Only (2) 20475 and (4) 555555 qualify.

Divisibility by 9: Sum of digits must be divisible by 9.

For 20475, sum = 2+0+4+7+5 = 18 (divisible by 9).

For 555555, sum = 5×6 = 30 (not divisible by 9).

∴ The correct answer is 20475.

Alternate Method

Given:

Numbers: 306999, 20475, 25436, 555555

Formula Used:

Divisibility rule of 45: A number N is divisible by 45 if N is divisible by 5 and 9 simultaneously (since 5 × 9 = 45 and HCF(5, 9) = 1).

Calculations:

⇒ Option (1) 306999: Last digit is 9. Not divisible by 5.

⇒ Option (3) 25436: Last digit is 6. Not divisible by 5.

⇒ Option (4) 555555: Last digit is 5. Sum = 5+5+5+5+5+5 = 30. 30 is not divisible by 9.

⇒ Option (2) 20475: Last digit is 5. Divisible by 5.

⇒ Sum of digits of 20475 = 2 + 0 + 4 + 7 + 5 = 18.

⇒ Since 18 is divisible by 9, 20475 is divisible by 9.

⇒ As 20475 is divisible by both 5 and 9, it must be divisible by 45.

∴ The correct answer is 20475.

Additional Information

Divisibility by 3

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

Divisibility by 4

A number is divisible by 4 if its last two digits form a number divisible by 4.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Shortcut Trick

The average of the first n odd natural numbers is always n.

Given value of n = 14.

∴ Average = 14.

∴ The correct answer is 14.

Alternate Method

Given:

Number of terms (n) = 14

Formula Used:

Sum of first n odd natural numbers = n²

Average = Sum of observations ÷ Total number of observations

⇒ First 14 odd numbers = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27

⇒ Sum = 14²

⇒ Sum = 196

⇒ Average = 196 ÷ 14

⇒ Average = 14

∴ The correct answer is 14.

Additional Information

Average of first n natural numbers

The average is given by the formula (n + 1) ÷ 2.

Average of first n even natural numbers

The average of the first n even natural numbers is (n + 1).

Average of Arithmetic Progression (AP)

For any sequence in AP, Average = (First term + Last term) ÷ 2.

Shortcut Trick

Remaining Distance = 3/4 × 40 = 30 km

Remaining Time = 2/3 × 5 = 10/3 hours

Required Speed = Remaining Distance ÷ Remaining Time

Speed = 30 ÷ (10/3) = (30 × 3) / 10 = 9 km/h

∴ The correct answer is 9.

Alternate Method

Given:

Total Distance = 40 km

Total Time = 5 hours

Formula Used:

Speed = Distance / Time

⇒ Total Distance = 40 km

⇒ Covered Distance = 40 × (1/4) = 10 km

⇒ Remaining Distance = 40 − 10 = 30 km

⇒ Total Time = 5 hours

⇒ Covered Time = 5 × (1/3) = 5/3 hours

⇒ Remaining Time = 5 − 5/3 = 10/3 hours

⇒ Required Speed = Remaining Distance ÷ Remaining Time

⇒ Required Speed = 30 ÷ (10/3)

⇒ Required Speed = (30 × 3) / 10

⇒ Required Speed = 9 km/h

∴ The correct answer is 9.

Additional Information

Average Speed

It is the ratio of total distance covered to the total time taken. Average Speed = Total Distance ÷ Total Time.

Relative Speed

When moving in the same direction, speed = S₁ − S₂. When moving in opposite directions, speed = S₁ + S₂.

Unit Conversion

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

Shortcut Trick

Volume of metal in a hollow pipe = π × (R² − r²) × h

Difference of squares (R² − r²) = (7 + 5) × (7 − 5) = 12 × 2 = 24.

Volume = π × 24 × 14

Volume = 336π cm³.

∴ The correct answer is 336π cm³.

Alternate Method

Given:

Outer radius (R) = 7 cm

Inner radius (r) = 5 cm

Length/Height (h) = 14 cm

Formula for the volume of metal in a hollow cylinder:

Volume = External Volume − Internal Volume

⇒ Volume = πR²h − πr²h

⇒ Volume = πh(R² − r²)

⇒ Volume = π × 14 × (7² − 5²)

⇒ Volume = π × 14 × (49 − 25)

⇒ Volume = π × 14 × 24

⇒ Volume = 336π cm³

∴ The correct answer is 336π cm³.

Additional Information

Volume of a Solid Cylinder

The volume of a solid cylinder with radius r and height h is given by V = πr²h.

Curved Surface Area (CSA) of Hollow Pipe

The total curved surface area (inner + outer) is 2πRh + 2πrh = 2πh(R + r).

Total Surface Area (TSA) of Hollow Pipe

TSA = CSA + 2 × Area of Base Rings = 2πh(R + r) + 2π(R² − r²) = 2π(R + r)(h + R − r).

Shortcut Trick

Efficiency Ratio (A : B) = 100 : 125 = 4 : 5

Total efficiency = 4 + 5 = 9 units

Total work = 9 units/hr × 8 hr = 72 units

Time for Pipe A = 72 ÷ 4 = 18 hours

Capacity = 18 × 60 min × 5 litres/min = 5400 litres

∴ The correct answer is 5400 litres.

Alternate Method

Given: Time (A+B) = 8 hours, Speed of A = 5 L/min

Efficiency of B = 125% of A ⇒ B = 1.25A

⇒ Let Efficiency of A = 4 units/hr and B = 5 units/hr

⇒ Total Efficiency (A + B) = 4 + 5 = 9 units/hr

⇒ Total Work = 9 × 8 = 72 units

⇒ Time taken by Pipe A = 72 ÷ 4 = 18 hours

⇒ Rate of Pipe A = 5 litres/min × 60 mins/hr = 300 litres/hr

⇒ Capacity of Cistern = Rate × Time = 300 × 18 = 5400 litres

∴ The correct answer is 5400 litres.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken. If the ratio of efficiencies is x:y, the ratio of time taken will be y:x.

Cistern Capacity Formula

Total Capacity = Rate of flow (litres/unit time) × Total time taken to fill the tank.

Pipes and Cisterns Work

Net work done = (Sum of work by inlet pipes) − (Sum of work by outlet pipes).

Shortcut Trick

Total Distance = 120 + 180 = 300 km

Total Time = (120 ÷ 50) + (180 ÷ 80) = 2.4 + 2.25 = 4.65 hours

Average Speed = Total Distance ÷ Total Time

Average Speed = 300 ÷ 4.65 ≈ 64.52 km/h

∴ The correct answer is 64.52 km/h.

Alternate Method

Given:

Distance 1 (d₁) = 120 km, Speed 1 (s₁) = 50 km/h

Distance 2 (d₂) = 180 km, Speed 2 (s₂) = 80 km/h

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

⇒ Time taken for first part (t₁) = 120 ÷ 50 = 2.4 hours

⇒ Time taken for second part (t₂) = 180 ÷ 80 = 2.25 hours

⇒ Total Distance = 120 + 180 = 300 km

⇒ Total Time = 2.4 + 2.25 = 4.65 hours

⇒ Average Speed = 300 ÷ 4.65

⇒ Average Speed = 64.516129... ≈ 64.52 km/h

∴ The correct answer is 64.52 km/h.

Additional Information

Average Speed for Equal Distances

When a journey covers two equal distances at speeds s₁ and s₂, Average Speed = (2 × s₁ × s₂) ÷ (s₁ + s₂).

Average Speed for Equal Time Intervals

If a person travels for equal time intervals at speeds s₁ and s₂, Average Speed = (s₁ + s₂) ÷ 2.

Unit Conversion Shortcut

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

Shortcut Trick

Factor out common term: tan²θ (tan²θ + 1)

Use identity: 1 + tan²θ = sec²θ

Substitute tan²θ = (sec²θ − 1)

Final Step: (sec²θ − 1) × sec²θ = sec⁴θ − sec²θ

∴ The correct answer is sec⁴θ − sec²θ.

Alternate Method

Given:

Expression: tan⁴θ + tan²θ

Formula Used:

Fundamental Trigonometric Identity: 1 + tan²θ = sec²θ

⇒ Expression = tan⁴θ + tan²θ

⇒ Taking tan²θ as common: tan²θ(tan²θ + 1)

⇒ From identity sec²θ = 1 + tan²θ, we have (tan²θ + 1) = sec²θ

⇒ Also, tan²θ = sec²θ − 1

⇒ Substituting these values: (sec²θ − 1) × sec²θ

⇒ Expanding the bracket: (sec²θ × sec²θ) − (1 × sec²θ)

⇒ Result = sec⁴θ − sec²θ

∴ The correct answer is sec⁴θ − sec²θ.

Additional Information

Standard Pythagorean Identities

There are three primary identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Reciprocal Relations

secθ = 1/cosθ and tanθ = sinθ/cosθ. Squaring these helps in converting between higher powers of different trigonometric ratios.

Algebraic Manipulations in Trigonometry

Always look for common factors like tan²θ or sin²θ to simplify complex powers into standard squared identities.

Shortcut Trick

Each additional side in a polygon adds 180° to the sum of interior angles.

Divide the total sum by 180° to find the value of (n − 2).

4500 ÷ 180 = 25

Number of sides (n) = 25 + 2 = 27

∴ The correct answer is 27.

Alternate Method

Given: Sum of interior angles = 4500°

Formula Used: Sum of interior angles of a polygon = (n − 2) × 180°

⇒ (n − 2) × 180 = 4500

⇒ n − 2 = 4500 ÷ 180

⇒ n − 2 = 25

⇒ n = 25 + 2

⇒ n = 27

∴ The correct answer is 27.

Additional Information

Exterior Angle Sum

The sum of the exterior angles of any convex polygon is always 360° regardless of the number of sides.

Regular Polygon Interior Angle

Each interior angle of a regular polygon is given by [(n − 2) × 180°] ÷ n.

Number of Diagonals

The number of diagonals in a polygon with n sides is calculated as [n × (n − 3)] ÷ 2.

Interior and Exterior Relation

At each vertex of a polygon, the sum of the interior angle and the corresponding exterior angle is 180°.

Shortcut Trick

Solve the nested expression from the innermost square root outwards.

Step 1: √(850 − 66) = √784 = 28

Step 2: √(1268 + 28) = √1296 = 36

Step 3: √(2061 − 36) = √2025 = 45

∴ The correct answer is 45.

Alternate Method

Given expression: √(2061 − √(1268 + √(850 − 66)))

Simplifying the innermost term:

⇒ 850 − 66 = 784

⇒ √784 = 28

Substituting back into the expression:

⇒ √(2061 − √(1268 + 28))

⇒ √(2061 − √1296)

Since 36² = 1296:

⇒ √(2061 − 36)

⇒ √2025

Since 45² = 2025:

⇒ 45

∴ The correct answer is 45.

Additional Information

Perfect Squares Ending in 5

To find the square of a number ending in 5 (n5), the result is [n × (n+1)] followed by 25. For example, 45² = [4 × 5]25 = 2025.

Order of Operations (VBODMAS)

In simplification problems with nested radicals or brackets, always evaluate from the innermost level to the outermost level.

Common Square Roots

Memorizing squares up to 50 is vital for competitive exams. √784 = 28 and √1296 = 36 are frequently appearing values.

Shortcut Trick

Successive Multiplier Method: MP × (100 − D)% × (100 + VAT)% = SP

x × (37/100) × (125/100) = 481

x × 0.37 × 1.25 = 481

x = 481 / 0.4625 = 1040

∴ The correct answer is 1,040.

Alternate Method

Given: Marked Price (MP) = ₹x, Discount = 63%, VAT = 25%, Final Selling Price = ₹481

⇒ Discounted Price (DP) = x − 63% of x = 37% of x = 0.37x

⇒ Final SP = DP + VAT = 0.37x + 25% of 0.37x

⇒ 481 = 0.37x × (1 + 25/100)

⇒ 481 = 0.37x × 1.25

⇒ 481 = 0.4625x

⇒ x = 481 / 0.4625

⇒ x = (481 × 10000) / 4625

⇒ x = 1,040

∴ The correct answer is 1,040.

Additional Information

Marked Price and Discount

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

VAT/Sales Tax

Value Added Tax is a percentage added to the net selling price. Final Price = SP × (100 + VAT%)/100.

Successive Percentage Change

If an initial value x undergoes multiple percentage changes a% and b%, final value = x(1 + a/100)(1 + b/100).

Shortcut Trick

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

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

Amount = 144% of ₹6,200 = 1.44 × 6200

Amount = ₹8,928

∴ The correct answer is ₹8,928.

Alternate Method

Given:

Principal (P) = ₹6,200

Rate (R) = 20% per annum

Time (n) = 2 years

Formula Used:

Amount (A) = P(1 + R/100)ⁿ

Calculations:

⇒ A = 6200 × (1 + 20/100)²

⇒ A = 6200 × (120/100)²

⇒ A = 6200 × (1.2)²

⇒ A = 6200 × 1.44

⇒ A = 8928

∴ The correct answer is ₹8,928.

Additional Information

CI vs SI Difference for 2 Years

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

Successive Percentage Change

For two successive changes of x% and y%, the effective change is [x + y + (xy/100)]%.

Quarterly Compounding

When interest is compounded quarterly, the rate becomes R/4 and the time becomes 4n.

Shortcut Trick

72% of voters = 8640

1% of voters = 8640 ÷ 72 = 120

Total voters next year (8% increase) = 100% + 8% = 108%

Next year total = 120 × 108 = 12,960

∴ The correct answer is 12,960.

Alternate Method

Given: Cast votes = 8640 (72% of current total) | Growth = 8%.

Formula: Current Total = (Given Value ÷ Current Percentage) × 100.

⇒ Let total current eligible voters be V_c.

⇒ 0.72 × V_c = 8640

⇒ V_c = 8640 ÷ 0.72 = 12,000

⇒ Next year growth = 8% of 12,000

⇒ Growth = 0.08 × 12,000 = 960

⇒ Total eligible voters next year = 12,000 + 960 = 12,960

∴ The correct answer is 12,960.

Additional Information

Percentage Increase Formula

New Value = Old Value × (1 + Growth Rate ÷ 100). For an 8% increase, multiply by 1.08.

Base Value Concept

When a percentage of a quantity is known, divide the value by its percentage to find 1% of the base, then multiply by the desired target percentage.

Multiplying Factor (MF)

A growth of x% means MF = (100 + x) ÷ 100. A decrease of x% means MF = (100 − x) ÷ 100.

Shortcut Trick

Ratio of shares = 7 : 13 : 9

Difference between largest (13) and smallest (7) = 6 units

Given: 6 units = ₹58,326

1 unit = 58,326 ÷ 6 = ₹9,721

Total units = 7 + 13 + 9 = 29 units

Total profit = 29 × 9,721 = ₹2,81,909

∴ The correct answer is ₹2,81,909.

Alternate Method

Given:

Ratio of shares of three partners = 7 : 13 : 9

Difference between the smallest and largest share = ₹58,326

Formula:

Individual Share = (Individual Ratio Part ÷ Total Ratio Parts) × Total Profit

Let the shares of the three partners be 7x, 13x, and 9x.

⇒ Smallest share = 7x

⇒ Largest share = 13x

⇒ Difference = 13x − 7x = 6x

⇒ 6x = 58,326

⇒ x = 58,326 ÷ 6

⇒ x = 9,721

⇒ Total profit = 7x + 13x + 9x = 29x

⇒ Total profit = 29 × 9,721

⇒ Total profit = 2,81,909

∴ The correct answer is ₹2,81,909.

Additional Information

Partnership Profit Ratio

Profit is always distributed in the ratio of the product of Capital and Time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Simple Partnership

When investments of all partners are for the same duration, the profit is distributed simply in the ratio of their investments.

Ratio Unitary Method

If the difference between two parts is D and their ratio parts are a and b, then 1 unit = D ÷ |a − b|. Total value = (Sum of all ratio parts) × (Value of 1 unit).

Shortcut Trick

Marked Price (100%) = ₹2,000

Discount = 10%

Selling Price = 100% − 10% = 90%

SP = 90% of 2,000 = 0.9 × 2,000 = ₹1,800

∴ The correct answer is ₹1,800.

Alternate Method

Given: Marked Price (MP) = ₹2,000 | Discount % = 10%

Formula Used: Selling Price (SP) = MP − Discount | Discount = (Discount% / 100) × MP

⇒ Discount = (10 / 100) × 2,000

⇒ Discount = 0.1 × 2,000 = ₹200

⇒ SP = 2,000 − 200

⇒ SP = ₹1,800

∴ The correct answer is ₹1,800.

Additional Information

Marked Price and Discount

Marked Price is the price printed on an item. Discount is always calculated on the Marked Price unless stated otherwise. Formula: SP = MP × (100 − Discount%)/100.

Successive Discounts

If two successive discounts of a% and b% are given, the effective single discount is (a + b − ab/100)%.

Markup Percentage

Markup is the extra value added to the Cost Price to arrive at the Marked Price. Markup % = (MP − CP)/CP × 100.

Shortcut Trick

To compare ratios P = 5/2 and Q = 3/7, use the cross-multiplication method.

Multiply the numerator of P by the denominator of Q: 5 × 7 = 35.

Multiply the numerator of Q by the denominator of P: 3 × 2 = 6.

Since 35 > 6, the first ratio P is greater than the second ratio Q.

∴ The correct answer is P > Q.

Alternate Method

Given: P = 5:2 and Q = 3:7

Convert the ratios into their decimal equivalents for direct comparison.

⇒ Value of P = 5 ÷ 2

⇒ P = 2.5

⇒ Value of Q = 3 ÷ 7

⇒ Q ≈ 0.428

⇒ Comparing the values: 2.5 > 0.428

⇒ P > Q

∴ The correct answer is P > Q.

Additional Information

Ratio Comparison Rule

For two ratios a:b and c:d, if (a × d) > (b × c), then a:b > c:d. This is known as the cross-product method.

Inverse Ratio

The inverse ratio of a:b is b:a. For example, the inverse of P (5:2) is 2:5.

Compounded Ratio

The compounded ratio of a:b and c:d is calculated as (a × c) : (b × d). Here, the compounded ratio of P and Q is 15:14.

Shortcut Trick

Total Expenditure = 1120 + 915 = ₹2035

Total Number of Books = 52 + 47 = 99

Average Price per Book = Total Expenditure ÷ Total Books

Average = 2035 ÷ 99 = 20.555...

Average ≈ 20.56

∴ The correct answer is 20.56.

Alternate Method

Given:

Price of 52 books = ₹1,120

Price of 47 books = ₹915

Formula Used:

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

Calculations:

⇒ Total books purchased by Keshav = 52 + 47 = 99

⇒ Total money spent by Keshav = 1120 + 915 = ₹2035

⇒ Average price per book = 2035 ÷ 99

⇒ Average price = 20.5555...

⇒ Rounding to two decimal places, we get 20.56

∴ The correct answer is 20.56.

Additional Information

Weighted Average

When groups have different sizes, use the formula A_w = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is quantity and x is the value.

Rounding Rules

To round to 2 decimal places, look at the 3^rd decimal. If it is 5 or more, increase the 2^nd decimal by 1.

Unitary Method

To find the value of one item, divide the total value by the total number of items.

Shortcut Trick

Total % of votes = 100%

Votes for C = 100% − (18% + 43%) = 39%

Ranking: B (43%) > C (39%) > A (18%) ∴ Runner up is C

Winning Margin = 43% − 39% = 4%

4% units = 6,996 → 1% unit = 1,749

Total votes (100%) = 1,749 × 100 = 1,74,900

∴ The correct answer is 1,74,900.

Alternate Method

Given:

Votes of A = 18%

Votes of B = 43%

Winning margin (B over Runner up) = 6,996

Formula Used:

Percentage of value = (Part ÷ Total) × 100

⇒ Let total number of votes be X.

⇒ Percentage of votes for C = 100% − (18% + 43%)

⇒ Percentage of votes for C = 100% − 61% = 39%

⇒ Comparison of votes: B (43%) > C (39%) > A (18%)

⇒ Since C is on second rank, B won the election over C.

⇒ Margin in percentage = 43% − 39% = 4%

⇒ According to question: 4% of X = 6,996

⇒ (4 ÷ 100) × X = 6,996

⇒ X = (6,996 × 100) ÷ 4

⇒ X = 1,749 × 100 = 1,74,900

∴ The correct answer is 1,74,900.

Additional Information

Percentage Difference

When comparing two percentages of the same base, the difference in their percentage values directly represents the difference in their absolute values relative to the base.

Total Votes Calculation

In election problems, Total Votes = Valid Votes + Invalid Votes. If not specified, all votes are considered valid and distributed among candidates.

Successive Percentage Formula

For two successive changes of a% and b%, the effective change is given by: [a + b + (ab ÷ 100)]%.

Shortcut Trick

Successive Discount Formula = (a + b − ab/100)%

Here a = 29 and b = 28

⇒ Equivalent Discount = (29 + 28 − (29 × 28)/100)%

⇒ 57 − 812/100 = 57 − 8.12

⇒ 48.88%

∴ The correct answer is 48.88%.

Alternate Method

Given:

First discount (d₁) = 29%

Second discount (d₂) = 28%

Formula Used:

Selling Price (SP) = Marked Price (MP) × (1 − d₁/100) × (1 − d₂/100)

⇒ Let Marked Price (MP) = 100

⇒ Price after first discount = 100 − (29% of 100) = 71

⇒ Second discount amount = 28% of 71 = (28/100) × 71

⇒ Second discount amount = 19.88

⇒ Final Selling Price (SP) = 71 − 19.88 = 51.12

⇒ Total Discount = MP − SP = 100 − 51.12 = 48.88

⇒ Single Equivalent Discount % = (48.88 / 100) × 100 = 48.88%

∴ The correct answer is 48.88%.

Additional Information

Successive Percentage Change

The net percentage change for two successive changes of x% and y% is given by the formula: (x + y + xy/100)%. For discounts, values of x and y are taken as negative.

Three Successive Discounts

For three discounts a%, b%, and c%, the single equivalent discount is found by calculating the equivalent of any two first, and then using that result with the third discount.

Relation between MP and SP

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

Shortcut Trick

Total amount to be saved = Target amount − Initial savings

Total savings required = ₹21,000 − ₹12,000 = ₹9,000

Number of weeks = Total savings required ÷ Weekly savings

Number of weeks = 9,000 ÷ 1,500 = 6

∴ The correct answer is 6 weeks.

Alternate Method

Given:

Initial savings = ₹12,000

Target amount = ₹21,000

Weekly savings rate = ₹1,500

Formula:

Total Amount = Initial Savings + (Weekly Savings × Number of Weeks)

Let the number of weeks be "n"

⇒ 21,000 = 12,000 + (1,500 × n)

⇒ 21,000 − 12,000 = 1,500n

⇒ 9,000 = 1,500n

⇒ n = 9,000 ÷ 1,500

⇒ n = 6

∴ The correct answer is 6 weeks.

Additional Information

Arithmetic Progression (A.P.)

This problem follows an A.P. where the first term a = 12,000 + 1,500 (after 1 week), the common difference d = 1,500, and the n^th term is 21,000.

Linear Equation Modeling

Any situation involving a fixed starting value and a constant rate of change can be modeled by y = mx + c, where c is the initial value and m is the rate.

Time and Savings Relation

Time = (Required Amount − Initial Amount) ÷ Savings per unit time. This logic is also used in Simple Interest where Interest = Principal × Rate × Time.

Shortcut Trick

Initial Income : Expenditure = 12 : 7

New Income = 12 × (100 + 40)% = 12 × 140

New Expenditure = 7 × (100 + 50)% = 7 × 150

New Ratio = (12 × 140) : (7 × 150) = 1680 : 1050

Simplifying by dividing by 210 ⇒ 8 : 5

∴ The correct answer is 8 : 5.

Alternate Method

Given: Initial Ratio of Income to Expenditure = 12 : 7

Formula: New Value = Old Value × [ (100 + Increase%) / 100 ]

⇒ Let Ratnesh’s monthly income = 1200 and expenditure = 700.

⇒ New Income = 1200 + (40 / 100) × 1200 = 1200 + 480 = 1680

⇒ New Expenditure = 700 + (50 / 100) × 700 = 700 + 350 = 1050

⇒ New Ratio = 1680 : 1050

⇒ Dividing both sides by 210: 1680/210 : 1050/210

⇒ New Ratio = 8 : 5

∴ The correct answer is 8 : 5.

Additional Information

Income-Expenditure Relationship

Total Income = Expenditure + Savings. If income increases more than expenditure, savings increase.

Percentage Change in Ratios

If ratio is a : b and terms increase by x% and y% respectively, new ratio = a(100+x) : b(100+y).

Effective Change in Savings

Change in Savings = New Income − New Expenditure. It can be expressed as a percentage of original savings.

Shortcut Trick

In an isosceles trapezium, height (h) = √[slant² − ((a−b)/2)²]

h = √[13² − ((20−10)/2)²] = √(169 − 25) = √144 = 12 cm

Area = 1/2 × (Sum of parallel sides) × h

Area = 1/2 × (20 + 10) × 12 = 15 × 12 = 180 cm²

∴ The correct answer is 180 cm².

Alternate Method

Given:

Parallel sides (a, b) = 20 cm, 10 cm

Non-parallel sides (c) = 13 cm each (Isosceles Trapezium)

Formula Used:

Area of Trapezium = 1/2 × (Sum of parallel sides) × Height

Pythagoras Theorem: h² + base² = hypotenuse²

⇒ Let the height be h. Since it is an isosceles trapezium, the perpendiculars from the top vertices divide the base into segments: 5 cm, 10 cm, and 5 cm.

⇒ Base of the right-angled triangle formed at the side = (20 − 10) ÷ 2 = 5 cm

⇒ In the right-angled triangle, height (h) = √(13² − 5²)

⇒ h = √(169 − 25) = √144 = 12 cm

⇒ Area of Trapezium = 1/2 × (20 + 10) × 12

⇒ Area = 1/2 × 30 × 12

⇒ Area = 15 × 12 = 180 cm²

∴ The correct answer is 180 cm².

Additional Information

Isosceles Trapezium Properties

In an isosceles trapezium, non-parallel sides are equal, base angles are equal, and diagonals are equal in length.

Cyclic Quadrilateral Property

Every isosceles trapezium is a cyclic quadrilateral, meaning all its vertices lie on a single circle.

Median of a Trapezium

The median (mid-segment) joins the midpoints of non-parallel sides; Median length = (a + b) ÷ 2.

Given: A, B, C, D, E, F and G are sitting in a row facing north.

F sits fifth from the right end of the row.

Only three people sit to the right of C.

D is the immediate neighbor of B and C.

A sits to the right of E but left of F

Now, People between G and A →  4 people

Hence, the correct answer is "Option 1".

The logic followed here is:

For MPLQ → SVRW

Now, for ORNS → UXTY

Now, applying the same logic for FIEJ:

So, FIEJ → LOKP

Hence, the correct answer is "Option 2".

The logic followed here is:

Given:

Rohit starts from Point A and drives 15 km towards the West.

He then takes a left turn, drives 11 km,

turns left and drives 18 km.

He then takes a left turn and drives 14 km.

He takes a final left turn, drives 3 km and stops at Point P.

​

So, To reach Point A, he must move 3 km towards the South

Hence, the correct answer is Option 2.

The logic followed here is:

Given:

‘leaves turn brown’ → ka gs rl

‘gather leaves daily’ → ny hf gs

So, the code for leaves = gs.

Hence, the correct answer is Option 3.

The logic followed here is:

For TOIL → BWQT

Now, for ZUOR → HCWZ

Now, applying the same logic for XSMP:

So, XSMP → FAUX

Hence, the correct answer is "Option 1".

The logic followed here is:

Given series: (Left) F J L T M N C R P W X Z K O H A Y Q D G E B (Right)

Logic: Count the number of letters appearing after Q and before E in the given series.

Series positions: Q D G E

Letters between Q and E are: D, G

Number of letters between Q and E = 2

Hence, the correct answer is Option 1.

Given: S, Q, U, A, R, E and D are sitting in a row facing north.

No one sits to the left of A.

Only four people sit between A and S.

Only three people sit to the right of R.

U sits to the immediate left of E.

D is not an immediate neighbor of R.

Now,

The rightmost end is position 7 → D

Hence, the correct answer is "Option 3".

Given:

Rekha is 13th from the left end and Ajay is 6th from the right end in a row of 51 people.

Here, Number of people between them = 51 − (13 + 6)

= 51 − 19

= 32

So, there are 32 people between Rekha and Ajay.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: Third Number = First Number + Second Number.

For, 77 - 3 - 80

⇒ 77 + 3 = 80

For, 12 - 4 - 16

⇒ 12 + 4 = 16

Now, checking the options:

Option 1) 20 - 13 - 43

⇒ 20 + 13 = 33 ≠ 43

Option 2) 11 - 1 - 23

⇒ 11 + 1 = 12 ≠ 23

Option 3) 14 - 9 - 25

⇒ 14 + 9 = 23 ≠ 25

Option 4) 6 - 5 - 11

⇒ 6 + 5 = 11 = 11

Therefore, the set '6 - 5 - 11' follows the same pattern.

Hence, the correct answer is Option 4.

Given: 435, 325, 123, 756, 264

Now, the third digit of the highest number and the second digit of the lowest number, then add them.

Highest number = 756

Third digit of highest number = 6

Lowest number = 123

Second digit of lowest number = 2

So, 6 + 2 = 8

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "53" will replace the question mark in the given series.

Hence, the correct option is "Option 2".

The logic followed here is:

Option 1) GN – LS

Option 2) XF – CK

Option 3) VC – BH

Option 4) MD – RI

Here, VC – BH is odd from the given options.

Hence, the correct answer is "Option 3".

The logic followed here is:

So, the missing term = XNG 62

Hence, the correct answer is Option 1.

The logic followed here is:

thus, "302" will replace the question mark in the given series.

Hence, the correct option is "Option 2'.

The logic followed here is:

Option 1) GU – KY

Option 2) LR – PV

Option 3) PC – TG

Option 4) EK – IP

Here, EK – IP is odd from the given options.

Hence, the correct answer is "Option 4".

The logic followed here is:

Given:

‘ride waves carefully’ → el uw cb

‘waves crash loud’ → sr cb vq

So, the code for waves = cb.

Hence, the correct answer is Option 1.

BODMAS Table: 

Decoding the given information:

 

Now, Checking given options by replacing the numbers and using BODMAS Rule, we get:

Option 1) 56 and 28

⇒ (28 ÷ 14) × 3 + 7 × 9 − 15 × 5 + 56

⇒ 2 × 3 + 63 − 75 + 56

⇒ 6 + 63 − 75 + 56

⇒ 69 − 75 + 56

⇒ −6 + 56

⇒ 50

 

Option 2) 7 and 14

⇒ (56 ÷ 7) × 3 + 14 × 9 − 15 × 5 + 28 = 103

⇒ 8 × 3 + 126 − 75 + 28 = 103

⇒ 24 + 126 − 75 + 28 = 103

⇒ 150 − 75 + 28 = 103

⇒ 75 + 28 = 103

⇒ 103 = 103

Option 3) 5 and 7

 

⇒ (56 ÷ 14) × 3 + 5 × 9 − 15 × 7 + 28

⇒ 4 × 3 + 45 − 105 + 28

⇒ 12 + 45 − 105 + 28

⇒ 57 − 105 + 28

⇒ −48 + 28

⇒ −20

Option 4) 28 and 14

⇒ (56 ÷ 28) × 3 + 7 × 9 − 15 × 5 + 14

⇒ 2 × 3 + 63 − 75 + 14

⇒ 6 + 63 − 75 + 14

⇒ 69 − 75 + 14

⇒ −6 + 14

⇒ 8

 

Hence, the correct option is "Option 2".

Given: G, H, I, Q, R and S are sitting in a row facing north.

Only two people sit to the right of Q.

Only one person sits between Q and S.

Only one person sits between H and R.

R sits to the immediate left of Q.

Only I sits to the right of G.

Now, People between H and G are:  3 people

Hence, the correct answer is "Option 3".

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

Now,

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

So, Needed pattern is: Even Digit - Even Digit - Even Digit

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

So, total valid even digits = 1

Hence, the correct answer is Option 1.

The logic followed here is:

For HOCX → SZNI

Now, for ZGUP → KRFA

Now, applying the same logic for GNBW:

So, GNBW → RYMH

Hence, the correct answer is "Option 2".

The logic followed here is:

Given:

Piyush starts from Point X and drives 6 km towards east.

He then takes a left turn, drives 3 km, turns right, and drives 5 km.

He then takes a right turn and drives 3 km.

Finally, he takes a right turn, drives 5 km, and stops at Point P.

To reach Point X, he must go 6 km towards the West

Hence, the correct answer is Option 3.

Family Chart:

Decoding the Symbols:

Given: L × O # V @ E + R

It means: L is the wife of O, O is the brother of V, V is the father of E, and E is the sister of R.

Since, L is the father’s brother’s wife of R.

Hence, the correct answer is "Option 3".

BODMAS Table: 

Decoding the given information:

<br>

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

⇒ 24 × 2 ÷ 4 + 5 − 3 = ?

⇒ 48 ÷ 4 + 5 − 3 = ?

⇒ 12 + 5 − 3 = ?

⇒ 17 − 3 = ?

⇒ 14

Hence, the correct option is "Option 4".

Given Statements:

All flutes are drums.

All drums are guitars.

The least possible Venn diagram for the given statement is:

Conclusions:

I. No guitar is a flute – Does not follow (Since all flutes are guitars, some guitars must be flutes, so this contradicts the statement).

II. Some guitars are flutes – Follows All flutes are guitars, so definitely some guitars are flutes).

So, only conclusion II follows.

Hence, the correct answer is "Option 4".

Given:

Miku is 12th from the right end and Pushkar sits 5 places to the left of Miku in a row of 50 students.

Here,

Pushkar’s position from the right end = 12 + 5 = 17th

Now, Pushkar’s position from the left end = 50 − 17 + 1 = 34th

So, Pushkar is 34th from the left end.

Hence, the correct answer is "Option 2".

BODMAS Table: 

Decoding the given information:

 

Now, Checking the given options by replacing the numbers and using BODMAS Rule, we get:

Option 1) 9 and 4

⇒ (36 − 28) × 3 + 4 × 5 − (108 ÷ 9) + 42

⇒ 8 × 3 + 20 − 12 + 42

⇒ 24 + 20 − 12 + 42

⇒ 44 − 12 + 42

⇒ 32 + 42

⇒ 74

 

Option 2) 4 and 3

⇒ (36 − 28) × 4 + 9 × 5 − (108 ÷ 3) + 42

⇒ 8 × 4 + 45 − 36 + 42

⇒ 32 + 45 − 36 + 42

⇒ 77 − 36 + 42

⇒ 41 + 42

⇒ 83

 

Option 3) 3 and 5

⇒ (36 − 28) × 5 + 9 × 3 − (108 ÷ 4) + 42

⇒ 8 × 5 + 27 − 27 + 42

⇒ 40 + 27 − 27 + 42

⇒ 67 − 27 + 42

⇒ 40 + 42

⇒ 82

 

Option 4) 36 and 42

⇒ (42 − 28) × 3 + 9 × 5 − (108 ÷ 4) + 36 = 96

⇒ 14 × 3 + 45 − 27 + 36 = 96

⇒ 42 + 45 − 27 + 36 = 96

⇒ 87 − 27 + 36 = 96

⇒ 60 + 36 = 96

⇒ 96 = 96

Hence, the correct option is "Option 4".

The logic followed here is:

Given series: (Left) 4 @ 3 % 5 $ 1 # 6 & 9 9 * 5 + 1 ? 6 + = 5 $ # 2 (Right)

Logic: Remove all the symbols from the series and find the 7th element from the right.

After removing all symbols: 4 3 5 1 6 9 5 1 6 5 2

So, the 7th element from the right = 6

Hence, the correct answer is Option 2.

The logic followed here is:

thus, "291" will replace the question mark in the given series.

Hence, the correct option is "Option 3".

Given Statements:

All books are pens.

Some pens are crayons.

The least possible Venn diagram for the given statement is:

Conclusions:

I. All crayons are pens – Does not follow  (Only “some pens are crayons” is given; it does not cover all crayons).

II. Some books are crayons – Does not follow   (Books are inside pens, but the overlap between pens and crayons may not include books).

So, neither conclusion I nor conclusion II follows.

Hence, the correct answer is "Option 3".

The logic followed here is:

So, the missing term = OLI 97

Hence, the correct answer is Option 1.