RRB Group D Official Paper (Held On: 02 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is loss of electrons.

Key Points

• Oxidation is a chemical process where an atom, ion, or molecule loses electrons during a reaction. This results in an increase in oxidation state.
• Oxidation is often paired with reduction, a process where another substance gains electrons. Together, these processes form a redox reaction.
• In terms of electron transfer, oxidation is defined as the loss of electrons, while reduction is the gain of electrons.
• Oxidation does not involve the loss of protons or gain of neutrons; these are unrelated processes. Hence, options "loss of protons" and "gain of neutrons" are incorrect.
• Similarly, oxidation does not involve "gain of electrons"; gaining electrons is characteristic of reduction, not oxidation.
• For example, in the reaction of sodium (Na) with chlorine (Cl₂), sodium loses electrons to form Na⁺, undergoing oxidation, while chlorine gains electrons to form Cl^-, undergoing reduction.

Additional Information

• Redox Reactions:
• Redox reactions are chemical reactions that involve the transfer of electrons between two species.
• Oxidation and reduction always occur simultaneously in redox reactions.
• Example: In the reaction 2Fe³⁺ + 2I^- → 2Fe²⁺ + I₂, Fe³⁺ is reduced to Fe²⁺, and I^- is oxidized to I₂.
• Oxidation State:
• The oxidation state (or oxidation number) is a measure of the degree of oxidation of an atom in a substance.
• An increase in oxidation state indicates oxidation, while a decrease indicates reduction.
• Example: In H₂O₂, oxygen has an oxidation state of -1, while in O₂, it has an oxidation state of 0.
• Examples of Oxidation:
• Rusting of iron: Fe + O₂ + H₂O → Fe₂O₃·xH₂O.
• Combustion of fuels: CH₄ + 2O₂ → CO₂ + 2H₂O.
• Oxidation of glucose in cellular respiration: C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O.
• Industrial Applications:
• Oxidation reactions are crucial in industries such as metallurgy, where metals are extracted from ores.
• Oxidation is also used in wastewater treatment to remove impurities.
• In the food industry, oxidation reactions are involved in processes like fermentation and food preservation.

Important Points

• Oxidizing Agents:
• An oxidizing agent (or oxidant) is a substance that causes oxidation by accepting electrons.
• Examples include oxygen (O₂), hydrogen peroxide (H₂O₂), and halogens (Cl₂, Br₂, etc.).
• Reducing Agents:
• A reducing agent (or reductant) is a substance that causes reduction by donating electrons.
• Examples include hydrogen (H₂), carbon monoxide (CO), and metals like Na and Zn.
• Electrochemical Cells:
• Oxidation occurs at the anode, and reduction occurs at the cathode in electrochemical cells.
• These cells are widely used in batteries and fuel cells.

The correct answer is Nature of the material.

Key Points

• Resistivity is a fundamental property of a material that determines how strongly it opposes the flow of electric current.
• The resistivity of a material depends solely on its nature and intrinsic properties, such as the type of atoms, arrangement of atoms, and the electron mobility within the material.
• Factors such as the current passing through the material, its shape, or its dimensions (length and area) do not affect the resistivity. These factors only influence resistance, not resistivity.
• Materials like metals (e.g., copper, aluminum) have low resistivity, making them good conductors, while insulators like rubber or glass have high resistivity.
• Temperature can also affect resistivity, as it changes the mobility of electrons within the material, but the intrinsic nature of the material remains the primary determinant.

Additional Information

• Definition of Resistivity:
• Resistivity (denoted as ρ) is defined as the resistance of a material of unit length and unit cross-sectional area.
• The formula for resistivity is: ρ = R × (A / L), where R is resistance, A is the cross-sectional area, and L is the length of the material.
• Units and Dimensions:
• The SI unit of resistivity is ohm-meter (Ω·m).
• It is a scalar quantity and has dimensions of [ML³T⁻³A⁻²].
• Factors Affecting Resistivity:
• Material's Nature: The type and arrangement of atoms in a material determine its resistivity.
• Temperature: For most conductors, resistivity increases with temperature, while for semiconductors, it decreases with temperature.
• Examples of Materials and Resistivity:
• Copper: Low resistivity (~1.68 × 10⁻⁸ Ω·m) - excellent conductor.
• Rubber: High resistivity (~10¹² to 10¹⁵ Ω·m) - excellent insulator.
• Silicon: Intermediate resistivity (~10⁻³ Ω·m) - semiconductor.

Important Points

• Difference between Resistance and Resistivity:
• Resistance (R): Depends on the material's resistivity, length, and cross-sectional area. Measured in ohms (Ω).
• Resistivity (ρ): An intrinsic property of the material, independent of its dimensions.
• Applications:
• Low-resistivity materials like copper are used in wires and electrical circuits.
• High-resistivity materials like tungsten are used in heating elements.
• Semiconductors like silicon are used in electronic devices due to their intermediate resistivity.
• Temperature Coefficient:
• The rate at which resistivity changes with temperature is called the temperature coefficient of resistivity.
• For metals, it is positive (resistivity increases with temperature), while for semiconductors, it is negative.

The correct answer is Big dams can help in flood control by regulating river flow..

Key Points

• Big dams are designed to regulate the flow of rivers by storing water during periods of excessive rainfall and releasing it gradually over time. This reduces the risk of sudden floods downstream.
• While big dams can significantly mitigate flood risks, they do not completely eliminate the possibility of floods, especially in cases of extreme rainfall or dam overflow.
• Dams are extensively used for flood control across the world, including in countries like India, the USA, and China. Examples include the Bhakra Dam in India and the Three Gorges Dam in China.
• Incorrect statements such as "Big dams are never used for flood control" and "Big dams completely eliminate the possibility of floods" disregard the partial but effective role of dams in managing floods.
• However, there are instances where dams can increase the risk of floods due to sudden releases or structural failures, making it crucial to manage them properly.

Additional Information

• Functions of Big Dams:
• Dams store large volumes of water in reservoirs, which can be released during dry periods, ensuring a steady supply of water for irrigation, drinking, and industrial purposes.
• They are used for hydroelectric power generation, contributing to renewable energy production.
• Dams play a key role in recreation and tourism, offering activities like boating, fishing, and sightseeing.
• They can also help prevent saltwater intrusion in coastal areas by maintaining freshwater levels in rivers.
• Examples of Dams Used for Flood Control:
• The Hoover Dam in the USA regulates the flow of the Colorado River and prevents flooding in downstream areas.
• India's Bhakra Dam on the Sutlej River provides flood control for the northern plains.
• The Three Gorges Dam in China reduces flooding along the Yangtze River.
• Limitations of Dams:
• Dams cannot fully prevent floods during extreme weather events like cyclones or heavy monsoons.
• The construction and maintenance of dams can lead to significant environmental and social impacts, including displacement of communities and alteration of ecosystems.
• Sudden releases of water from dams, either due to operational decisions or emergencies, can increase flood risks downstream.
• Global Flood Control Measures:
• Countries implement multiple flood control measures such as levees, floodwalls, stormwater management systems, and afforestation along riverbanks.
• Integrated water management approaches combine dams with other technologies like flood forecasting and early warning systems to enhance flood mitigation.

Important Points

• Environmental Impacts:
• Dams can disrupt aquatic ecosystems, affecting fish migration and river biodiversity.
• Reservoirs created by dams submerge large areas, leading to loss of forests and agricultural land.
• Social Impacts:
• Displacement of people due to dam construction can lead to social and economic challenges.
• Compensation and rehabilitation for affected communities are critical aspects of dam projects.
• Technological Advances:
• Modern dams incorporate advanced materials and designs to enhance safety and efficiency.
• Automation and remote monitoring systems are increasingly being used to manage dam operations.

The correct answer is C60.

Key Points

• Fullerene is a unique allotrope of carbon discovered in 1985 by scientists Harold Kroto, Robert Curl, and Richard Smalley, who later won the Nobel Prize in Chemistry in 1996.
• The chemical formula of fullerene is C60, which represents a molecule made of 60 carbon atoms arranged in a spherical shape resembling a soccer ball.
• Fullerenes are also called buckminsterfullerenes or “buckyballs,” named after architect Buckminster Fuller, whose geodesic domes inspired their structure.
• C60 consists of pentagonal and hexagonal rings, making it a highly symmetric and stable molecule.
• It has unique properties such as high thermal stability, electrical conductivity, and the ability to act as a superconducting material under certain conditions.
• The discovery of fullerenes opened up new avenues in nanotechnology, material science, and medicine.

Additional Information

• Other Allotropes of Carbon:
• Diamond: A crystalline allotrope where each carbon atom is bonded to four others in a tetrahedral structure, known for its extreme hardness and used in cutting tools and jewelry.
• Graphite: Composed of carbon atoms arranged in hexagonal layers, it is a good conductor of electricity and widely used in pencils and electrodes.
• Graphene: A single layer of graphite with remarkable strength, flexibility, and electrical conductivity, used in advanced electronics and materials.
• Applications of Fullerenes:
• Used in drug delivery systems due to their ability to encapsulate molecules.
• Applied in solar cells and lightweight batteries for renewable energy solutions.
• Utilized in lubricants and catalysts due to their spherical structure.
• Explored for antioxidant properties and potential in therapeutic treatments.
• Important Discoveries:
• Fullerenes were first synthesized using the laser vaporization of graphite.
• Their spherical structure was confirmed via X-ray crystallography.
• Since the discovery of C60, larger fullerenes like C70, C76, and C84 have also been identified.
• Interesting Facts:
• C60 has a diameter of approximately 0.7 nanometers, making it suitable for nanoscale applications.
• It is highly resistant to radiation and heat, making it suitable for use in extreme environments.
• The term “buckyballs” is often used interchangeably with fullerenes due to the spherical shape of C60 molecules.

The correct answer is 9J.

Key Points

• Kinetic Energy (KE) is the energy possessed by a body due to its motion. It is given by the formula: KE = (1/2)mv², where:
• m is the mass of the body (in kilograms).
• v is the velocity of the body (in meters per second).
• Here, the given values are:
• Mass (m) = 2 kg.
• Velocity (v) = 3 m/s.
• Substituting the values in the formula:
• KE = (1/2) × 2 × (3)².
• KE = 1 × 9.
• KE = 9 J.
• Thus, the kinetic energy of the body is 9 Joules.

Additional Information

• Energy is the capacity to do work. It exists in various forms such as kinetic energy, potential energy, thermal energy, and more.
• Kinetic Energy:
• Depends on the mass and velocity of the object.
• The unit of kinetic energy in the International System of Units (SI) is the joule (J).
• An object with greater velocity will have higher kinetic energy if the mass remains constant.
• Potential Energy:
• This is the energy a body possesses due to its position or configuration.
• For example, a raised object has gravitational potential energy.
• Formula: PE = mgh, where:
• m = mass of the object.
• g = acceleration due to gravity (9.8 m/s²).
• h = height of the object above the reference point.
• Work-Energy Theorem:
• The work done on an object is equal to the change in its kinetic energy.
• For instance, applying a force on a stationary object can increase its velocity, thereby increasing its kinetic energy.
• Applications of Kinetic Energy:
• Kinetic energy is used in various engineering applications, such as designing vehicles, wind turbines, and projectiles.
• It is also a key concept in sports, such as calculating the impact of a moving ball or athlete.

Important Points

• SI Unit of Energy: The standard unit of energy is the joule (J), where 1 joule = 1 kg·m²/s².
• Conservation of Energy: Energy can neither be created nor destroyed; it can only be transformed from one form to another.
• Relation Between Kinetic and Potential Energy:
• In a closed system, the total mechanical energy (sum of kinetic and potential energy) remains constant.
• For example, in a pendulum, energy oscillates between kinetic and potential forms.
• Practical Example: A car moving at a speed of 60 km/h has significant kinetic energy due to its mass and velocity. This energy must be dissipated (e.g., through brakes) to bring the car to rest.

The correct answer is Uterus lining.

Key Points

• After fertilization, the zygote travels through the oviduct (fallopian tube) towards the uterus.
• By the time it reaches the uterus, the zygote has developed into a structure called a blastocyst.
• The uterus lining, also known as the endometrium, is prepared during the menstrual cycle to receive and nourish the blastocyst. It thickens and becomes rich in blood vessels under the influence of hormones like estrogen and progesterone.
• Implantation occurs in the uterus lining, where the blastocyst attaches itself and begins to grow, forming a connection with the mother’s blood supply.
• The endometrium provides all the necessary nutrients and support to the developing embryo during the early stages of pregnancy.
• Other structures like the oviduct, ovary, and cervix do not play a role in implantation. These structures serve different functions in the reproductive process.

Additional Information

• Oviduct (Fallopian Tube):
• The oviduct is the site of fertilization, where the sperm meets the egg.
• It plays no role in implantation but helps transport the fertilized egg (zygote) to the uterus.
• Blockage or dysfunction in the oviduct can lead to complications such as ectopic pregnancy, where the embryo implants outside the uterus.
• Ovary:
• The ovaries produce ova (eggs), which are necessary for reproduction.
• They also produce hormones like estrogen and progesterone that regulate the menstrual cycle and prepare the uterus for potential pregnancy.
• However, the ovary does not contribute to the implantation process.
• Cervix:
• The cervix is the lower part of the uterus that opens into the vagina.
• It serves as a passage for sperm to travel to the uterus and for menstrual blood to exit the body.
• The cervix does not play any direct role in the process of implantation.
• Endometrium:
• The endometrium undergoes cyclic changes during the menstrual cycle to prepare for potential implantation.
• If implantation does not occur, the endometrium is shed during menstruation.
• If implantation occurs, the endometrium provides the growing embryo with nutrients and support until the placenta forms.
• Hormonal Regulation:
• Hormones like estrogen and progesterone are essential for preparing the endometrium for implantation.
• These hormones are primarily produced by the corpus luteum in the ovary during the luteal phase of the menstrual cycle.
• After implantation, the developing embryo secretes human chorionic gonadotropin (hCG), which helps maintain the corpus luteum and continued progesterone production.

Important Points

• Conditions for Successful Implantation:
• A healthy and receptive uterus lining is crucial for successful implantation.
• Hormonal imbalances, infections, or structural abnormalities in the uterus can hinder implantation.
• Ectopic Pregnancy:
• If implantation occurs outside the uterus, such as in the oviduct, it leads to a condition called ectopic pregnancy.
• This is a medical emergency that requires immediate attention.
• Role of Placenta:
• After implantation, the placenta begins to form and takes over the role of providing nutrients and oxygen to the growing fetus.
• The placenta also removes waste products and produces hormones to sustain pregnancy.

The correct answer is O₂.

Key Points

• O₂ is a diatomic molecule formed by a non-metallic element, specifically oxygen. It is composed of two oxygen atoms bonded together by a covalent bond.
• The term "diatomic" refers to molecules composed of two atoms, which may be of the same or different chemical elements. In the case of O₂, it is homonuclear, meaning both atoms are the same.
• Oxygen is a non-metallic element found in group 16 of the periodic table and is essential for life processes such as respiration and combustion.
• The electron configuration of oxygen allows it to form a double covalent bond with another oxygen atom, resulting in the stable diatomic molecule O₂.
• Other options such as He, Mg, and Na are not diatomic molecules formed by non-metals. Helium (He) is a noble gas, magnesium (Mg) and sodium (Na) are metals, and none of these form diatomic molecules under normal conditions.

Additional Information

• Helium (He):
• Helium is a noble gas and is monoatomic, meaning it exists as single atoms rather than forming diatomic molecules.
• It is chemically inert due to its complete outer electron shell, which prevents it from forming bonds with other atoms.
• Helium has applications in cryogenics, balloon inflation, and as a coolant in nuclear reactors.
• Magnesium (Mg):
• Magnesium is a metal located in group 2 of the periodic table and does not form diatomic molecules.
• It reacts with oxygen to form magnesium oxide (MgO), a compound rather than a diatomic molecule.
• Magnesium is used in alloy formation, electronics, and as a lightweight material in aerospace industries.
• Sodium (Na):
• Sodium is a metal from group 1 of the periodic table and does not form diatomic molecules either.
• It reacts readily with other elements such as chlorine to form sodium chloride (NaCl), a compound.
• Sodium is extensively used in chemical industries, food preservation (as table salt), and glass manufacturing.
• Diatomic Molecules:
• Diatomic molecules like O₂, N₂, H₂, and Cl₂ are common examples of molecules formed by non-metallic elements.
• These molecules are crucial in various natural and industrial processes such as photosynthesis, respiration, and chemical synthesis.

Important Points

• Oxygen (O₂): Oxygen is abundant in Earth's atmosphere, comprising approximately 21% of the air by volume. It is vital for life processes and plays a key role in various chemical reactions.
• Non-metallic Elements: Non-metallic elements like oxygen, nitrogen, and hydrogen tend to form covalent bonds and often exist as diatomic molecules.
• Periodic Table Groups: Group 16 elements, including oxygen, are known as chalcogens and exhibit properties conducive to forming diatomic molecules.
• Applications: Diatomic oxygen (O₂) is extensively used in medical treatments, industrial processes, and combustion reactions.
• Bond Nature: The double bond in O₂ contributes to its stability and reactivity, making it a highly versatile molecule.

The correct answer is Isotopes have the same atomic number but differ in their number of neutrons, which affects physical properties like mass..

Key Points

• Isotopes: Isotopes are variants of a particular chemical element that have the same number of protons (and hence the same atomic number) but a different number of neutrons, which affects their atomic mass.
• Chemical Properties: The chemical properties of an atom are primarily determined by its electron configuration, which is dictated by the atomic number (number of protons). Since isotopes of an element have the same atomic number, their chemical properties are identical.
• Physical Properties: Physical properties, such as density, melting point, and boiling point, can vary between isotopes because these properties are influenced by the mass of the atom. The differing number of neutrons causes a difference in atomic mass.
• Example: Hydrogen has three isotopes: protium (1 proton, 0 neutrons), deuterium (1 proton, 1 neutron), and tritium (1 proton, 2 neutrons). While their chemical properties are the same, their physical properties, like mass, differ significantly.
• Correct Statement: The statement that isotopes have the same atomic number but differ in their number of neutrons, which affects physical properties like mass, is accurate. This is why isotopes exhibit identical chemical behavior but different physical characteristics.

Additional Information

• Definition of Isotopes: Isotopes are atoms of the same element that have the same number of protons but a different number of neutrons. This difference in neutrons leads to variations in atomic mass.
• Applications of Isotopes:
• Carbon Dating: The isotope Carbon-14 is used to determine the age of archaeological specimens through radiocarbon dating.
• Medical Imaging: Isotopes like Technetium-99m are widely used in medical imaging techniques like SPECT scans.
• Nuclear Energy: Uranium isotopes, such as Uranium-235, are used as fuel in nuclear reactors.
• Industrial Applications: Isotopes like Cobalt-60 are used for sterilization and radiation therapy.
• Key Characteristics of Isotopes:
• Stability: Some isotopes are stable, while others are radioactive and decay over time.
• Mass Number: The mass number of an isotope is the sum of protons and neutrons in its nucleus.
• Physical Differences: Isotopes can exhibit differences in physical properties, such as:
• Density: Heavier isotopes of an element are denser.
• Boiling and Melting Points: These can vary slightly based on mass differences.
• Examples of Isotopes:
• Chlorine: Exists as Chlorine-35 and Chlorine-37.
• Oxygen: Has isotopes like Oxygen-16, Oxygen-17, and Oxygen-18.

Important Points

• Misconceptions Addressed:
• Statement 2: Incorrect as isotopes have the same atomic structure except for the number of neutrons.
• Statement 3: Incorrect because isotopes do not differ in the number of protons; they differ in neutrons.
• Statement 4: Incorrect as isotopes have the same number of protons but differ in neutrons, not electrons.
• Relevance in Science: Understanding isotopes is crucial for fields like nuclear physics, chemistry, biology, and archaeology.
• Periodic Table: Isotopes do not change the placement of elements in the periodic table since placement is based on the atomic number (protons).

The correct answer is It decreases with altitude.

Key Points

• Acceleration due to gravity (g) decreases as altitude increases. This is because gravity is inversely proportional to the square of the distance from the Earth's center. Hence, as we move further away from the Earth's surface, the value of g reduces.
• At the Earth's surface, the value of acceleration due to gravity is approximately 9.8 m/s². However, as altitude increases, this value decreases.
• The mathematical relationship between gravity and altitude is given by the formula: g' = g (R / (R + h))², where R is the radius of the Earth and h is the height above the surface.
• The statement "It increases as we go inside the Earth" is incorrect. Gravity actually decreases as we go deeper into the Earth beyond a certain point because the mass attracting you is not entirely beneath you but distributed around you.
• The statement "It is the same everywhere on Earth" is incorrect because the value of gravity varies with altitude, latitude, and geological formations.
• The statement "It remains constant with depth" is incorrect as gravity decreases with depth due to the reduction in the effective mass beneath the point of observation.

Additional Information

• Factors Affecting Gravity:
• Altitude: As we move to higher altitudes, the distance from the Earth's center increases, leading to a decrease in gravity. For example, gravity at the top of Mount Everest is slightly less than at sea level.
• Depth: Inside the Earth, gravity decreases as we move towards the core because the effective mass attracting the object decreases.
• Latitude: Gravity is slightly stronger at the poles and weaker at the equator due to the Earth's oblate shape (flattened at the poles and bulging at the equator).
• Local Geological Variations: Variations in the density of rocks and other subsurface structures can cause small changes in gravity.
• Gravitational Formula: Gravity at a height h above the Earth's surface can be calculated using g' = g (R / (R + h))². Here:
• g = Gravity at sea level
• R = Radius of the Earth (approximately 6,371 km)
• h = Height above the Earth's surface
• Gravity and Depth: As we go deeper into the Earth, gravity decreases linearly with depth in the upper layers, assuming uniform density. Beyond a certain depth, the relationship becomes more complex due to variations in density.
• Gravity on Other Celestial Bodies: Gravity varies significantly on other planets and celestial bodies due to differences in their masses and radii. For example, gravity on the Moon is about 1/6th of that on Earth.

Important Points

• Variation in Gravity on Earth:
• Gravity at the equator is approximately 9.78 m/s², while at the poles, it is about 9.83 m/s².
• Gravity decreases by about 0.3086 mGal per meter increase in altitude.
• Effects of Gravity Variations:
• Astronauts experience microgravity in space because they are in free fall around the Earth.
• Variations in gravity are used in geological surveys to detect subsurface mineral and oil deposits.

The correct answer is Lungs.

Key Points

• The lungs are the primary organs of the respiratory system and are located in the chest cavity, enclosed by the rib cage.
• The chest cavity is also known as the thoracic cavity, and it houses vital organs like the lungs, heart, trachea, and esophagus.
• The lungs are responsible for the process of gas exchange, where oxygen is absorbed into the blood and carbon dioxide is expelled from the body.
• They are protected by the pleura, a double-layered membrane that reduces friction during breathing movements.
• The lungs are connected to the trachea via the bronchi, which branch further into smaller bronchioles and end in alveoli for effective gas exchange.

Additional Information

• Pharynx:
• The pharynx is a part of the digestive and respiratory systems.
• It is located in the neck and throat region and not in the chest cavity.
• The pharynx acts as a passageway for food to the esophagus and air to the larynx.
• Pancreas:
• The pancreas is located in the abdominal cavity, behind the stomach.
• It plays a vital role in both the digestive system (producing enzymes) and the endocrine system (producing insulin).
• Liver:
• The liver is the largest gland in the human body and is situated in the upper right abdomen.
• It performs essential functions like detoxification, production of bile, and protein synthesis.
• Chest Cavity:
• The chest cavity contains the lungs and heart, protected by the rib cage and diaphragm.
• The diaphragm is a dome-shaped muscle that aids in respiration by contracting and relaxing during breathing.

Important Points

• Human Respiratory System:
• The primary function of the respiratory system is gas exchange – oxygen is absorbed, and carbon dioxide is expelled.
• The system includes the nasal cavity, pharynx, larynx, trachea, bronchi, and lungs.
• Protection of Chest Cavity:
• The rib cage protects organs in the chest cavity, including the lungs and heart.
• The pleural membrane reduces friction during respiratory movements.
• Significance of the Lungs:
• The lungs are crucial for maintaining blood oxygen levels necessary for cellular functions.
• Diseases such as pneumonia, asthma, and lung cancer can impair their function significantly.

The correct answer is Ciliary muscles.

Key Points

• Ciliary muscles are responsible for modifying the curvature of the eye lens, enabling it to adjust its focal length for clear vision at different distances. This process is known as accommodation.
• When the ciliary muscles contract, the tension on the suspensory ligaments decreases, causing the lens to become thicker and more convex, which is ideal for focusing on nearby objects.
• When the ciliary muscles relax, the tension on the suspensory ligaments increases, causing the lens to become thinner and less convex, which is suitable for focusing on distant objects.
• This adjustment mechanism plays a vital role in ensuring the light rays focus directly on the retina for clear vision.
• The ciliary muscles are a part of the uveal tract of the eye and work in coordination with other parts of the eye, such as the cornea and retina, to facilitate vision.

Additional Information

• Retina:
• The retina is the light-sensitive layer at the back of the eye that converts light signals into neural signals.
• It is not involved in modifying the curvature of the lens but plays a crucial role in image perception by transmitting signals to the brain via the optic nerve.
• Iris:
• The iris is the colored part of the eye that controls the size of the pupil, regulating the amount of light entering the eye.
• It does not directly influence the lens's curvature or focal length.
• Pupil:
• The pupil is the opening in the center of the iris that allows light to enter the eye.
• While its size is controlled by the iris, it has no role in changing the lens's curvature.
• Lens:
• The lens is a transparent, flexible, and biconvex structure that focuses light on the retina by changing its shape.
• However, the ability to change shape is facilitated by the action of the ciliary muscles.
• Accommodation Process:
• This is the process by which the lens changes its shape to focus on objects at varying distances.
• The ciliary muscles, suspensory ligaments, and lens work together to achieve this.

Important Points

• The ciliary muscles are controlled by the parasympathetic nervous system via the oculomotor nerve (cranial nerve III).
• Presbyopia, a common age-related condition, occurs when the lens loses its flexibility, and the ciliary muscles weaken, leading to difficulty focusing on nearby objects.
• Myopia (nearsightedness) and hyperopia (farsightedness) are refractive errors that can occur due to improper focusing of light on the retina, though they are not directly related to the ciliary muscles.
• Surgical interventions such as lens replacement or laser surgeries do not involve the ciliary muscles but aim to correct refractive errors by modifying the cornea or replacing the lens.

The correct answer is 2Al + 6HCl → 2AlCl₃ + 3H₂.

Key Points

• Reaction of Aluminum with Hydrochloric Acid: The reaction of aluminum (Al) with hydrochloric acid (HCl) is a classic example of a metal reacting with an acid to produce a salt and hydrogen gas. The correct balanced equation is 2Al + 6HCl → 2AlCl₃ + 3H₂.
• Stoichiometry of the Reaction: In this reaction, 2 moles of aluminum react with 6 moles of hydrochloric acid to produce 2 moles of aluminum chloride (AlCl₃) and 3 moles of hydrogen gas (H₂). This balanced equation ensures that the law of conservation of mass is satisfied.
• Formation of Aluminum Chloride: Aluminum forms aluminum chloride (AlCl₃) as it reacts with hydrochloric acid. Aluminum chloride is an ionic compound where aluminum donates electrons to chlorine, forming strong ionic bonds.
• Release of Hydrogen Gas: During the reaction, hydrogen gas (H₂) is released as a byproduct. This can be observed as bubbles forming in the reaction mixture.
• Importance of the Reaction: This reaction is significant in understanding the behavior of metals with acids and is commonly used in laboratory experiments to study chemical reactivity and gas evolution.

Additional Information

• Aluminum:
• Aluminum is a highly reactive metal with the atomic number 13.
• It is lightweight, corrosion-resistant, and widely used in industries such as aerospace, construction, and packaging.
• Its high reactivity allows it to react with acids like HCl to produce hydrogen gas and salts such as aluminum chloride.
• Hydrochloric Acid (HCl):
• Hydrochloric acid is a strong acid commonly used in laboratories and industrial processes.
• It is composed of hydrogen chloride dissolved in water and has the chemical formula HCl.
• In reactions, HCl donates hydrogen ions (H⁺) to form salts and liberates gases in reactions with metals.
• Applications of the Reaction:
• This reaction is utilized in industrial processes for hydrogen gas production and the preparation of aluminum salts.
• It is also used in chemical demonstrations to showcase metal reactivity with acids.
• Key Characteristics of Hydrogen Gas:
• Hydrogen gas (H₂) is colorless, odorless, and highly flammable.
• It is the lightest element and is used in applications such as fuel cells, welding, and as a reducing agent in chemical reactions.
• The "pop test" is used to confirm the presence of hydrogen gas, where a burning splint produces a characteristic popping sound.

Important Points

• Balanced Chemical Equations: A balanced equation ensures the law of conservation of mass is maintained, with equal numbers of each atom on both sides of the reaction.
• Common Errors: Incorrect stoichiometry or omission of products like hydrogen gas can lead to errors in writing the chemical reaction.
• Industrial Use: Aluminum chloride produced from this reaction is used in industries for applications such as water purification and as a catalyst in organic synthesis.
• Safety Precautions: Hydrochloric acid is corrosive, and proper safety measures, such as wearing gloves and goggles, should be followed during the reaction to avoid harm.

The correct answer is The optical centre coincides with the centre of curvature..

Key Points

• Every spherical lens has two centres of curvature:
• A spherical lens is formed by two spherical surfaces. Each of these surfaces has a center of the sphere from which it is a part.
• These centers are called the centres of curvature of the lens.
• Hence, statement 1 is correct.
• The line joining the two centres of curvature is called the principal axis:
• The straight line passing through both centres of curvature is known as the principal axis of the lens.
• This axis is crucial for understanding the optical properties of the lens.
• Hence, statement 2 is correct.
• The optical centre coincides with the centre of curvature:
• The optical centre of a lens is a point on its principal axis where a light ray passing through does not deviate from its path.
• However, this point does not coincide with either of the centres of curvature. The optical center is a distinct point between the two spherical surfaces.
• Hence, statement 3 is incorrect.
• A concave lens also possesses two centres of curvature:
• Just like a convex lens, a concave lens also has two spherical surfaces, each with its own centre of curvature.
• This property is common to all spherical lenses, irrespective of their type.
• Hence, statement 4 is correct.

Additional Information

• Key terms related to spherical lenses:
• Principal axis: It is the straight line that passes through the two centres of curvature of a lens.
• Optical centre: A point inside the lens through which light passes without any deviation.
• Focal length: The distance between the optical centre and the focal point of the lens.
• Centre of curvature: The centre of the sphere of which a spherical surface of the lens is a part.
• Types of spherical lenses:
• Convex lens: This lens is thicker at the center and thinner at the edges. It converges light rays.
• Concave lens: This lens is thinner at the center and thicker at the edges. It diverges light rays.
• Applications of spherical lenses:
• Convex lenses: Used in magnifying glasses, cameras, and eyeglasses for correcting hypermetropia.
• Concave lenses: Used in eyeglasses for correcting myopia and in optical instruments like telescopes.
• Important properties of lenses:
• Lenses follow the lens formula: 1/f = 1/v - 1/u, where f is the focal length, v is the image distance, and u is the object distance.
• The magnification produced by a lens is given by the formula: Magnification = Height of image / Height of object = v/u.

Important Points

• Diverging and converging lenses: A concave lens is a diverging lens, while a convex lens is a converging lens.
• Uses in daily life: Lenses are used in spectacles, microscopes, telescopes, and cameras for various optical purposes.
• Real and virtual images: Convex lenses can form both real and virtual images, whereas concave lenses generally form virtual images.

The correct answer is Neuron.

Key Points

• Neuron is the fundamental unit of the nervous system and is categorized as a cell rather than a tissue. Tissues are made up of similar cells working together to perform specific functions, while neurons are specialized cells that transmit nerve signals. Hence, Neuron is not an example of tissue.
• Muscle is a type of tissue composed of muscle cells that work together to perform movement-related functions. It qualifies as a tissue. Hence, Muscle is an example of tissue.
• Blood is a connective tissue that transports oxygen, nutrients, and waste products throughout the body. It is classified as a tissue. Hence, Blood is an example of tissue.
• Bone is a specialized connective tissue that provides structural support and protection to the body. It is considered a tissue. Hence, Bone is an example of tissue.
• In biological terms, tissues are groups of similar cells that perform a specific function. Since neurons operate as individual cells and not as a collective tissue, they are distinct from the examples mentioned. Hence, Neuron does not qualify as a tissue.

Additional Information

• Tissues in the Human Body:
• Epithelial Tissue: Forms the lining of organs and structures, providing protection and secretion. Examples include the skin and lining of the digestive tract.
• Connective Tissue: Includes blood, bone, and cartilage. It provides structural support and connects various parts of the body.
• Muscle Tissue: Responsible for movement in the body. It includes skeletal, cardiac, and smooth muscle tissues.
• Nervous Tissue: Composed of neurons and glial cells, it facilitates communication and coordination within the body.
• Neuron Characteristics:
• Neurons are specialized cells that transmit electrical signals in the body.
• They consist of three main parts: the cell body, dendrites, and axon.
• Neurons are involved in processing and transmitting information in the brain and throughout the nervous system.
• Unlike tissues, neurons function individually to relay messages rather than forming collective structures.
• Examples of Tissues:
• Blood: A liquid connective tissue that circulates oxygen, nutrients, and waste.
• Bone: A rigid connective tissue that provides structural support.
• Muscle: Tissue responsible for movement and locomotion.
• Epithelial Tissue: Protects organs and regulates permeability.
• Functions of Tissues:
• Protection: Epithelial tissue protects the body from harmful external factors.
• Support: Connective tissues like bones provide structural support.
• Movement: Muscle tissue facilitates movement.
• Communication: Nervous tissue enables communication within the body.

Important Points

• Neuron vs Tissue:
• Neuron: A single specialized cell that transmits electrical impulses.
• Tissue: A group of similar cells working together to perform specific functions.
• Classification of Tissues:
• The human body has four types of tissues: epithelial, connective, muscle, and nervous.
• Each type performs distinct functions essential for survival.
• Biological Significance:
• Tissues play a pivotal role in maintaining homeostasis and ensuring proper organ function.
• Understanding the difference between cells like neurons and tissues is crucial in biology.

The correct answer is Asexual reproduction.

Key Points

• Asexual reproduction ensures a rapid increase in population as it does not require the involvement of two parents.
• This method involves a single organism reproducing offspring that are genetically identical to itself, ensuring quicker and continuous reproduction.
• Common modes of asexual reproduction include binary fission, budding, fragmentation, and vegetative propagation.
• Examples of organisms that reproduce asexually include bacteria (binary fission), yeast (budding), hydra (budding), and some plants (vegetative propagation).
• In contrast to sexual reproduction, which requires fertilization and genetic recombination, asexual reproduction is faster as it skips these complex processes.

Additional Information

• Self-pollination:
• Self-pollination occurs in plants where pollen from the same flower or plant fertilizes its ovules.
• This method is slower in population growth compared to asexual reproduction and requires specific conditions for fertilization.
• Internal fertilization:
• In this method, fertilization occurs inside the body of an organism, typically seen in animals like mammals, birds, and reptiles.
• It involves genetic recombination, which increases genetic diversity but is relatively slow compared to asexual reproduction.
• Cross-pollination:
• Cross-pollination refers to the transfer of pollen from one plant to another, ensuring genetic diversity in plants.
• While it benefits genetic variation, it is not as rapid as asexual reproduction in increasing population.
• Advantages of Asexual Reproduction:
• It is highly efficient and allows organisms to rapidly colonize an environment.
• Energy and time are conserved as there is no need to find a mate.
• It is useful in stable environments where genetic variation is less critical.
• Examples of Asexual Reproduction:
• Bacteria: Reproduce through binary fission, leading to exponential population growth.
• Hydra: Reproduces through budding, creating genetically identical offspring.
• Plants: Methods like vegetative propagation (e.g., potato tubers, strawberry runners) allow for quick reproduction.

Important Points

• Binary Fission: A form of asexual reproduction where a single organism divides into two identical cells, commonly seen in bacteria.
• Budding: A new organism develops from a bud or outgrowth of a parent, as seen in yeast and hydra.
• Fragmentation: An organism breaks into fragments, each capable of growing into a new organism, as in starfish and some algae.
• Vegetative Propagation: Plants reproduce using structures like runners, tubers, and bulbs, ensuring rapid propagation in agriculture and horticulture.

The correct answer is – 4000 N.

Key Points

• The car is moving at an initial speed of 720 km/hr. To convert this into meters per second (m/s), use the formula:
  

Speed (m/s) = Speed (km/hr) × (1000/3600).

Hence, 720 km/hr = 720 × (1000/3600) = 200 m/s.

• The car comes to a stop in 4 seconds. This indicates that the final velocity (v) is 0 m/s, and the initial velocity (u) is 200 m/s.
• The deceleration (a) can be calculated using the formula:
  

v = u + at, where:

• v = Final velocity = 0 m/s
• u = Initial velocity = 200 m/s
• t = Time = 4 seconds

Rearranging the formula: a = (v - u) / t. Substituting the values:

a = (0 - 200) / 4 = -50 m/s².

• The total mass of the car and passengers is given as 80 kg.
• The force applied by the brakes can be calculated using Newton's second law of motion:
  

F = m × a, where:

• m = Mass = 80 kg
• a = Acceleration = -50 m/s²

Substituting the values:

F = 80 × (-50) = -4000 N.

• The negative sign indicates that the force is applied in the opposite direction of the car's motion to bring it to a stop.

Additional Information

• Newton's Second Law of Motion:
• This law states that the force acting on an object is equal to the rate of change of its momentum.
• The formula is: F = m × a, where F is force, m is mass, and a is acceleration.
• It explains how the velocity of an object changes when it is subjected to an external force.
• Deceleration:
• Deceleration is negative acceleration, where the velocity of an object decreases over time.
• It occurs when the force applied is opposite to the direction of motion.
• Units of Force:
• Force is measured in Newtons (N).
• 1 Newton is defined as the force required to accelerate a mass of 1 kg by 1 m/s².
• Braking Force:
• Braking force is the resistive force applied to stop or slow down a moving vehicle.
• The magnitude of braking force depends on the mass of the vehicle, its velocity, and the time taken to stop.

Important Points

• SI Units:
• Speed is measured in m/s.
• Mass is measured in kilograms (kg).
• Acceleration is measured in m/s².
• Force is measured in Newtons (N).
• Stopping Distance and Time:
• The stopping distance of a vehicle depends on its initial velocity and the braking force applied.
• The time to stop is inversely proportional to the deceleration.
• Practical Applications:
• Understanding braking force is crucial for designing safe and efficient braking systems in vehicles.
• It helps in accident prevention and ensures the safety of passengers.

The correct answer is Tubectomy.

Key Points

• Tubectomy, also known as tubal ligation, is a permanent method of contraception for females. It involves the surgical blocking or sealing of the fallopian tubes to prevent the egg from reaching the uterus for fertilization.
• Vasectomy, which is the surgical cutting or sealing of the vas deferens, is the male counterpart to tubectomy. It is a permanent contraception method for males, preventing the release of sperm during ejaculation.
• Both procedures are considered permanent and are performed on individuals or couples who do not wish to have more children.
• Unlike other contraceptive methods like Copper-T or cervical caps, tubectomy is irreversible in most cases, making it equivalent to the permanency of vasectomy in males.
• Tubectomy is typically recommended for women who are certain about not having more children, as it involves surgical intervention with a recovery period.

Additional Information

• Coitus Interruptus:
• This is a natural method of contraception, also known as the withdrawal method, where the male withdraws his penis from the female's vagina before ejaculation to prevent sperm from entering the uterus.
• It is not a reliable method of contraception as it has a high failure rate due to the possibility of pre-ejaculate containing sperm.
• It is a temporary contraceptive method and does not provide protection against sexually transmitted infections (STIs).
• Copper-T:
• The Copper-T is an intrauterine device (IUD) inserted into the uterus to prevent pregnancy.
• It releases copper ions, which are toxic to sperm, and prevents the fertilization of the egg.
• It is a temporary, reversible method of contraception and can be used for long-term protection, typically lasting 5-10 years depending on the type.
• It is suitable for women who want effective contraception without undergoing surgery.
• Cervical Caps:
• Cervical caps are barrier methods of contraception that are inserted into the vagina to cover the cervix and block sperm entry.
• They are used with spermicide for increased effectiveness and are reusable for a certain period.
• They are not as effective as permanent methods like tubectomy or vasectomy and require proper fitting and placement.
• Tubectomy:
• It is a surgical procedure performed under anesthesia, where the fallopian tubes are cut, tied, or sealed to prevent eggs from reaching the uterus.
• It is a permanent method of contraception and is highly effective in preventing pregnancy.
• Tubectomy does not affect hormonal balance, menstrual cycles, or sexual activity of the woman.
• It is often performed postpartum or as an elective procedure for family planning.
• Important Points:
• Both tubectomy and vasectomy are considered safe and effective permanent contraception methods when performed under proper medical supervision.
• These methods are recommended for individuals who are certain about not wanting more children, as reversal procedures are complex and often not successful.
• Permanent methods like tubectomy and vasectomy do not provide protection against sexually transmitted infections (STIs). For such protection, barrier methods like condoms are recommended.
• Consultation with a healthcare provider is essential before opting for permanent contraception to ensure the individual or couple is fully informed about the procedure, risks, and consequences.

The correct answer is Option 4 (0.1 s).

Key Points

• The speed of sound in air is approximately 340 m/s. This is the standard speed used in physics calculations unless stated otherwise.
• The distance to the reflecting surface is given as 17 m. Since the echo must travel to the surface and back, the total distance covered by the sound wave is 2 × 17 = 34 m.
• The formula to calculate time is: Time = Distance / Speed. Substituting the total distance and speed, we get:
  

Time = 34 m / 340 m/s = 0.1 s.

• This implies that the echo will return to the source in 0.1 seconds.
• Hence, the correct answer is Option 4 (0.1 s).

Additional Information

• Echo:
• An echo is a sound that is reflected off a surface (such as a wall or cliff) and heard again by the original source.
• The time delay in hearing an echo depends on the distance to the reflecting surface and the speed of sound.
• For an echo to be distinctly heard, the reflecting surface should typically be at least 17 m away from the source because the human ear cannot distinguish sounds separated by less than 0.1 s.
• Speed of Sound:
• The speed of sound in air is approximately 340 m/s under standard conditions (at room temperature and normal atmospheric pressure).
• The speed of sound varies depending on the medium (e.g., air, water, or solid) and the temperature. For example, sound travels faster in water (approximately 1480 m/s) and even faster in solids.
• Distance Calculation:
• When calculating the distance traveled by sound for an echo, it's important to remember that sound travels to the reflecting surface and back. Hence, the total distance is 2 × the distance to the reflecting surface.
• Applications of Echo:
• Echoes are used in various fields, including sonar systems for underwater detection and ultrasound imaging in medical diagnostics.
• They are also used in measuring distances in geological surveys and testing the acoustics of auditoriums.

Important Points

• Practical Conditions for Echo:
• For an echo to be heard clearly, the reflecting surface must be large and rigid, such as a wall, cliff, or building.
• The minimum distance required for an echo to be heard is approximately 17 m, considering the speed of sound and the human ear's ability to distinguish sounds.
• Factors Affecting Speed of Sound:
• The speed of sound increases with temperature. For example, at 0°C, the speed is approximately 331 m/s, while at 20°C, it increases to about 343 m/s.
• Sound travels faster in denser media, such as water and solids, compared to air.

The correct answer is Xylem transports water and minerals, phloem transports food..

Key Points

• Xylem is a vascular tissue in plants primarily responsible for transporting water and dissolved minerals from the roots to the rest of the plant. It supports upward movement through processes like root pressure, capillary action, and transpiration pull.
• Phloem, on the other hand, transports organic nutrients, primarily sugar (sucrose), which is produced during photosynthesis in the leaves, to different parts of the plant, including growing regions and storage organs.
• The transport process in xylem is unidirectional (from roots to the aerial parts of the plant), whereas the transport process in phloem is bidirectional (both upward and downward, depending on the plant's needs).
• Key structural components of xylem include tracheids, vessels, xylem parenchyma, and xylem fibers, while phloem is composed of sieve tubes, companion cells, phloem parenchyma, and phloem fibers.
• This distinction in the primary functions of xylem and phloem is critical for maintaining the overall growth and metabolic activities of the plant.

Additional Information

• Xylem Details:
• Water Transport Mechanism: Water movement in xylem occurs due to physical forces like transpiration pull, capillary action, and root pressure.
• Structural Adaptations: Xylem vessels and tracheids have thick lignified walls that prevent collapse under water pressure.
• Example: Xylem vessels are prominent in angiosperms, while tracheids are the primary conducting cells in gymnosperms.
• Phloem Details:
• Translocation Process: Transport in phloem (called translocation) involves the movement of sugars and other organic compounds using energy (an active process).
• Role of Companion Cells: These cells are essential for maintaining the function of sieve tubes by providing energy and support for translocation.
• Example: Phloem is abundant in the vascular bundles of leaves, stems, and roots, ensuring sugar distribution throughout the plant.
• Comparative Features:
• Direction of Transport: Xylem transports substances only upward, while phloem can transport substances in multiple directions.
• Living vs. Non-Living: Xylem tissues (except xylem parenchyma) are non-living, whereas phloem tissues (except phloem fibers) are living.
• Energy Requirement: Xylem transport is a passive process, while phloem transport requires energy in the form of ATP.
• Important Functions:
• Water Balance: Xylem helps maintain water balance in the plant by replacing water lost during transpiration.
• Growth Support: Phloem ensures that growing parts (like buds, fruits, and flowers) receive the necessary nutrients for development.

Important Points

• Adaptations for Transport: Xylem vessels are large and hollow to facilitate water movement, while sieve tubes in phloem have perforated walls for efficient sugar transport.
• Energy Dependency: The xylem's water transport is passive, relying on physical forces, whereas phloem transport requires energy to actively move nutrients.
• Critical to Plant Survival: Both tissues are vital to the plant's survival, as they ensure the distribution of water, minerals, and food throughout the plant body.

The correct answer is Current in its turns produces a magnetic field.

Key Points

• A solenoid is a long coil of wire wound in the shape of a cylinder. When an electric current flows through its turns, it produces a magnetic field.
• The magnetic field inside a solenoid is nearly uniform and strong, resembling the magnetic field of a bar magnet. This property makes it behave like a magnet.
• The strength of the magnetic field produced depends on factors such as the number of turns in the solenoid, the current passing through it, and the material of the core (if any).
• When the solenoid is wound around a magnetic material like iron, its magnetic field becomes even stronger due to the material's high magnetic permeability.
• This principle of solenoids is widely used in electromagnets, which are employed in various applications like electric bells, relays, and magnetic locks.

Additional Information

• Magnetic Field in a Solenoid:
• The magnetic field inside a solenoid is given by the formula: B = μ₀ × n × I, where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current.
• The field lines inside the solenoid are parallel, indicating a uniform magnetic field.
• Applications of Solenoids:
• Solenoids are used to create controlled magnetic fields for scientific experiments.
• They are employed in electromagnets, which are used in devices like electric motors, loudspeakers, and MRI machines.
• In industrial applications, solenoids are used in valves, actuators, and relays.
• Difference Between Permanent Magnets and Solenoids:
• Permanent magnets produce a constant magnetic field, while solenoids generate a magnetic field only when electric current flows through them.
• Solenoids can be turned on or off and their strength can be controlled by varying the current, making them more versatile for applications.
• Iron Core in a Solenoid:
• Adding an iron core to the solenoid increases the magnetic field's strength due to the core's magnetic properties.
• This configuration is used in electromagnets to achieve high magnetic field strengths for specific applications.

Important Points

• Iron Core vs. Air Core:
• Solenoids with an iron core (also called ferromagnetic core) produce much stronger magnetic fields compared to solenoids with an air core.
• Air core solenoids are used when a uniform magnetic field is required without interference from the core material.
• Electromagnets vs. Permanent Magnets:
• Electromagnets, which use solenoids, can be switched on and off and their magnetic strength is adjustable, unlike permanent magnets.
• They are commonly used in applications requiring dynamic control, such as electric motors and transformers.
• Discovery:
• The magnetic effect of electric current was discovered by Hans Christian Ørsted in 1820, laying the foundation for the development of solenoids and electromagnets.

The correct answer is Gas.

Key Points

• Matter exists in three primary states: solids, liquids, and gases, distinguished by the arrangement and energy of their particles.
• In a Gas, particles possess the highest speed of rapid random motion due to their maximum kinetic energy and weakest intermolecular forces. These particles move freely in all directions at very high velocities.
• In a Solid, particles are held together by strong intermolecular forces in a fixed lattice. They vibrate about their mean positions, resulting in the lowest speed of motion. Hence, this statement is incorrect.
• In a Liquid, particles have moderate kinetic energy and can slide over one another, allowing the liquid to flow. Their speed is higher than solids but lower than gases. Hence, this statement is incorrect.
• The statement that both solids and liquids possess the highest speed of motion is incorrect, as gases clearly dominate in terms of particle speed due to minimal intermolecular forces.

Additional Information

• Kinetic Molecular Theory:
• This theory explains that matter consists of small particles in constant motion, with the motion proportional to the temperature of the substance.
• As temperature increases, the average kinetic energy of particles increases, causing them to move faster.
• Brownian Motion:
• Brownian motion refers to the random, zig-zag movement of microscopic particles in liquids or gases caused by collisions with fast-moving atoms or molecules.
• It was first observed by Robert Brown in 1827.
• Compressibility and Density:
• Gases are highly compressible due to large spaces between particles.
• Solids are incompressible and have high density because of negligible spaces between particles.
• Liquids have fixed volume but no fixed shape, taking the shape of their container.
• Diffusion:
• Diffusion is the process by which particles spread out to fill a space.
• The rate of diffusion is fastest in gases due to high speed and large spaces between particles, and slowest in solids.

Important Points

• Plasma:
• Plasma is considered the fourth state of matter, consisting of super-heated, ionized gas.
• Particles in plasma move at even higher speeds due to extreme energy levels, as found in stars or lightning bolts.
• Absolute Zero:
• At 0 Kelvin (or -273.15 degrees Celsius), the motion of particles theoretically stops.
• This is the lowest possible temperature in the universe.
• Intermolecular Forces vs. Thermal Energy:
• The state of matter is determined by the competition between intermolecular forces (which keep particles together) and thermal energy (which drives them apart).
• In gases, thermal energy dominates, allowing particles to move freely and rapidly.

The correct answer is Alcohol.

Key Points

• The functional group -OH, known as the hydroxyl group, is characteristic of alcohols in organic chemistry.
• Alcohols are organic compounds where the -OH group is bonded to a saturated carbon atom (sp³ hybridized).
• The presence of the hydroxyl group in alcohols makes them capable of forming hydrogen bonds, contributing to their high boiling points compared to hydrocarbons of similar molecular weight.
• Alcohols are classified into three types based on the number of alkyl groups attached to the carbon bonded to the hydroxyl group: primary (1°), secondary (2°), and tertiary (3°) alcohols.
• Examples of alcohols include methanol (CH₃OH), ethanol (C₂H₅OH), and isopropanol (C₃H₇OH), which are widely used in industries and households.

Additional Information

• Carboxylic Acid:
• Carboxylic acids contain the functional group -COOH, which is composed of a carbonyl group (C=O) and a hydroxyl group (-OH).
• Carboxylic acids are weak acids and commonly found in nature, such as in acetic acid (vinegar) and citric acid.
• They are used in the production of polymers, pharmaceuticals, and food preservatives.
• Aldehyde:
• Aldehydes contain the functional group -CHO, where a carbonyl group (C=O) is bonded to a hydrogen atom and a carbon atom.
• They are commonly used in industries for manufacturing resins, dyes, and fragrances.
• Examples include formaldehyde (HCHO) and acetaldehyde (CH₃CHO).
• Ketone:
• Ketones have the functional group C=O bonded to two carbon atoms.
• Unlike aldehydes, ketones do not have a hydrogen atom attached directly to the carbonyl group.
• Common examples include acetone (CH₃COCH₃), which is widely used as a solvent.
• Alcohol:
• Alcohols are used as solvents, fuels, and disinfectants. Ethanol is a key component in alcoholic beverages.
• They are also used in the manufacture of pharmaceuticals and cosmetics.
• Alcohols exhibit properties such as hydrophilicity due to the hydroxyl group, making them miscible with water.

Important Points

• Hydrogen Bonding:
• The hydroxyl group (-OH) in alcohols forms hydrogen bonds with water molecules, enhancing their solubility in water.
• This property is responsible for the higher boiling points of alcohols compared to similar molecular weight hydrocarbons.
• Nomenclature:
• Alcohols are named using the IUPAC system by replacing the suffix "-e" in alkanes with "-ol." For example, methane becomes methanol.
• Industrial Applications:
• Alcohols are extensively used in industries for the production of plastics, synthetic fibers, and antifreeze.
• They also serve as intermediates in chemical synthesis, particularly for esters and ethers.

The correct answer is Unbalanced force.

Key Points

• Unbalanced force is a force that causes a change in the state of motion of an object. It is responsible for accelerating or decelerating an object, changing its direction, or altering its shape.
• An unbalanced force occurs when the net force acting on an object is not equal to zero. This results in a change in the object's velocity, which could mean a change in speed, direction, or both.
• For example, when a person pushes a stationary object with enough force to overcome friction and inertia, the object moves. This motion occurs due to the unbalanced force acting on it.
• In the absence of unbalanced forces, an object will either remain stationary (if initially at rest) or continue moving in a straight line at constant speed (if initially in motion). This is in accordance with Newton's First Law of Motion.
• Unbalanced forces are fundamental to many phenomena in daily life, such as driving a car, playing sports, or the movement of celestial bodies.

Additional Information

• Balanced Force:
• When the net force acting on an object is zero, the forces are termed as balanced forces.
• Balanced forces do not change the state of motion of an object; they keep it at rest or moving uniformly in a straight line.
• Example: A book resting on a table experiences balanced forces – the force of gravity pulling it down is balanced by the upward normal force exerted by the table.
• Gravitational Force:
• This is the attractive force exerted by a massive body, such as Earth, on other objects.
• It is responsible for phenomena like objects falling to the ground and the orbits of planets around the Sun.
• Gravitational force acts on all objects with mass and is proportional to their masses and inversely proportional to the square of the distance between them.
• Frictional Force:
• Friction is a force that opposes the relative motion between two surfaces in contact.
• It acts parallel to the surfaces and can either prevent motion (static friction) or resist motion (kinetic friction).
• Frictional force plays a crucial role in enabling walking, driving, and holding objects but also causes wear and heat generation.
• Newton's Laws of Motion:
• Newton's First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by an unbalanced force.
• Newton's Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma).
• Newton's Third Law: For every action, there is an equal and opposite reaction.
• Applications of Unbalanced Forces:
• Unbalanced forces are responsible for starting, stopping, and changing the direction of motion in vehicles.
• They are crucial in sports, where players apply forces to accelerate, decelerate, or change the direction of balls or themselves.
• Rocket launches utilize unbalanced forces generated by the expulsion of gases to propel themselves upward.

Important Points

• Force:
• Force is defined as a push or pull that can cause an object to accelerate, decelerate, or change its shape or direction.
• It is a vector quantity, meaning it has both magnitude and direction.
• The SI unit of force is the Newton (N).
• Examples of Unbalanced Forces:
• A car accelerating on a highway experiences unbalanced forces due to the engine's thrust overcoming air resistance and friction.
• When a ball is kicked, the force applied by the foot overcomes the balanced forces of gravity and friction, causing motion.
• A spacecraft maneuvering in outer space applies unbalanced forces via thrusters to change its velocity or direction.
• Practical Implications:
• Understanding unbalanced forces helps in designing efficient vehicles, aircraft, and machinery.
• It aids in predicting and controlling the motion of objects in sports and engineering applications.

The correct answer is It decreases..

Key Points

• pH is a measure of the acidity or alkalinity of a solution, determined by the concentration of hydrogen ions (H⁺) present in it.
• When the concentration of H⁺ ions increases, the solution becomes more acidic, resulting in a decrease in pH.
• The pH scale ranges from 0 to 14, where a pH of 7 is neutral, below 7 indicates acidity, and above 7 indicates alkalinity.
• The relationship between pH and hydrogen ion concentration is logarithmic: pH = -log[H⁺]. As [H⁺] increases, the logarithmic value becomes smaller, causing the pH to decrease.
• For example, if the concentration of H⁺ ions increases from 10⁻⁷ M to 10⁻⁴ M, the pH decreases from 7 to 4, indicating a much stronger acidic environment.

Additional Information

• Acidity and Alkalinity:
• Acidity refers to the presence of a higher concentration of H⁺ ions, while alkalinity indicates a higher concentration of hydroxide ions (OH⁻).
• Highly acidic solutions have a pH closer to 0, while highly alkaline solutions have a pH closer to 14.
• pH Indicators:
• pH can be measured using pH meters, litmus paper, or universal indicators.
• Litmus paper turns red in acidic solutions and blue in alkaline solutions.
• Applications of pH:
• pH is crucial in industries like agriculture, medicine, and water treatment for monitoring the acidity or alkalinity of solutions.
• For example, maintaining the optimal pH of soil is essential for plant growth.
• Effect of pH on Biological Systems:
• Human blood has a pH range of 7.35 to 7.45, and deviations from this range can lead to severe health issues.
• The functioning of enzymes and metabolic processes in organisms is highly dependent on maintaining a stable pH.

Important Points

• Logarithmic Nature: A small change in H⁺ concentration results in a significant shift in pH due to the logarithmic scale.
• Environmental Impact: Acid rain, caused by pollutants like sulfur dioxide and nitrogen oxides, decreases the pH of soil and water bodies, harming ecosystems.
• Food Science: pH is a critical factor in food preservation, fermentation, and flavor development in products like yogurt and pickles.
• Buffer Systems: Buffer solutions resist changes in pH when small amounts of acid or base are added, playing a vital role in maintaining biological and chemical stability.

The correct answer is Fore-brain.

Key Points

• The fore-brain, also known as the cerebrum, is the main thinking part of the human brain. It is responsible for higher cognitive functions such as reasoning, decision-making, memory, and emotions.
• The fore-brain consists of two cerebral hemispheres connected by a structure called the corpus callosum. This allows communication between the left and right hemispheres of the brain.
• It plays a key role in sensory perception, including sight, hearing, touch, taste, and smell, by processing sensory input received from the environment.
• Key regions of the fore-brain include the thalamus (a relay center for sensory and motor signals) and the hypothalamus (which regulates homeostasis, hunger, thirst, and emotions).
• The cerebral cortex, the outermost layer of the fore-brain, is responsible for advanced functions such as language, abstract thinking, and voluntary motor control.

Additional Information

• Medulla:
• The medulla is a part of the hind-brain and is responsible for controlling vital autonomic functions such as breathing, heart rate, and blood pressure.
• It acts as a relay center for nerve signals between the brain and the spinal cord.
• Spinal Cord:
• The spinal cord is a long, cylindrical structure of nervous tissue that extends from the medulla and is enclosed within the vertebral column.
• It serves as the main pathway for transmitting information between the brain and the rest of the body, including motor commands and sensory data.
• Mid-brain:
• The mid-brain is located between the fore-brain and hind-brain and is involved in functions such as eye movement, hearing, and motor control.
• It contains structures like the tectum and tegmentum, which play roles in reflex actions and the regulation of alertness.
• Fore-brain:
• The fore-brain is the largest and most complex part of the brain. It is divided into four lobes: frontal, parietal, temporal, and occipital, each specialized in different functions.
• It integrates sensory information, coordinates voluntary movements, and facilitates complex thought processes.
• Disorders of the fore-brain can affect memory, language, and emotions.

Important Points

• Brain Structure:
• The human brain is divided into three main parts: the fore-brain, mid-brain, and hind-brain.
• Each part has specialized functions, with the fore-brain being the most advanced and evolved.
• Functions of the Fore-brain:
• It controls advanced cognitive functions like thinking, problem-solving, and memory.
• It is also responsible for processing sensory information and initiating voluntary muscle movements.
• Clinical Relevance:
• Damage to the fore-brain can result in disorders such as Alzheimer’s disease, Parkinson’s disease, or stroke-related impairments.

Shortcut Trick

Side of Square (a) = 16 cm (since diagonal = a√2 = 16√2).

Perimeter of Square = 4 × 16 = 64 cm.

Side of Triangle (s) = (3/5 × 64) ÷ 3 = 64/5 cm.

Ratio of Areas = [√3/4 × (64/5)²] : [16²] = [(√3/4 × 4096)/25] : 256.

Ratio = [1024√3 / 25] : 256 = 4√3 : 25.

∴ The correct answer is 4√3 : 25.

Alternate Method

Given:

Diagonal of square = 16√2 cm.

Perimeter of equilateral triangle = 3/5 × Perimeter of square.

Formula Used:

1. Diagonal of square = Side × √2
2. Area of square = Side²
3. Area of equilateral triangle = (√3/4) × Side²

Calculations:

⇒ Side of square (a) × √2 = 16√2

⇒ a = 16 cm

⇒ Perimeter of square = 4 × 16 = 64 cm

⇒ Area of square = 16 × 16 = 256 cm²

⇒ Perimeter of triangle = 3/5 × 64 = 192/5 cm

⇒ Side of triangle (s) = (192/5) ÷ 3 = 64/5 cm

⇒ Area of triangle = (√3/4) × (64/5)²

⇒ Area of triangle = (√3/4) × (4096/25) = 1024√3/25 cm²

⇒ Ratio = (1024√3/25) : 256

⇒ Ratio = (1024√3) : (256 × 25) = 4√3 : 25

∴ The correct answer is 4√3 : 25.

Additional Information

Equilateral Triangle Properties

All sides are equal. Height (h) = (√3/2) × side. Area = (√3/4) × side² = h²/√3.

Square Properties

Area = side² = (diagonal²)/2. Perimeter = 4 × side. Diagonal = side × √2.

Ratio of Areas for Similar Polygons

For two similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions (sides, perimeters, or diagonals).

Shortcut Trick

Candidate A votes = Total Voters × Cast % × Valid % × A share %

Candidate A votes = 50,000 × 80/100 × 90/100 × 70/100

⇒ 50,000 × 0.8 × 0.9 × 0.7 = 25,200

∴ The correct answer is 25200 votes.

Alternate Method

Given:

Total eligible voters = 50,000

Votes cast = 80%

Invalid votes = 10% (So, Valid votes = 90%)

Candidate A share = 70% of valid votes

Step-by-step Calculation:

⇒ Total votes cast = 50,000 × (80/100) = 40,000

⇒ Total valid votes = 40,000 × (90/100) = 36,000

⇒ Votes secured by Candidate A = 36,000 × (70/100)

⇒ Votes secured by Candidate A = 360 × 70 = 25,200

∴ The correct answer is 25200 votes.

Additional Information

Percentage Change Base

Always ensure the percentage is calculated on the correct base. Here, invalid votes are a percentage of cast votes, not total voters.

Successive Percentage

For a series of percentage reductions or shares, the final value = Initial Value × (p₁/100) × (p₂/100) × ...

Winning Margin

If Candidate A gets 70% of valid votes, then Candidate B gets 30%. The winning margin is (70% − 30%) = 40% of valid votes.

Shortcut Trick

Apply the successive percentage change formula: Net Change = a + b + (a × b)/100

Here, Markup (a) = +20% and Discount (b) = −10%.

Net Profit % = 20 − 10 + [20 × (−10)]/100

Net Profit % = 10 − 2 = 8%.

∴ The correct answer is 8%.

Alternate Method

Given: Cost Price (CP) = ₹2,600, Markup = 20%, Discount = 10%.

⇒ Marked Price (MP) = CP × (100 + Markup %)/100

⇒ MP = 2600 × (120/100) = ₹3,120

⇒ Selling Price (SP) = MP × (100 − Discount %)/100

⇒ SP = 3120 × (90/100) = ₹2,808

⇒ Profit = SP − CP = 2808 − 2600 = ₹208

⇒ Profit % = (Profit/CP) × 100

⇒ Profit % = (208/2600) × 100 = 8%

∴ The correct answer is 8%.

Additional Information

Marked Price and CP Relation

The relationship between CP and MP is given by: MP/CP = (100 + Profit%)/(100 − Discount%).

Successive Discounts

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

Dishonest Dealer

Profit % = [Error / (True Value − Error)] × 100, when a dealer uses false weights.

Shortcut Trick

To find the smaller value (Raj's share) when the sum and difference are known:

Raj's spending = (Total Sum − Difference) ÷ 2

Raj's spending = (380 − 80) ÷ 2

Raj's spending = 300 ÷ 2 = ₹150

∴ The correct answer is ₹150.

Alternate Method

Given:

Total expenditure = ₹380

Spending difference (Ravi − Raj) = ₹80

Let Raj's spending be x.

Then, Ravi's spending = x + 80

⇒ x + (x + 80) = 380

⇒ 2x + 80 = 380

⇒ 2x = 380 − 80

⇒ 2x = 300

⇒ x = 300 ÷ 2

⇒ x = 150

∴ The correct answer is ₹150.

Additional Information

Linear Equations in One Variable

Equations of the form ax + b = c where x is the unknown variable to be solved by transposing constants.

Sum and Difference Formula

If A + B = S and A − B = D, then the Larger value A = (S + D) ÷ 2 and Smaller value B = (S − D) ÷ 2.

Shortcut Trick

Total Distance = 80 + 32 = 112 km

Total Time = (80 ÷ 40) + (32 ÷ 16) = 2 + 2 = 4 hours

Average Speed = Total Distance ÷ Total Time = 112 ÷ 4 = 28 kmph

∴ The correct answer is 28 km/hr.

Alternate Method

Given:

Distance 1 (d₁) = 80 km, Speed 1 (s₁) = 40 km/hr

Distance 2 (d₂) = 32 km, Speed 2 (s₂) = 16 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Time = Distance ÷ Speed

Calculations:

⇒ Time taken for first part (t₁) = 80 ÷ 40 = 2 hours

⇒ Time taken for second part (t₂) = 32 ÷ 16 = 2 hours

⇒ Total Time = t₁ + t₂ = 2 + 2 = 4 hours

⇒ Total Distance = d₁ + d₂ = 80 + 32 = 112 km

⇒ Average Speed = 112 ÷ 4 = 28 km/hr

∴ The correct answer is 28 km/hr.

Additional Information

Average Speed (Equal Distances)

When a body covers two equal distances at speeds x and y, the average speed is 2xy/(x + y).

Relative Speed

For objects moving in the same direction, relative speed is |s₁ − s₂|; in opposite directions, it is s₁ + s₂.

Unit Conversion

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

Shortcut Trick

Cost Price (CP) : Selling Price (SP) ratio = 12000 : 13500 = 8 : 9.

Profit = 9 − 8 = 1 unit.

Profit % = (1 ÷ 8) × 100 = 12.5%.

∴ The correct answer is 12.5%.

Alternate Method

Given:

Cost Price (CP) = ₹12,000

Selling Price (SP) = ₹13,500

Formula Used:

Profit = SP − CP

Profit % = (Profit ÷ CP) × 100

Calculations:

⇒ Profit = 13,500 − 12,000 = ₹1,500

⇒ Profit % = (1,500 ÷ 12,000) × 100

⇒ Profit % = (1 ÷ 8) × 100

⇒ Profit % = 12.5%

∴ The correct answer is 12.5%.

Additional Information

Loss Percentage

When CP > SP, Loss % = [(CP − SP) ÷ CP] × 100.

Relation between SP and CP

SP = CP × [ (100 + Profit %) ÷ 100 ] or SP = CP × [ (100 − Loss %) ÷ 100 ].

Marked Price and Discount

Discount is always calculated on the Marked Price (MP). SP = MP − Discount.

Shortcut Trick

In a 30°-60°-90° triangle, the side opposite to 30° is 1/√3 times the side opposite to 60°.

Base (opposite to 60°) = 20 m.

Height (opposite to 30°) = 20 / √3 m.

Using the given approximation √3 ≈ 2:

Height = 20 ÷ 2 = 10 m.

∴ The correct answer is 10 m.

Alternate Method

Given: Distance from base (B) = 20 m, Angle of elevation (θ) = 30°.

Formula Used: tan(θ) = Perpendicular / Base

⇒ tan(30°) = Height / 20

⇒ 1/√3 = Height / 20

⇒ Height = 20 / √3

⇒ Since √3 ≈ 2 is given:

⇒ Height = 20 ÷ 2

⇒ Height = 10 m

∴ The correct answer is 10 m.

Additional Information

Trigonometric Ratios

Standard values for tan include tan(30°) = 1/√3, tan(45°) = 1, and tan(60°) = √3.

Angle of Elevation

It is the angle between the horizontal line of sight and the line of sight up to an object.

Right Triangle Properties

Pythagoras theorem states that Base² + Perpendicular² = Hypotenuse².

Shortcut Trick

Teacher's Age = New Average + (Initial Number of Students × Increase in Average)

Increase in average = 34 − 33 = 1 year.

Teacher's Age = 34 + (25 × 1) = 34 + 25 = 59 years.

∴ The correct answer is 59 years.

Alternate Method

Given:

Average age of 25 students = 33 years.

Average age of 26 people (including teacher) = 34 years.

Formula Used:

Total = Average × Number of entities

Calculations:

⇒ Total age of 25 students = 25 × 33

⇒ Total age of 25 students = 825 years

⇒ Total age including teacher (26 people) = 26 × 34

⇒ Total age including teacher = 884 years

⇒ Teacher's Age = Total age of 26 people − Total age of 25 students

⇒ Teacher's Age = 884 − 825 = 59 years

∴ The correct answer is 59 years.

Additional Information

Including a New Member

When a new person joins a group, the Age of New Member = New Average + (Old Member Count × Change in Average).

Excluding an Existing Member

When a person leaves, the Age of Outgoing Member = Old Average + (New Member Count × Change in Average).

Replacing a Member

If one person is replaced by another, the Age of New Person = Age of Replaced Person + (Total Count × Change in Average).

Given:

$6 \times 6 + 6 \times (8 - 5)^2 \div 3 \times 9$

Concept Used:

$\text{BODMAS rule}$

Calculation:

$\Rightarrow 6 \times 6 + 6 \times (3)^2 \div 3 \times 9$

$\Rightarrow 6 \times 6 + 6 \times 9 \div 3 \times 9$

$\Rightarrow 36 + 54 \div 3 \times 9$

$\Rightarrow 36 + 18 \times 9$

$\Rightarrow 36 + 162$

$\Rightarrow 198$

∴ The correct answer is $198$

Shortcut Trick

Net multiplying factor = 1.12 × 0.92 × 1.08

0.92 × 1.08 = (1 − 0.08)(1 + 0.08) = 1 − 0.0064 = 0.9936

Population in 2018 = 9580 × 1.12 × 0.9936

Population ≈ 10729.6 × 0.9936 ≈ 10661

∴ The correct answer is 10661.

Alternate Method

Given:

Initial population (P₀) = 9580

Rate of change: r₁ = +12%, r₂ = −8%, r₃ = +8%

Calculations:

⇒ Population in 2018 = 9580 × (1 + 12/100) × (1 − 8/100) × (1 + 8/100)

⇒ Population = 9580 × (112/100) × (92/100) × (108/100)

⇒ Population = 9580 × 1.12 × 0.9936

⇒ Population = 10729.6 × 0.9936

⇒ Population = 10660.93... ≈ 10661

∴ The correct answer is 10661.

Additional Information

Successive Percentage Change

Total change = [a + b + (a × b)/100]% where a and b are successive changes. For three changes, apply the formula twice.

Multiplying Factor Method

Final Value = Initial Value × (100 ± r₁)/100 × (100 ± r₂)/100 ... and so on.

Algebraic Identity Shortcut

When a value increases and then decreases by the same x%, there is always a net decrease of (x²/100)%.

Shortcut Trick

Use the direct relationship formula: MP / CP = (100 + Profit%) / (100 − Discount%)

MP / CP = (100 + 32) / (100 − 20) = 132 / 80 = 33 / 20

Markup % = [(33 − 20) / 20] × 100 = 13/20 × 100 = 65%

∴ The correct answer is 65%.

Alternate Method

Given:Discount = 20%, Profit = 32%

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

⇒ Let Cost Price (CP) = 100

⇒ Selling Price (SP) = CP + 32% of CP = 100 + 32 = 132

⇒ Marked Price (MP) × (100 − 20)% = SP

⇒ MP × 80/100 = 132

⇒ MP = (132 × 100) / 80 = 132 × 1.25 = 165

⇒ Markup Value = MP − CP = 165 − 100 = 65

⇒ Markup % = (Markup Value / CP) × 100 = (65 / 100) × 100 = 65%

∴ The correct answer is 65%.

Additional Information

Markup Percentage

It is the percentage by which the cost price is increased to reach the marked price: Markup % = [(MP − CP) / CP] × 100.

Effective Discount Formula

For successive discounts a% and b%, Net Discount = [a + b − (a × b)/100]%.

Profit and Markup Relation

Profit % = Markup % − Discount % − [(Markup % × Discount %) / 100]. This is derived from the successive percentage change formula.

Shortcut Trick

Total Work (LCM of 3, 9, 18) = 18 units.

Combined efficiency of A, B, and C = (18 ÷ 3) + (18 ÷ 9) + (18 ÷ 18) = 6 + 2 + 1 = 9 units/day.

Work to be completed = 3 × 18 = 54 units.

Time required = 54 ÷ 9 = 6 days.

∴ The correct answer is 6 days.

Alternate Method

Given: Time taken by A = 3 days, B = 9 days, and C = 18 days.

⇒ Combined Efficiency of (A + B + C) = 6 + 2 + 1 = 9 units/day.

⇒ Original Work = 18 units.

⇒ New Work (Thrice the original) = 3 × 18 = 54 units.

⇒ Time required = Total New Work ÷ Combined Efficiency

⇒ Time = 54 ÷ 9

⇒ Time = 6 days.

∴ The correct answer is 6 days.

Additional Information

Work and Efficiency

Efficiency is inversely proportional to the time taken to complete a work. Total Work = Efficiency × Time.

Combined Work Formula

For three people taking x, y, and z days, combined time T = (x × y × z) ÷ (xy + yz + zx).

Multiple Work Case

If the work is increased by 'n' times, the time required at the same efficiency also increases by 'n' times.

Shortcut Trick

Speed of the train is the ratio of difference in tunnel lengths to the difference in time taken.

Speed = (2708 − 2195) ÷ (88 − 76) = 513 ÷ 12 = 42.75 m/s.

Length of train = (Speed × Time₁) − Tunnel Length₁ = (42.75 × 88) − 2708.

Length of train = 3762 − 2708 = 1054 m.

∴ The correct answer is 1054 m.

Alternate Method

Given: Length of Tunnel 1 (L₁) = 2708 m, Time 1 (T₁) = 88 s; Length of Tunnel 2 (L₂) = 2195 m, Time 2 (T₂) = 76 s.

Formula Used: Speed = (Length of train + Length of tunnel) ÷ Time

⇒ Let the length of the train be L.

⇒ Speed = (L + 2708) ÷ 88 = (L + 2195) ÷ 76

⇒ 76 × (L + 2708) = 88 × (L + 2195)

⇒ 19 × (L + 2708) = 22 × (L + 2195) (Dividing both sides by 4)

⇒ 19L + 51452 = 22L + 48290

⇒ 22L − 19L = 51452 − 48290

⇒ 3L = 3162

⇒ L = 1054 m

∴ The correct answer is 1054 m.

Additional Information

Train Crossing a Static Object with Length

When a train crosses a platform, bridge, or tunnel, the total distance covered is the sum of the length of the train and the length of the object.

Relative Speed

When two objects move in the same direction, relative speed is the difference (S₁ − S₂). In the opposite direction, it is the sum (S₁ + S₂).

Unit Conversion

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

Shortcut Trick

Total observations n = 10 (Even).

Median = Average of the 5^th and 6^th terms = 12.

Sum of middle terms = 12 × 2 = 24.

Arranging known values: 4, 6, 7, 9, 11, 15, 16, 20, 23.

11 + (K + 5) = 24 ⇒ K + 16 = 24 ⇒ K = 8.

∴ The correct answer is 8.

Alternate Method

Given: Data = 4, 23, 11, 7, 6, 9, (K + 5), 15, 20, 16 and Median = 12.

Formula: For even n, Median = [(n/2)^th term + (n/2 + 1)^th term] / 2.

Arranging data in ascending order: 4, 6, 7, 9, 11, (K + 5), 15, 16, 20, 23.

⇒ 5^th term = 11

⇒ 6^th term = K + 5

⇒ Median = (11 + K + 5) / 2

⇒ 12 = (K + 16) / 2

⇒ 24 = K + 16

⇒ K = 24 − 16

⇒ K = 8

∴ The correct answer is 8.

Additional Information

Median for Odd Number of Observations

If n is odd, after arranging in ascending/descending order, Median = [(n + 1) / 2]^th term.

Empirical Relationship

The relationship between central tendencies is given by: Mode = 3 Median − 2 Mean.

Effect of Constant Addition

If a constant 'c' is added to every observation in a data set, the new median will be Old Median + c.

Shortcut Trick

Use the standard algebraic identity: (x + a)(x + b) = x² + (a + b)x + ab.

Here, a = 2 and b = 3.

Sum of constants (a + b) = 2 + 3 = 5.

Product of constants (ab) = 2 × 3 = 6.

Substituting these: x² + 5x + 6.

∴ The correct answer is x² + 5x + 6.

Alternate Method

Given Expression: (x + 2) (x + 3)

To simplify, apply the Distributive Property (FOIL method) by multiplying each term in the first binomial by each term in the second binomial.

⇒ x(x + 3) + 2(x + 3)

⇒ (x × x) + (x × 3) + (2 × x) + (2 × 3)

⇒ x² + 3x + 2x + 6

Now, combine the like terms (3x and 2x):

⇒ x² + 5x + 6

∴ The correct answer is x² + 5x + 6.

Additional Information

Algebraic Identity for Binomials

The product of two binomials with a common variable is (x + a)(x + b) = x² + (a + b)x + ab, where (a+b) is the sum and ab is the product of the constants.

Standard Quadratic Form

A simplified expression in the form ax² + bx + c is known as a quadratic trinomial, where a, b, and c are constants.

FOIL Rule

FOIL stands for First, Outer, Inner, Last. It is a mnemonic used to remember the steps to multiply two binomials correctly.

Shortcut Trick

The ratio of radii (k) = 10.5 ÷ 1.5 = 7.

When a sphere is melted into smaller ones, the number of balls (n) = k³ = 7³ = 343.

The surface area of one small ball = S ÷ k² = S ÷ 7² = S/49.

Total Surface Area = n × (Surface area of 1 small ball) = 343 × (S/49) = 7S.

∴ The correct answer is 7S.

Alternate Method

Given:

Radius of large sphere (R) = 10.5 cm

Radius of small ball (r) = 1.5 cm

Original Surface Area = S

Formula Used:

Volume of Sphere = 4/3 × π × r³

Surface Area of Sphere = 4 × π × r²

Calculations:

⇒ Volume of large sphere = 4/3 × π × (10.5)³

⇒ Volume of one small ball = 4/3 × π × (1.5)³

⇒ Number of small balls (n) = [4/3 × π × (10.5)³] ÷ [4/3 × π × (1.5)³]

⇒ n = (10.5 / 1.5)³ = 7³ = 343

⇒ Original Surface Area (S) = 4 × π × (10.5)²

⇒ Surface Area of 1 small ball = 4 × π × (1.5)²

⇒ Total Surface Area (TSA) = 343 × 4 × π × (1.5)²

⇒ Since 1.5 = 10.5/7, TSA = 343 × 4 × π × (10.5 / 7)²

⇒ TSA = 343 × [4 × π × (10.5)²] ÷ 49

⇒ TSA = (343 / 49) × S = 7S

∴ The correct answer is 7S.

Additional Information

Volume and Surface Area Relationship

When an object is melted and recast, its total volume remains constant, but the total surface area usually increases as the number of pieces increases.

Sphere Scaling Properties

If the radius is changed by a factor of k, the surface area changes by k² and the volume changes by k³.

Hemisphere Formulas

Curved Surface Area = 2πr² | Total Surface Area = 3πr² | Volume = 2/3 πr³.

Shortcut Trick

To check divisibility by 41, use the oscillator method where the multiplier is 4.

Subtract 4 times the last digit from the remaining number: 713297 − (5 × 4) = 713277.

Repeat: 71327 − (7 × 4) = 71299 → 7129 − (9 × 4) = 7093 → 709 − (3 × 4) = 697 → 69 − (7 × 4) = 41.

Since 41 is divisible by 41, the whole number is divisible by 41.

∴ The correct answer is 7132975.

Alternate Method

Given: Numbers 7575438, 7132975, 7397665, 7497194.

Method: Perform direct division for each option to check for a zero remainder.

⇒ Option 1: 7575438 ÷ 41 = 184766.78 (Not divisible)

⇒ Option 2: 7132975 ÷ 41

⇒ 41 × 1 = 41; 71 − 41 = 30; Carry 3 → 303

⇒ 41 × 7 = 287; 303 − 287 = 16; Carry 2 → 162

⇒ 41 × 3 = 123; 162 − 123 = 39; Carry 9 → 399

⇒ 41 × 9 = 369; 399 − 369 = 30; Carry 7 → 307

⇒ 41 × 7 = 287; 307 − 287 = 20; Carry 5 → 205

⇒ 41 × 5 = 205; 205 − 205 = 0

⇒ 7132975 ÷ 41 = 173975

∴ The correct answer is 7132975.

Additional Information

Divisibility Rule for 41

Subtract 4 times the last digit from the number formed by the remaining digits. If the result is 0 or divisible by 41, the original number is divisible by 41.

Oscillators for Prime Numbers

For primes ending in 1 (like 31, 41, 61), the negative oscillator is used. For 41, the oscillator is 4.

General Divisibility Shortcut

If a number is divisible by n, then any multiple of that number added or subtracted from it will also be divisible by n.

Shortcut Trick

Product of fractions: 1/2 × 7/10 × 1/3 = 7/60

Equating to value: 7 units = 539

1 unit = 77

60 units = 77 × 60 = 4620

∴ The correct answer is 4620.

Alternate Method

Let the number be x.

Given: 50% × 70% × (2/6) × x = 539

⇒ (50/100) × (70/100) × (1/3) × x = 539

⇒ (1/2) × (7/10) × (1/3) × x = 539

⇒ (7/60) × x = 539

⇒ x = (539 × 60) ÷ 7

⇒ x = 77 × 60

⇒ x = 4620

∴ The correct answer is 4620.

Additional Information

Percentage to Fraction Conversion

To convert percentage to fraction, divide by 100. Example: 50% = 50/100 = 1/2 and 70% = 70/100 = 7/10.

Successive Percentage Calculation

When multiple percentages are applied to a number, they can be multiplied as fractions directly to find the final effective multiplier.

Reverse Calculation Method

To find the original number when the final value is known, multiply the result by the reciprocal of each fraction in reverse order.

Shortcut Trick

Combine the ratio: 1^st : 2^nd : 3^rd = (20 × 21) : (18 × 21) : (18 × 12).

Simplified Ratio = 420 : 378 : 216 → Divide by 6 → 70 : 63 : 36.

Total units = 70 + 63 + 36 = 169 units.

169 units = 8788 ⇒ 1 unit = 52.

Difference (1^st − 3^rd) = (70 − 36) × 52 = 34 × 52 = 1768.

∴ The correct answer is 1768.

Alternate Method

Given: Sum of numbers = 8788. Ratio A : B = 20 : 18, Ratio B : C = 21 : 12.

Formula Used: Combined Ratio = A/B × B/C.

⇒ Simplify Ratio A : B = 10 : 9.

⇒ Simplify Ratio B : C = 7 : 4.

⇒ To make B common, multiply A : B by 7 and B : C by 9.

⇒ A : B : C = (10 × 7) : (9 × 7) : (4 × 9) = 70 : 63 : 36.

⇒ Let numbers be 70x, 63x, and 36x.

⇒ 70x + 63x + 36x = 8788

⇒ 169x = 8788

⇒ x = 8788 ÷ 169 = 52

⇒ Difference = 70x − 36x = 34x

⇒ Difference = 34 × 52 = 1768.

∴ The correct answer is 1768.

Additional Information

Compound Ratio

When two or more ratios are multiplied together, the result is the compound ratio. For A : B and B : C, the combined ratio is A : B : C.

Proportion Rule

If a : b :: c : d, then the product of extremes equals the product of means (a × d = b × c).

Duplicate Ratio

The duplicate ratio of a : b is a² : b².

Shortcut Trick

Present Ratio = 3 : 4 (Difference = 1)

Ratio after 25 years = 8 : 9 (Difference = 1)

Increase in ratio units = 8 − 3 = 5 units (or 9 − 4 = 5 units)

Since 5 units = 25 years, then 1 unit = 5 years.

Present age of Sunil = 3 units = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Alternate Method

Given: Present ratio of Sunil and Shyam = 3 : 4

Formula Used: Ratio and Proportions for Age Problems

⇒ Let present age of Sunil be 3x and Shyam be 4x.

⇒ According to the question: (3x + 25) / (4x + 25) = 8 / 9

⇒ 9 × (3x + 25) = 8 × (4x + 25)

⇒ 27x + 225 = 32x + 200

⇒ 32x − 27x = 225 − 200

⇒ 5x = 25

⇒ x = 5

⇒ Present age of Sunil = 3x = 3 × 5 = 15 years.

∴ The correct answer is 15 years.

Additional Information

Constant Age Difference

The difference between the ages of two people always remains constant over time. If the ratio differences are not equal, multiply to make them equal.

Ratio Balancing Method

If present ratio is a:b and future ratio is c:d, then calculate the difference (b-a) and (d-c). Multiply ratios by the other difference to equalize them.

Linear Equation Setup

Age after n years = (Present Age + n); Age n years ago = (Present Age − n).

Shortcut Trick

Find the semi-perimeter s = (41 + 21 + 50) ÷ 2 = 56 cm.

Calculate Area = √(56 × (56−41) × (56−21) × (56−50)) = √(56 × 15 × 35 × 6) = 420 cm².

Altitude (h) corresponding to side 21 cm: h = (2 × Area) ÷ Base = (2 × 420) ÷ 21.

h = 840 ÷ 21 = 40 cm.

∴ The correct answer is 40 cm.

Alternate Method

Given:

Sides of the triangle: a = 41 cm, b = 21 cm, c = 50 cm.

Formula used:

1. Heron's Formula: Area = √(s(s − a)(s − b)(s − c))
2. Area of Triangle: Area = 1/2 × base × height

Calculations:

⇒ Semi-perimeter s = (41 + 21 + 50) ÷ 2

⇒ s = 112 ÷ 2 = 56 cm

⇒ Area = √(56 × (56 − 41) × (56 − 21) × (56 − 50))

⇒ Area = √(56 × 15 × 35 × 6)

⇒ Area = √(8 × 7 × 3 × 5 × 7 × 5 × 2 × 3)

⇒ Area = √(16 × 9 × 25 × 49) = 4 × 3 × 5 × 7

⇒ Area = 420 cm²

Now, using the area to find the altitude (h) on the side of 21 cm:

⇒ 420 = 1/2 × 21 × h

⇒ 840 = 21 × h

⇒ h = 840 ÷ 21

⇒ h = 40 cm

∴ The correct answer is 40 cm.

Additional Information

Heron's Formula

It is used to calculate the area of any triangle when all three sides are known: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter.

Altitudes of a Triangle

Every triangle has three altitudes. The length of an altitude is inversely proportional to the length of the corresponding base.

Area of Right-Angled Triangle

For a right-angled triangle, the area is simply 1/2 × product of the two sides containing the right angle.

Shortcut Trick

Equate total work: (19M + 13W) × 12 = (17M + 9W) × 14

⇒ 6(19M + 13W) = 7(17M + 9W)

⇒ 114M + 78W = 119M + 63W

⇒ 119M − 114M = 78W − 63W

⇒ 5M = 15W ⇒ 1M = 3W

∴ The correct answer is 3 women.

Alternate Method

Given: 19 Men (M) and 13 Women (W) take 12 days. 17 Men and 9 Women take 14 days.

Formula Used: Total Work = (Efficiency × Time)

Calculations:

⇒ 12 × (19M + 13W) = 14 × (17M + 9W)

⇒ 228M + 156W = 238M + 126W

⇒ 238M − 228M = 156W − 126W

⇒ 10M = 30W

⇒ 1M = 3W

∴ The correct answer is 3 women.

Additional Information

Work Equality Principle

When different groups complete the same work, we use M₁D₁ = M₂D₂ to find the ratio of efficiencies.

Man-Days Concept

Total work is often expressed in terms of Man-days or Woman-days, which is the product of the number of people and the time taken.

Efficiency Ratio

If 1 man is equivalent to 'n' women, the efficiency ratio M:W is n:1.

Shortcut Trick

Difference in Simple Interest = Principal × Time × (Difference in Rates) ÷ 100

Difference in Rates = 7.5% − 3.5% = 4%

Difference in Interest = 2,40,000 × 3 × 4 ÷ 100

Difference in Interest = 2,400 × 12 = 28,800

∴ The correct answer is ₹28800.

Alternate Method

Given:

Principal (P) = ₹2,40,000

Rate for Bank A (R₁) = 3.5% p.a.

Rate for Bank B (R₂) = 7.5% p.a.

Time (T) = 3 years

Formula Used:

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

Calculations:

⇒ SI paid to Bank A = (2,40,000 × 3.5 × 3) ÷ 100

⇒ SI_A = 2,400 × 10.5 = ₹25,200

⇒ SI paid to Bank B = (2,40,000 × 7.5 × 3) ÷ 100

⇒ SI_B = 2,400 × 22.5 = ₹54,000

⇒ Positive Difference = SI_B − SI_A

⇒ Difference = 54,000 − 25,200 = 28,800

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Shortcut Trick

List the first 16 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53.

Group them for easy addition: (2+53) + (3+47) + (5+43) + (7+41) + (11+37) + (13+31) + (17+29) + (19+23).

Sum = 55 + 50 + 48 + 48 + 48 + 44 + 46 + 42 = 381.

∴ The correct answer is 381.

Alternate Method

Given: Need to find the sum of the first 16 prime numbers.

Step-by-step addition:

⇒ 2 + 3 = 5

⇒ 5 + 5 = 10

⇒ 10 + 7 = 17

⇒ 17 + 11 = 28

⇒ 28 + 13 = 41

⇒ 41 + 17 = 58

⇒ 58 + 19 = 77

⇒ 77 + 23 = 100

⇒ 100 + 29 = 129

⇒ 129 + 31 = 160

⇒ 160 + 37 = 197

⇒ 197 + 41 = 238

⇒ 238 + 43 = 281

⇒ 281 + 47 = 328

⇒ 328 + 53 = 381

∴ The correct answer is 381.

Additional Information

Definition of Prime Numbers

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

Smallest Prime Number

2 is the only even prime number and the smallest prime number.

Prime Numbers up to 100

There are 25 prime numbers between 1 and 100.

Shortcut Trick

For two numbers a and b, the third proportional x is calculated as x = b² ÷ a.

Given: a = 8 and b = 56.

⇒ x = (56 × 56) ÷ 8

⇒ x = 7 × 56 = 392.

∴ The correct answer is 392.

Alternate Method

Given: First number (a) = 8, Second number (b) = 56, Third proportional = x.

Formula Used: If a, b, and x are in continued proportion, then a : b = b : x.

Calculations:

⇒ 8 / 56 = 56 / x

⇒ 1 / 7 = 56 / x

⇒ x = 56 × 7

⇒ x = 392

∴ The correct answer is 392.

Additional Information

Mean Proportional

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

Fourth Proportional

For three numbers a, b, and c, the fourth proportional d is given by d = (b × c) ÷ a.

Compounded Ratio

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

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

According to the question:

Required condition: Odd number - Odd number - Even number

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

Thus, there are five such odd numbers which are immediately preceded by an odd number and also immediately followed by an even number.

Hence, "Option 1" is the correct answer.

Decoding signs:

Given: 12 C 16 B 15 A 8 D 14 = ?3

12 C 16 B 15 A 8 D 14 = ? ⇒ 12 + 16 × 15 ÷ 8 - 14 = ?3

⇒ 12 + 16 × 15 ÷ 8 - 14 = ?3

⇒ 12 + 16 × 1.875 - 14 = ?3

⇒ 12 + 30 - 14 = ?3

⇒ 42 - 14 = ?3

⇒ 28 = ?

Hence, "Option 1" is the correct answer.

Given series: (Left) $9$ 6 7 8 £ 5 1 8 2 £ 9 3 4 6 9 # 8 8 7 7 (Right)

According to the question:

Required condition: Symbol - Number - Symbol

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

Thus, there is one such numbers which is immediately preceded by a symbol and also immediately followed by a symbol.

Hence, "Option 2" is the correct answer.

Decoding signs:

Given: 8 D 14 A 2 C 3 B 9 = ?3

8 D 14 A 2 C 3 B 9 = ? ⇒ 8 - 14 ÷ 2 + 3 × 9 = ?

⇒ 8 - 14 ÷ 2 + 3 × 9 = ?

⇒ 8 - 7 + 3 × 9 = ?

⇒ 8 + 27 - 7 = ?

⇒ 35 - 7 = ?3

⇒ 28 = ?

Hence, "Option 3" is the correct answer.

The logic followed here is:

SG 11 is related to VH −6

And,

NM 19 is related to QN 2

Similarly,

ON 2 is related to

Hence, "Option 3" is the correct answer.

According to the question:

Thus, 'queen' is coded as 'ng'.

Hence, "Option 2" is the correct answer.

According to the question:

Thus, 'A' is coded as 'ng'.

Hence, "Option 3" is the correct answer.

The logic followed here is:

PO 8 is related to SM -5

And,

QW 12 is related to TU -1

Similarly,

CJ 18 is related to

Hence, "Option 1" is the correct answer.

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

1) Only two people sit between F and B when counted from the right of F.

2. Only two people sit between B and A when counted from the right of B.

3. Only three people sit between F and E when counted from the right of F.

4. D sits to the immediate left of C.

The remaining position will be occupied by the only person left i.e. G.

Thus, one person sit between F and G when counted from the left of F.

Hence, "Option 4" is the correct answer.

Given:

Naveen starts from Point A and drives 11 km towards the south.

He then takes a left turn, drives 5 km, turns left and drives 4 km.

He then takes a right turn and drives 6 km.

He takes a final left turn, drives 7 km and stops at Point P.

Here, Naveen should drive 11 km in the west direction in order to reach Point A again.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Hence, "Option 2" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

(I) Some professionals are teachers → Follow (As some teachers are writers and all writers are professionals. As some part of teachers is common with writers which whole comes in professionals, therefore it is true.)

(II) All teachers are professionals → Does not follow (As some teachers are writers and all writers are professionals. So it is possible but not definite.)

∴ Here, Only conclusion (I) follows.​

Hence, the correct answer is "Option 2".

Decoding symbols:

Given: T + K ÷ S × U - J = ?

T + K → T is the sister of K.

K ÷ S → K is the brother of S.

S × U → S is the husband of U.

U - J → U is the mother of J.

Here, T is J's father's sister.

Hence, "Option 2" is the correct answer.

Given: (Left) 458 569 248 671 349 (Right)

According to the question:

Thus, according to the final arrangement, in only one number will the first digit be exactly divisible by the second digit.

Hence, "Option 1" is the correct answer.

The logic followed here is:

Hence, "Option 1" is the correct answer.

Given:

Rahul starts from Point A and drives 9 km towards North.

He then takes a right turn, drives 6 km, turns right and drives 14 km.

He then takes a right turn and drives 9 km.

He takes a final right turn, drives 5 km and stops at Point P.

Here, Rahul should drive 3 km in the east direction in order to reach Point A again.

Hence, "Option 3" is the correct answer.

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

1) Only two boxes are kept below I.

2) Only one box is kept above J.

3) Only one box is kept between J and K.

4) B is kept immediately above C.

5. D is kept at some place below A.

Thus, four boxes are kept between A and B.

Hence, "Option 2" is the correct answer.

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

(I): Some candies are sprouts → Follow (As all fruits are candies and all fruits are sprouts. As whole fruits comes in candies and sprouts, therefore it is true.)

(II): All sprouts are fruits → Does not follow (As all fruits are sprouts. So it is possible but not definite.)

∴ Here, Only conclusion (I) follows.​

Hence, the correct answer is "Option 1".