RRB Group D Official Paper (Held On: 17 Dec, 2025 Shift 1) (Previous Year Paper)

The correct answer is salt, carbon dioxide, and water.

Key Points

• When an acid reacts with a metal carbonate, the chemical reaction consistently produces three distinct products: a salt, carbon dioxide gas, and water.
• The general chemical word equation for this interaction is: Acid + Metal Carbonate → Salt + Water + Carbon Dioxide.
• The formation of carbon dioxide is a characteristic feature of this reaction, usually manifesting as rapid effervescence or bubbling when the acid makes contact with the carbonate.
• The specific salt produced depends on the metal present in the carbonate and the anion of the acid used; for instance, Hydrochloric Acid reacting with Calcium Carbonate produces Calcium Chloride. Hence, statement 4 is correct.
• A base and water are typically products of a reaction between an acid and a metal oxide or hydroxide (neutralization), not a carbonate. Hence, statement 1 is incorrect.
• Hydrogen gas is released when an acid reacts with a metal, but not when it reacts with a metal carbonate. Hence, statement 2 is incorrect.
• While a gas is produced, the other products (salt and water) are usually in aqueous or liquid states, not necessarily a solid precipitate in all cases. Hence, statement 3 is incorrect.

Additional Information

• Chemical Identification of CO2: The carbon dioxide gas produced in this reaction can be identified using the lime water test. When the gas is passed through lime water (calcium hydroxide), it turns milky due to the formation of insoluble calcium carbonate.
• Metal Hydrogen Carbonates: Also known as bicarbonates, these compounds react with acids in the exact same manner as carbonates, yielding a salt, water, and carbon dioxide. An example is the reaction of Sodium Bicarbonate (Baking Soda) with citric acid or vinegar.
• Salt Formation: If Sulfuric Acid is used, the resulting salt will be a sulfate. If Nitric Acid is used, the salt will be a nitrate. The metal component comes directly from the metal carbonate reactant.
• Nature of Carbonates: Metal carbonates are considered basic in nature. Therefore, their reaction with acids is technically a form of neutralization, though the presence of the carbonate group (CO3) leads to the additional byproduct of carbon dioxide.
• Real-world Examples: This chemical principle is used in fire extinguishers (where acid and carbonate react to release CO2 to smother flames) and is also the reason why acid rain causes the erosion of statues and buildings made of limestone or marble, which are forms of calcium carbonate.

Important Points

• Equation for Calcium Carbonate: A standard laboratory example is 2HCl + CaCO3 → CaCl2 + H2O + CO2. This reaction involves Hydrochloric Acid and Calcium Carbonate (found in eggshells and chalk).
• Effervescence: The term effervescence refers to the escape of gas from an aqueous solution and the foaming or fizzing that results from that release.
• Limestone Weathering: In nature, when rainwater (which is slightly acidic due to dissolved CO2) reacts over long periods with limestone rocks, it leads to the formation of caves and sinkholes through this specific chemical breakdown.

The correct answer is Protium and deuterium have same the mass number..

Key Points

• The Mass Number (represented by the symbol A) is defined as the total number of protons and neutrons present in the nucleus of an atom. These particles located in the nucleus are collectively known as nucleons.
• Since the total number of nucleons (protons + neutrons) determines the mass number, the statement that the number of nucleons is the same as the mass number is scientifically accurate. Hence, statement 1 is correct.
• The mathematical relationship for an atom is Mass Number (A) = Atomic Number (Z) + Number of Neutrons (n). This confirms that the mass number is indeed the sum of protons and neutrons. Hence, statement 2 is correct.
• Isobars are atoms of different chemical elements that have the same mass number but different atomic numbers. Because they share the same total count of nucleons, they occupy different positions in the Periodic Table but have identical mass values. Hence, statement 3 is correct.
• Protium and Deuterium are isotopes of Hydrogen. Protium (1H1) has a mass number of 1 (1 proton, 0 neutrons), while Deuterium (1H2) has a mass number of 2 (1 proton, 1 neutron). Therefore, they do not have the same mass number. Hence, statement 4 is incorrect.
• Hence, the correct answer is Protium and deuterium have same the mass number.

Additional Information

• Isotopes:
• These are atoms of the same element that possess the same Atomic Number (Z) but different Mass Numbers (A).
• This difference arises because they have a different number of neutrons in their nuclei. For example, Carbon-12 and Carbon-14 are isotopes of carbon.
• Atomic Number (Z):
• The Atomic Number refers specifically to the number of protons in the nucleus of an atom.
• In a neutral atom, the number of protons is equal to the number of electrons. This number defines the identity of the element.
• Isotones:
• These are atoms of different elements that have the same number of neutrons but different mass numbers and atomic numbers.
• An example includes Boron-12 and Carbon-13, both of which contain 7 neutrons.
• Isotopes of Hydrogen:
• Hydrogen is unique as its isotopes have specific names: Protium (mass 1), Deuterium (mass 2, also called heavy hydrogen), and Tritium (mass 3, which is radioactive).
• While they share the same chemical properties due to having the same number of electrons, their physical properties differ due to the varying mass.

Important Points

• Notation of Elements: In scientific notation, the mass number (A) is written as a superscript on the left side of the element symbol, and the atomic number (Z) is written as a subscript.
• Mass of Subatomic Particles: The mass of an electron is approximately 1/1840 times the mass of a proton, which is why the mass number only accounts for protons and neutrons.
• Applications of Deuterium: Deuterium is widely used in nuclear reactors as a moderator in the form of heavy water (D2O) to slow down fast-moving neutrons.
• Tritium: It is the rarest isotope of hydrogen, with a nucleus containing 1 proton and 2 neutrons, making it unstable and subject to beta decay.

The correct answer is nucleolus.

Key Points

• The nucleolus is a distinct, dense, and spherical structure located inside the nucleus of eukaryotic cells. It is not surrounded by a membrane, which allows it to exist as a concentrated region of chromatin, RNA, and proteins.
• Its primary and most crucial function is the biosynthesis of ribosomal RNA (rRNA). It serves as the site where rRNA is transcribed and processed from DNA templates.
• The nucleolus is responsible for the assembly of ribosomal subunits. These subunits are formed by combining rRNA with specific proteins that are imported from the cytoplasm.
• Once assembled, these large and small ribosomal subunits exit the nucleus through nuclear pores to reach the cytoplasm, where they unite to carry out protein synthesis.
• The size and number of nucleoli within a nucleus can vary depending on the metabolic activity of the cell; cells that require high levels of protein production typically have larger or multiple nucleoli.
• During the process of cell division (specifically in mitosis), the nucleolus undergoes a cycle of disassembly and reassembly, usually disappearing during prophase and reappearing during telophase.

Additional Information

• Centrosome:
• The centrosome is an organelle found primarily in animal cells that serves as the main microtubule-organizing center (MTOC).
• It consists of two centrioles arranged at right angles to each other. During cell division, it facilitates the organization of spindle fibers, which are essential for the proper separation of chromosomes.
• Lysosome:
• Known as the "suicidal bags" of the cell, lysosomes are membrane-bound vesicles containing hydrolytic enzymes.
• Their function is to digest macromolecules, old organelles, and foreign substances like bacteria. They play a vital role in cellular waste management and autophagy.
• Mitochondria:
• Often referred to as the "powerhouse of the cell," mitochondria are double-membrane bound organelles.
• They are the site of aerobic respiration and the production of ATP (Adenosine Triphosphate), which is the primary energy currency of the cell. They also contain their own mitochondrial DNA and ribosomes.
• Nucleus:
• The nucleus acts as the control center of the eukaryotic cell. It houses the cell's genetic material (DNA) in the form of chromosomes or chromatin.
• The nuclear envelope, a double membrane, protects the DNA and regulates the exchange of materials between the nucleus and the cytoplasm through nuclear pores.

The correct answer is Petroleum is a non-renewable resource.

Key Points

• Petroleum, also known as crude oil, is a naturally occurring liquid found beneath the Earth's surface. It is classified as a fossil fuel because it is formed from the remains of ancient marine organisms.
• Conservation is critical because petroleum is a non-renewable resource. This means that once it is consumed, it cannot be replaced or replenished within a human timeframe, as its formation process takes millions of years. Hence, statement 1 is correct.
• Unlike renewable resources like solar or wind energy, which are inexhaustible, the global supply of petroleum is finite. The rate at which humanity extracts and uses it far exceeds the geological rate of its creation. Hence, statement 2 is incorrect.
• While petroleum reserves are found in various parts of the world, they are not abundant in an infinite sense; they are depletable assets that are being rapidly exhausted. Hence, statement 3 is incorrect.
• The extraction, refining, and combustion of petroleum have significant negative impacts on the environment, including air pollution, oil spills, and the release of greenhouse gases. Hence, statement 4 is incorrect.

Additional Information

• Formation Process: Petroleum is formed through the decomposition of organic matter (mostly zooplankton and algae) buried under layers of sedimentary rock. Over millions of years, high pressure and heat transform this matter into liquid hydrocarbons.
• Economic Value: Often referred to as "Black Gold," petroleum is the backbone of the modern global economy. It is the primary source of energy for transportation and acts as a vital raw material for the petrochemical industry.
• Environmental Consequences: Burning petroleum-based fuels like gasoline and diesel releases carbon dioxide (CO2), a major contributor to global warming and climate change. It also produces pollutants like sulfur dioxide and nitrogen oxides.
• Global Distribution: The largest proven reserves of petroleum are concentrated in regions such as the Middle East (e.g., Saudi Arabia, Iran, Iraq), Venezuela, Russia, and the United States.

Important Points

• Refining: Crude oil is processed in refineries through a method called fractional distillation. This process separates the oil into various products like petrol, kerosene, lubricating oil, paraffin wax, and asphalt.
• Conservation Methods: Reducing petroleum dependency involves shifting to electric vehicles (EVs), improving public transportation, and increasing the use of biofuels or renewable energy sources.
• Sustainable Development: Conserving fossil fuels aligns with the principles of sustainable development, which aims to meet the needs of the present without compromising the ability of future generations to meet their own needs.
• By-products: Beyond fuel, petroleum is used to manufacture plastics, synthetic rubbers, chemical fertilizers, pesticides, and even some pharmaceuticals.

The correct answer is Oral contraceptive pills.

Key Points

• Oral contraceptive pills are a popular method of birth control used by females, which involves the oral administration of small doses of either progestogens or progestogen-estrogen combinations.
• These pills function by inhibiting ovulation and implantation, and they also alter the quality of cervical mucus to prevent or retard the entry of sperms into the uterus.
• Because they introduce exogenous hormones into the female body to override the natural menstrual cycle, they can lead to a significant hormonal imbalance.
• Common health issues or side effects associated with these pills include nausea, abdominal pain, breakthrough bleeding, irregular menstrual cycles, breast tenderness, and weight gain. Hence, the correct answer is Oral contraceptive pills.
• Other methods like condoms or Copper-T do not primarily function by altering the systemic hormonal levels of the user, making oral pills the most likely cause of such imbalances.

Additional Information

• Tubectomy:
• It is a surgical method (also known as sterilization) performed on females to prevent pregnancy permanently.
• In this procedure, a small part of the fallopian tube is removed or tied up through a small incision in the abdomen or through the vagina.
• This blocks gamete transport and prevents conception without affecting the hormonal balance or the production of estrogen and progesterone.
• Condoms:
• These are barrier methods made of thin rubber or latex sheath used to cover the penis in the male or the vagina and cervix in the female.
• They prevent the physical meeting of the sperm and the ovum.
• A major advantage of using condoms (such as the popular brand Nirodh) is that they protect the user from Sexually Transmitted Infections (STIs) and AIDS.
• Copper-T (CuT):
• It is a type of Intrauterine Device (IUD) that is inserted by doctors or expert nurses in the uterus through the vagina.
• The CuT releases copper ions which suppress sperm motility and the fertilising capacity of sperms.
• While it is an effective contraceptive, it is a non-hormonal IUD (unlike LNG-20 or Progestasert), meaning it does not cause hormonal imbalances.
• Saheli:
• It is a unique oral contraceptive for females developed by the Central Drug Research Institute (CDRI) in Lucknow, India.
• It is a non-steroidal preparation and is taken as a 'once-a-week' pill.
• It has high contraceptive value with very few side effects compared to traditional steroidal hormonal pills.

Important Points

• Classification of Contraceptives: Contraceptive methods are broadly grouped into Natural/Traditional, Barrier, IUDs, Oral Contraceptives, Injectables, Implants, and Surgical Methods.
• Medical Supervision: It is important to note that the selection of a suitable contraceptive method should always be done in consultation with qualified medical professionals to minimize health risks.
• Emergency Contraceptives: Administration of progestogens or progestogen-estrogen combinations within 72 hours of coitus is used as an emergency contraceptive to avoid possible pregnancy due to rape or casual unprotected intercourse.

The correct answer is 0.1 s.

Key Points

• An echo is the phenomenon where a sound is heard again after it is reflected back from a hard surface like a wall, mountain, or cliff.
• To hear a distinct echo, the sensation of the original sound must have faded from the human brain. The persistence of hearing for a human ear is approximately 0.1 seconds.
• This means that the time interval between the original sound and the reflected sound must be at least 0.1 s for the human brain to distinguish them as two separate sounds.
• If the reflected sound arrives in less than 0.1 s, the two sounds merge, and we hear a prolonged sound rather than a distinct echo.
• The total distance covered by the sound (from the source to the obstacle and back) must be such that the time taken is 0.1 s or more. At a standard speed of sound in air (approx. 344 m/s at 22°C), this total distance is 34.4 meters.

Additional Information

• Calculation of Minimum Distance:
• The minimum distance required to hear an echo can be calculated using the formula: Distance = (Speed × Time) / 2.
• Using the standard speed of 344 m/s and the minimum time of 0.1 s, the distance is (344 × 0.1) / 2 = 17.2 meters.
• Therefore, to hear a clear echo, the obstacle must be at least 17.2 meters away from the source of the sound.
• Reverberation:
• When sound is reflected multiple times in a closed space (like a large hall), the reflections persist and blur the original sound; this is known as reverberation.
• Unlike an echo, which is a single distinct reflection, reverberation is the persistence of sound due to repeated reflections arriving in less than 0.1 s.
• Factors Affecting Speed of Sound:
• The speed of sound varies with the temperature of the medium. For every 1 degree Celsius rise in temperature, the speed of sound increases by approximately 0.6 m/s.
• As the speed of sound increases with temperature, the minimum distance required to hear an echo also increases on a hot day.

Important Points

• Nature of Surfaces: Echoes are best heard when the reflecting surface is large, hard, and distant. Soft surfaces like curtains or carpets absorb sound and prevent echoes.
• Applications of Reflection: The principle of sound reflection and echo is used in SONAR (Sound Navigation and Ranging) to measure the depth of the sea and in Ultrasound imaging for medical diagnostics.
• Speed in Media: Sound travels fastest in solids, followed by liquids, and slowest in gases. In water, the speed of sound is roughly 1500 m/s, which would significantly increase the minimum distance needed for an echo compared to air.

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

Key Points

• The power of accommodation of the eye refers to the ability of the eye lens to adjust its focal length to see objects clearly at various distances. This process is primarily controlled by the ciliary muscles.
• When the eye is focusing on a near object, the ciliary muscles must contract. This contraction reduces the tension on the suspensory ligaments, allowing the lens to become thicker and more convex.
• To achieve maximum power of accommodation, the eye is focusing on its near point. At this stage, the ciliary muscles are fully contracted to provide the maximum possible curvature to the lens.
• As the curvature of the eye lens increases, its focal length decreases, thereby increasing the converging power of the eye to focus the diverging rays from near objects onto the retina.
• Hence, the correct answer is Fully contracted.

Additional Information

• Near Point and Far Point:
• For a normal human eye, the near point (the minimum distance at which an object can be seen distinctly without strain) is 25 cm. At this distance, the accommodation is at its maximum.
• The far point of a normal eye is at infinity. To view distant objects, the ciliary muscles are fully relaxed, making the lens thin and increasing the focal length.
• Components of Accommodation:
• Ciliary Muscles: Circular smooth muscles in the ciliary body of the eye that change the shape of the lens.
• Suspensory Ligaments (Zonules of Zinn): Fibers that connect the ciliary body with the lens. When the muscles contract, these ligaments loosen.
• Crystalline Lens: A transparent, biconvex structure that provides the secondary refractive power needed for fine-tuning the focus.
• Optical Physics of the Eye:
• The refractive power (P) of the lens is inversely proportional to its focal length (f), denoted by the formula P = 1/f.
• To focus on near objects, a higher power is required, which necessitates a shorter focal length achieved through increased convexity.

Important Points

• Presbyopia:
• With aging, the ciliary muscles weaken and the flexibility of the eye lens decreases. This leads to a reduction in the power of accommodation, a condition known as Presbyopia.
• People with this condition often require bifocal lenses or convex reading glasses to assist with near vision.
• Summary of Muscle States:
• Distant Vision: Ciliary muscles relax, lens becomes flat/thin, focal length is maximum, power is minimum.
• Near Vision: Ciliary muscles contract, lens becomes round/thick, focal length is minimum, power is maximum.

The correct answer is Epithelial tissues.

Key Points

• Epithelial tissues are the covering or protective tissues in the animal body. They cover most organs and cavities within the body and form a barrier to keep different body systems separate. Hence, the statement regarding Epithelial tissues is correct.
• These tissues are tightly packed and form a continuous sheet with almost no intercellular spaces between the cells. Any material entering or leaving the body must cross at least one layer of epithelium.
• Sclerenchyma tissues are permanent plant tissues that make the plant hard and stiff. They consist of dead cells with thick walls due to lignin deposition. Hence, the statement regarding Sclerenchyma tissues is incorrect as it is a plant tissue.
• Nervous tissues are specialized for conduction of electrochemical impulses and are found in the brain, spinal cord, and nerves. While they are animal tissues, their primary function is communication and control, not covering. Hence, the statement regarding Nervous tissues is incorrect.
• Parenchyma tissues are the most common simple permanent tissues in plants, consisting of relatively unspecialized living cells with thin cell walls. Hence, the statement regarding Parenchyma tissues is incorrect as it is a plant tissue.

Additional Information

• Types of Epithelial Tissues:
• Simple Squamous Epithelium: These are extremely thin and flat and form a delicate lining. They are found in the esophagus and the lining of the mouth.
• Stratified Squamous Epithelium: Skin epithelial cells are arranged in many layers to prevent wear and tear; since they are arranged in a pattern of layers, the epithelium is called stratified.
• Columnar Epithelium: Found where absorption and secretion occur, such as in the inner lining of the intestine. These tall epithelial cells facilitate movement across the epithelial barrier.
• Cuboidal Epithelium: These cube-shaped cells form the lining of kidney tubules and ducts of salivary glands, providing mechanical support.
• Glandular Epithelium: Sometimes a portion of the epithelial tissue folds inward, and a multicellular gland is formed. This is used for secreting substances at the epithelial surface.
• Plant Tissues Comparison:
• Parenchyma cells are usually isodiametric and serve functions like photosynthesis (chlorenchyma) and buoyancy in aquatic plants (aerenchyma).
• Sclerenchyma provides strength to the plant parts; it is found in stems, around vascular bundles, and in the hard covering of seeds and nuts (like the coconut husk).

Important Points

• All epithelium is usually separated from the underlying tissue by an extracellular fibrous basement membrane.
• The permeability of the cells of various epithelia plays a vital role in regulating the exchange of materials between the body and the external environment, as well as between different parts of the body.
• In the respiratory tract, the columnar epithelial tissue also has cilia, which are hair-like projections on the outer surfaces of epithelial cells. These cilia can move, and their movement pushes the mucus forward to clear it. This type of epithelium is ciliated columnar epithelium.

The correct answer is Alpha particles.

Key Points

• Alpha particles are identified as doubly charged helium ions because they consist of two protons and two neutrons, effectively making them a helium nucleus (He²⁺) without any electrons.
• In 1911, the physicist Ernest Rutherford utilized these particles in his landmark scattering experiment to investigate the internal structure of the atom.
• Because an alpha particle has a mass of approximately 4 atomic mass units (u) and a positive charge of +2, it was an ideal probe for interacting with the dense parts of the atom.
• The experiment involved bombarding a very thin gold foil with high-energy alpha particles and observing their deflection patterns using a zinc sulfide screen.
• Rutherford's observation that most particles passed through the foil while a few were deflected at large angles proved that the positive charge was not distributed but concentrated in a tiny nucleus.

Additional Information

• Beta particles: These are high-speed electrons (beta-minus) or positrons (beta-plus) emitted during radioactive decay. They carry a single negative or positive charge and have much less mass than alpha particles.
• Gamma particles (Gamma rays): These are high-energy electromagnetic radiations (photons) that have zero mass and zero electric charge. They possess the highest penetrating power among the three types of radiation.
• Delta particles: In physics, these usually refer to secondary electrons with enough energy to escape an atom and cause further ionization in a medium when hit by a primary ionizing particle.
• Rutherford's Atomic Model: Often referred to as the planetary model, it proposed that electrons revolve around a small, dense, positively charged nucleus in specific paths, similar to how planets orbit the sun.

Important Points

• Ionizing Power: Due to their double positive charge and relatively large mass, alpha particles have a very high ionizing power compared to beta and gamma radiation.
• Penetration Depth: Alpha particles have the lowest penetrating power; they can be stopped by a simple sheet of paper or even the human skin, although they are dangerous if ingested or inhaled.
• Velocity: In radioactive emissions, alpha particles typically move at speeds ranging from 1.4 × 10⁷ m/s to 2.1 × 10⁷ m/s, which is about 5% of the speed of light.
• Discovery of the Nucleus: Rutherford's Alpha-scattering experiment led to the conclusion that the nucleus occupies a volume about 10⁻¹⁵ times the total volume of the atom, meaning the atom is mostly empty space.
• Charge Magnitude: The absolute charge of an alpha particle is +3.202 × 10⁻¹⁹ Coulombs, which is exactly twice the elementary charge of a proton.

The correct answer is In acidic solutions, onion extract retains its characteristic smell.

Key Points

• Olfactory indicators are substances whose odour or smell changes depending on whether they are mixed with an acidic or a basic solution.
• Onion extract is a common olfactory indicator. When it is treated with an acid (like dilute Hydrochloric acid), its characteristic pungent smell does not go away and remains detectable. Hence, statement 4 is correct.
• In contrast, when onion extract is treated with a base (like Sodium Hydroxide), its characteristic smell disappears or cannot be detected. Hence, statement 3 is incorrect.
• Vanilla essence also acts as an olfactory indicator; it retains its sweet smell in acidic conditions but loses it completely in basic solutions. Hence, statement 2 is incorrect.
• Both onion and vanilla do not lose their smell in acidic conditions; they only lose their smell when exposed to alkaline or basic substances. Hence, statement 1 is incorrect.
• Hence, the correct answer is In acidic solutions, onion extract retains its characteristic smell.

Additional Information

• Types of Indicators:
• Natural Indicators: These are obtained from natural sources like plants. Examples include Litmus (extracted from Lichens), Turmeric, and Red Cabbage extract. Turmeric stays yellow in acid but turns reddish-brown in base.
• Synthetic Indicators: These are man-made chemical substances. Phenolphthalein remains colorless in acidic solutions but turns pink in basic solutions. Methyl orange turns red in acid and yellow in base.
• Olfactory Indicators: These are particularly useful in chemistry laboratories for visually impaired students, as they can identify the nature of a substance by smell rather than color change.
• Clove Oil: It is another prominent olfactory indicator. Like onion and vanilla, the smell of clove oil can be detected in the presence of an acid, but it vanishes when mixed with a base.
• Practical Application: To test if a substance is a base using an olfactory indicator, one can rub onion juice on a strip of cloth and then add the test solution. If the smell of the onion disappears, the solution is basic in nature.

Important Points

• Acids: These are substances that produce Hydrogen ions (H+) in aqueous solutions. They have a sour taste and turn blue litmus red.
• Bases: These are substances that produce Hydroxide ions (OH-) in aqueous solutions. They have a bitter taste, a soapy feel, and turn red litmus blue.
• Neutralization: When an acid and a base react, they produce salt and water. This reaction often eliminates the indicator changes associated with the individual pH levels.
• The pH scale ranges from 0 to 14, where 7 is neutral, values less than 7 are acidic, and values greater than 7 are basic.

The correct answer is Hugging of infected person.

Key Points

• HIV (Human Immunodeficiency Virus) is a virus that attacks the body's immune system. It is transmitted through specific bodily fluids such as blood, semen, vaginal fluids, and breast milk.
• Hugging is considered a form of casual contact. The virus cannot survive for long outside the human body and is not transmitted through skin-to-skin contact that does not involve the exchange of the aforementioned fluids. Hence, hugging is NOT responsible for the transmission of HIV.
• Sexual contact with an infected person is one of the most common modes of HIV transmission. It involves the exchange of sexual fluids through mucous membranes. Hence, statement 1 is a method of transmission.
• Transfusion of contaminated blood or blood products is a highly efficient way of transmitting the virus directly into the bloodstream. However, in modern medical facilities, blood is rigorously screened for HIV-1 and HIV-2 antibodies. Hence, statement 3 is a method of transmission.
• Sharing infected needles or syringes, particularly among intravenous drug users, provides a direct route for the virus to enter the body through contaminated blood remaining in the needle. Hence, statement 4 is a method of transmission.

Additional Information

• Mechanism of Infection: HIV targets and destroys CD4 cells (also known as T-helper cells), which are a type of white blood cell crucial for the immune response. As the CD4 count drops, the person becomes more susceptible to opportunistic infections and certain types of cancer.
• AIDS (Acquired Immunodeficiency Syndrome): This is the most advanced stage of HIV infection. A person is diagnosed with AIDS when their CD4 cell count falls below 200 cells/mm3 or when they develop specific clinical conditions associated with severe immune deficiency.
• Non-Transmission Myths: It is scientifically established that HIV is NOT spread through mosquito bites, ticks, or other insects. It is also not spread through saliva, tears, sweat, sharing toilet seats, drinking fountains, or swimming pools.
• Diagnostic Testing: The primary method for detecting HIV is the ELISA (Enzyme-Linked Immunosorbent Assay) test, which identifies antibodies. If a person tests positive, a confirmatory test like the Western Blot or a Nucleic Acid Test (NAT) is performed to ensure accuracy.
• Treatment (ART): There is currently no cure for HIV, but it can be managed with Antiretroviral Therapy (ART). ART involves taking a combination of medicines every day that prevent the virus from multiplying, effectively reducing the viral load to undetectable levels.

Important Points

• Prevention Strategies: Methods to prevent HIV include the use of condoms, Pre-Exposure Prophylaxis (PrEP) for high-risk individuals, and Post-Exposure Prophylaxis (PEP) if taken within 72 hours of possible exposure.
• Mother-to-Child Transmission: HIV can be passed from a mother to her child during pregnancy, childbirth, or breastfeeding. However, with proper ART, the risk of transmission can be reduced to less than 1%.
• Global Impact: Since the start of the epidemic in the early 1980s, millions of people have been infected. Organizations like UNAIDS and the World Health Organization (WHO) work globally to provide education, funding, and healthcare to combat the spread of the virus.

The correct answer is Manufacturing proteins.

Key Points

• Structure 'A' in the diagram represents ribosomes, which are the fundamental cellular organelles responsible for protein synthesis.
• Ribosomes are often found attached to the Rough Endoplasmic Reticulum (RER), giving it a distinctive "rough" or granular appearance under an electron microscope.
• The primary role of these attached ribosomes is to translate messenger RNA (mRNA) into polypeptide chains, which then enter the lumen of the RER for proper folding and processing. Hence, the option regarding manufacturing proteins is correct.
• These proteins produced on the RER are typically destined for secretion out of the cell, incorporation into the plasma membrane, or for use within lysosomes.
• Ribosomes function by linking amino acids together in a specific order dictated by the genetic code, a process known as translation.

Additional Information

• Endoplasmic Reticulum (ER):
• The ER is a continuous membrane system that forms a series of flattened sacs within the cytoplasm of eukaryotic cells.
• It is divided into two types: Rough ER (studded with ribosomes) and Smooth ER (lacking ribosomes).
• The Smooth Endoplasmic Reticulum (SER) is primarily involved in the production of fat molecules (lipids), steroids, and the detoxification of metabolic byproducts.
• Nucleus:
• The nucleus is the "control center" of the cell and is responsible for storing genetic material in the form of DNA (Deoxyribonucleic acid).
• It coordinates cell activities like growth, intermediary metabolism, and reproduction (cell division).
• Lysosomes:
• These are membrane-bound organelles containing digestive enzymes (hydrolases) responsible for the digestion of food particles, damaged organelles, and foreign substances like bacteria.
• They are often referred to as the "suicide bags" of the cell because they can digest the cell itself if it becomes damaged or dies.

Important Points

• Discovery: Ribosomes were first identified in the mid-1950s by George Emil Palade, who used electron microscopy to observe them as dense particles or granules.
• Structure: A ribosome is composed of two subunits—the large subunit and the small subunit—which come together during the process of protein manufacturing.
• Composition: They are chemically composed of ribosomal RNA (rRNA) and dozens of distinct proteins.
• Location: While many are attached to the Rough ER, "free" ribosomes also exist in the cytosol, where they synthesize proteins that function within the cytoplasm itself.

The correct answer is Propane.

Key Points

• Saturated hydrocarbons are organic compounds where the carbon atoms are connected to each other only by single bonds. These are also known as alkanes.
• Propane is an alkane with the chemical formula C3H8. Since it contains only single covalent bonds between its three carbon atoms, it is classified as a saturated hydrocarbon. Hence, this option is correct.
• Ethene (C2H4) belongs to the group of alkenes, which are unsaturated hydrocarbons containing at least one double bond between carbon atoms. Hence, this option is incorrect.
• Ethyne (C2H2) and Butyne (C4H6) belong to the group of alkynes. These are unsaturated hydrocarbons containing at least one triple bond between carbon atoms. Hence, these options are incorrect.
• The general formula for saturated hydrocarbons (alkanes) is CnH2n+2, where n represents the number of carbon atoms in the molecule.

Additional Information

• Classification of Hydrocarbons:
• Alkanes: These are saturated compounds with the general formula CnH2n+2. Common examples include Methane (CH4), Ethane (C2H6), and Butane (C4H10). They are primarily used as fuels (e.g., LPG consists mostly of propane and butane).
• Alkenes: These are unsaturated compounds featuring at least one double bond with the formula CnH2n. Ethene is the simplest alkene, often used in the production of plastics like polyethylene.
• Alkynes: These are unsaturated compounds featuring at least one triple bond with the formula CnH2n-2. Ethyne, commonly known as acetylene, is used in welding torches.
• Chemical Reactivity:
• Saturated hydrocarbons are relatively unreactive under normal conditions and are often referred to as paraffins (meaning "little affinity"). They undergo substitution reactions.
• Unsaturated hydrocarbons are more chemically reactive because of the presence of double or triple bonds, which can easily break to form new bonds in addition reactions.
• Structure and Hybridization:
• In alkanes (saturated), carbon atoms are sp3 hybridized, leading to a tetrahedral geometry.
• In alkenes, the double-bonded carbons are sp2 hybridized, while in alkynes, the triple-bonded carbons are sp hybridized.

Important Points

• Nomenclature Prefixes: The names of hydrocarbons are based on the number of carbon atoms: 1 (meth-), 2 (eth-), 3 (prop-), 4 (but-), 5 (pent-), etc.
• Physical State: The first four alkanes (C1 to C4) are gases at room temperature, while those with 5 to 17 carbons are liquids, and those with 18 or more carbons are solids.
• Combustion: All hydrocarbons burn in the presence of oxygen to produce carbon dioxide, water, and heat energy, making them excellent energy sources.

The correct answer is Increase in wind speed.

Key Points

• Evaporation is a surface phenomenon where liquid changes into vapor at any temperature below its boiling point.
• Increase in wind speed: With an increase in wind speed, the particles of water vapor move away with the wind. This decreases the amount of water vapor in the surrounding atmosphere, thereby increasing the rate of evaporation. Hence, statement 3 is correct.
• Humidity: Humidity is the amount of water vapor present in the air. The air around us cannot hold more than a definite amount of water vapor at a given temperature. If the amount of water in the air is already high (high humidity), the rate of evaporation decreases. Hence, statement 1 is incorrect.
• Temperature: With an increase in temperature, more particles get enough kinetic energy to go into the vapor state. Therefore, a decrease in temperature leads to a slower rate of evaporation. Hence, statement 2 is incorrect.
• Surface Area: Since evaporation is a surface phenomenon, if the surface area is increased, the rate of evaporation increases. For example, while putting clothes for drying up, we spread them out. A decrease in surface area will reduce the rate. Hence, statement 4 is incorrect.

Additional Information

• Mechanism of Evaporation:
• In a liquid, a small fraction of particles at the surface, having higher kinetic energy, is able to break away from the forces of attraction of other particles and gets converted into vapor.
• Unlike boiling, which is a bulk phenomenon occurring at a specific boiling point (e.g., 100 degrees Celsius for water at 1 atmospheric pressure), evaporation occurs at all temperatures.
• Evaporation Causes Cooling:
• The particles of liquid absorb energy from the surrounding to regain the energy lost during evaporation. This absorption of energy from the surroundings makes the surroundings cold.
• Example 1: During summer, people sprinkle water on the roof or open ground because the large latent heat of vaporization of water helps to cool the hot surface.
• Example 2: We wear cotton clothes in summer because cotton, being a good absorber of water, helps in absorbing sweat and exposing it to the atmosphere for easy evaporation, which cools our body.
• Latent Heat of Vaporization:
• This is the heat energy required to change 1 kg of a liquid to gas at atmospheric pressure at its boiling point.
• During evaporation, the liquid takes this latent heat from the surroundings, resulting in the cooling effect.

Important Points

• Intermolecular Forces: The rate of evaporation also depends on the nature of the liquid. Liquids with weaker intermolecular forces (like alcohol or petrol) evaporate much faster than liquids with stronger forces (like water or glycerin) at the same temperature.
• Pressure: The rate of evaporation is inversely proportional to the atmospheric pressure. When the pressure is low, it is easier for molecules to escape into the air.
• Practical Application: Desert coolers work better on a hot dry day because the high temperature and low humidity increase the rate of evaporation of water, resulting in better cooling.

The correct answer is a combination of multiple long and short lines.

Key Points

• An electric battery is defined as a device consisting of two or more electric cells connected together to serve as a source of electrical energy.
• In an electric circuit diagram, the symbol for a battery is represented by a series of alternating long and short parallel lines, reflecting that it is composed of multiple cells connected in a sequence. Hence, statement 3 is correct.
• The longer vertical line in the symbol represents the positive terminal, while the shorter, thicker vertical line represents the negative terminal of the cell.
• To form a functional battery, the positive terminal of one cell is typically connected to the negative terminal of the next cell; this arrangement is known as a series connection.
• Batteries are essential for providing the electromotive force (EMF) or voltage required to drive an electric current through a closed circuit.

Additional Information

• Electric Cell:
• A single electric cell is a device that converts chemical energy into electrical energy.
• The symbol for a single cell consists of only one long and one short line. Hence, statement 1 describes a cell, not a battery.
• Types of Batteries:
• Primary Batteries: These are non-rechargeable units, such as the alkaline batteries used in wall clocks and TV remotes.
• Secondary Batteries: These are rechargeable units, such as the Lithium-ion (Li-ion) batteries found in smartphones or Lead-acid batteries used in automobiles.
• Historical Context:
• The first true battery was invented by the Italian physicist Alessandro Volta in the year 1800, known as the Voltaic Pile.
• The unit of electric potential, the Volt (V), is named in his honor.
• Voltage Addition:
• When cells are connected in series, their individual voltages are added together. For example, connecting four 1.5V cells in series results in a 6V battery.

Important Points

• Circuit Symbols of Other Components:
• Option 2 (A circle with a dot): This symbol is often used to represent a plug key or a switch in the closed (on) position. If the dot is absent, it represents an open switch.
• Option 4 (A rectangle with arrows): A rectangle usually represents a resistor or a generic load. If arrows are pointing towards it, it may represent a Light Dependent Resistor (LDR), and if arrows point away, it represents a Light Emitting Diode (LED). Hence, statement 4 is incorrect for a battery.
• Safety and Polarity:
• It is crucial to observe polarity when joining cells; connecting the positive terminal of one cell to the positive terminal of another (in a simple series circuit) will result in the voltages opposing each other, and the device will not function.

The correct answer is 3.6 x 10^6 Joule.

Key Points

• The kilowatt-hour (kWh) is the commercial unit of electrical energy. It is commonly used by electricity providers to bill consumers for the total amount of energy consumed over a period.
• To find the value of 1 kWh in Joules, we use the relationship Energy = Power x Time. Here, the power is 1 kilowatt and the time is 1 hour.
• 1 kilowatt (kW) is equal to 1000 Watts. Since 1 Watt is defined as 1 Joule per second (J/s), 1000 Watts is equivalent to 1000 Joules per second.
• 1 hour consists of 60 minutes, and each minute has 60 seconds. Therefore, 1 hour equals 3600 seconds (60 x 60).
• The calculation for conversion is: 1 kWh = 1000 J/s x 3600 s, which results in 3,600,000 Joules. In scientific notation, this is expressed as 3.6 x 10^6 Joules. Hence, the value of 1 kWh is 3.6 x 10^6 Joules.

Additional Information

• Electrical Energy: This refers to the energy derived from electric potential energy or kinetic energy. The SI unit of energy is the Joule (J), named after the English physicist James Prescott Joule in the 19th century.
• Power: It is defined as the rate at which work is done or energy is transferred. The SI unit of power is the Watt (W), named after James Watt. 1 Watt is equal to 1 Joule per second.
• Need for Commercial Units: Because the Joule is a very small unit of energy, it is inconvenient for expressing large quantities of energy used in homes, industries, and commercial establishments. For these purposes, the larger unit kilowatt-hour is used.
• Measuring Consumption: Household electricity meters measure energy in "units". In this context, 1 unit of electricity is exactly equal to 1 kilowatt-hour.

Important Points

• 1 Joule: It is the work done when a force of one Newton displaces an object by one meter in the direction of the force. It is also the energy dissipated as heat when an electric current of one Ampere passes through a resistance of one Ohm for one second.
• History of the Unit: The term Board of Trade (B.O.T.) unit was formerly used in the United Kingdom as a synonym for kilowatt-hour. This was the standard for electrical billing starting in the year 1882.
• Conversions: While 1 kWh is 3.6 x 10^6 Joules, other related units include the calorie (approx 4.184 Joules) and the British Thermal Unit (BTU) (approx 1055 Joules).
• Efficiency: Understanding kWh helps in calculating the efficiency of electrical appliances by determining how many Joules of energy they consume to perform a specific amount of work.

The correct answer is It stays the same no matter where the object is.

Key Points

• Mass is a fundamental property of matter that represents the total amount of matter or atoms contained within an object.
• Statement 1: It depends on gravity. This statement is incorrect. Mass is independent of gravitational pull. It is weight that depends on the acceleration due to gravity (g).
• Statement 2: It stays the same no matter where the object is. This statement is correct because mass is an intrinsic property. Whether the object is on Earth, the Moon, or in outer space, the quantity of matter it contains does not change. Hence, statement 2 is correct.
• Statement 3: It increases at pole. This statement is incorrect. While the weight of an object increases at the Earth's poles because the Earth is not a perfect sphere and the gravitational force is stronger there, the mass remains constant.
• Statement 4: It changes from place to place. This statement is incorrect. This is a characteristic of weight, which varies depending on the altitude, latitude, or the celestial body the object is placed upon.
• Mass is a scalar quantity, meaning it only has magnitude and no direction. It can never be zero for a physical object.

Additional Information

• Measurement Units:
• The SI unit of mass is the kilogram (kg).
• Other common units include grams (g), milligrams (mg), and tonnes (1,000 kg).
• Relationship with Weight:
• Weight is the force exerted by gravity on an object. It is calculated using the formula W = m × g, where m is mass and g is acceleration due to gravity (approximately 9.8 m/s² on Earth).
• Since g varies at different locations, weight changes, but mass (m) remains the same.
• Inertia:
• Mass is often described as a measure of inertia. Inertia is the tendency of an object to resist changes in its state of motion.
• An object with a larger mass has greater inertia, making it harder to start moving, stop, or change its direction.
• Measuring Instruments:
• Mass is typically measured using a beam balance or a physical balance, which compares an unknown mass against standard masses.
• Weight is measured using a spring balance, which measures the force of gravity pulling on the object.

Important Points

• Conservation of Mass: In classical physics and chemical reactions, the Law of Conservation of Mass states that mass can neither be created nor destroyed; it only changes forms.
• Difference on the Moon: The gravity on the Moon is only 1/6th of that on Earth. If an object has a mass of 60 kg, it will still have a mass of 60 kg on the Moon, but its weight will drop from 588 N (on Earth) to approximately 98 N.
• Atomic Mass: On a microscopic scale, the mass of an atom is measured in Atomic Mass Units (amu), defined as 1/12th of the mass of a carbon-12 atom.
• Relativistic Mass: According to Einstein's Theory of Relativity, as an object approaches the speed of light, its relativistic mass increases, though its rest mass remains unchanged.

The correct answer is Isaac Newton.

Key Points

• The phenomenon of dispersion of light was first explained by the English physicist and mathematician Sir Isaac Newton in the year 1666.
• Through his landmark experiments with glass prisms, Newton demonstrated that white light is not a single entity but is actually composed of a spectrum of different colors.
• He observed that when a narrow beam of sunlight passes through a triangular glass prism, it splits into a band of seven colors, commonly known as VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange, and Red).
• Newton concluded that different colors of light bend through different angles with respect to the incident beam as they pass through the prism. Violet light undergoes the maximum deviation, while red light undergoes the minimum deviation.
• To confirm that the prism was not adding color to the light, he placed a second identical prism in an inverted position with respect to the first one, which allowed the separated colors to recombine into white light.

Additional Information

• Dispersion: This is defined as the splitting of a composite beam of light into its constituent colors when it passes through a transparent medium like a prism or a water droplet.
• Refractive Index and Wavelength: The cause of dispersion is that the Refractive Index of a material varies for different wavelengths. Since each color has a specific wavelength, they travel at different speeds within the medium, leading to different degrees of refraction.
• Christiaan Huygens: A Dutch scientist who proposed the Wave Theory of Light in 1678. He suggested that light travels in the form of longitudinal waves through a hypothetical medium called "ether."
• James Clerk Maxwell: A Scottish scientist who developed the theory of Electromagnetism in the 1860s. He mathematically proved that light is an electromagnetic wave that can travel through a vacuum.
• Albert Einstein: A German-born physicist who explained the Photoelectric Effect in 1905, for which he received the Nobel Prize in Physics in 1921. He introduced the concept of photons, establishing the particle nature of light.

Important Points

• Natural Dispersion: A rainbow is a prime example of dispersion of sunlight caused by water droplets in the atmosphere. This process involves refraction, dispersion, and Total Internal Reflection (TIR).
• Visible Spectrum: The range of wavelengths visible to the human eye is approximately 400 nm to 700 nm. Violet has the shortest wavelength and highest frequency, while red has the longest wavelength and lowest frequency.
• Newton's Opticks: Isaac Newton published his detailed findings on light and color in his famous book titled "Opticks" in 1704, which laid the foundation for modern physical optics.

The correct answer is Medulla oblongata controls reflex actions like pulling hand away from a hot object..

Key Points

• The Medulla oblongata is a part of the hindbrain responsible for controlling involuntary actions such as breathing, heart rate, blood pressure, and digestion. However, sudden reflex actions involving the limbs, such as pulling a hand away from a hot object, are primarily coordinated by the spinal cord to ensure a rapid response without immediate brain processing. Hence, statement 1 is incorrect.
• The Cerebellum, often referred to as the "little brain," is located at the back of the skull. Its primary function is to coordinate precision, fine motor skills, and maintain body posture and equilibrium. It ensures that movements are smooth and balanced. Hence, statement 2 is correct.
• The Cerebrum is the largest part of the human brain, making up about 80% of its weight. It is divided into two cerebral hemispheres and is the seat of intelligence, memory, reasoning, and the control of voluntary actions like walking or speech. Hence, statement 3 is correct.
• The Spinal cord serves as the main communication link between the brain and the peripheral nervous system. It acts as a relay station for sensory and motor impulses and is the integration center for spinal reflexes, which occur instantly to protect the body from harm. Hence, statement 4 is correct.
• Hence, the incorrect statement is Medulla oblongata controls reflex actions like pulling hand away from a hot object.

Additional Information

• The Human Brain:
• The brain is protected by the cranium (skull) and surrounded by three layers of membranes called meninges. It is divided into three main regions: the Forebrain, Midbrain, and Hindbrain.
• The Forebrain includes the Cerebrum, Thalamus, and Hypothalamus. The Hypothalamus is crucial for thermoregulation, hunger, and thirst.
• The Midbrain connects the forebrain to the hindbrain and controls visual and auditory reflexes.
• The Hindbrain consists of the Cerebellum, Pons, and Medulla Oblongata. The Pons acts as a bridge between various parts of the brain and helps regulate respiration.
• Reflex Action and Reflex Arc:
• A Reflex action is an extremely quick, automatic response to a stimulus.
• The path taken by nerve impulses in a reflex action is called a Reflex Arc. It involves a receptor, sensory neuron, spinal cord (interneuron), motor neuron, and effector organ (muscle or gland).
• By bypassing the conscious brain (cerebrum) during the initial response, the body saves milliseconds, which can be critical in preventing tissue damage.

Important Points

• Brainstem: The Midbrain, Pons, and Medulla oblongata collectively form the brainstem, which connects the brain to the spinal cord and handles vital life-support functions.
• Weight and Composition: An average adult human brain weighs approximately 1.3 to 1.4 kg and contains nearly 86 billion neurons.
• Voluntary vs Involuntary: Voluntary actions are under conscious control (e.g., writing with a pen), while Involuntary actions occur without conscious thought (e.g., the beating of the heart or the peristaltic movement of the gut).
• Medulla's Specific Functions: Apart from vital centers, the Medulla also contains centers for swallowing, coughing, sneezing, and vomiting.

The correct answer is They act on two different objects at the same time.

Key Points

• Newton’s Third Law of Motion states that for every action, there is an equal and opposite reaction. This law describes the nature of forces between interacting objects.
• Action and reaction forces always occur in pairs. A single, isolated force cannot exist in nature.
• A fundamental characteristic of these force pairs is that they act on two different objects. If object A exerts a force on object B, then object B simultaneously exerts a force back on object A. Hence, statement 3 is correct.
• The forces act in exactly opposite directions. Statement 1 claims they act in the same direction, which contradicts the law. Hence, statement 1 is incorrect.
• These forces are simultaneous, meaning they exist at the exact same moment. There is no time delay between the action and its reaction. Hence, statement 2 is incorrect.
• Since action and reaction forces act on different bodies, they never cancel each other out. For forces to cancel and result in equilibrium, they must act on the same object. Hence, statement 4 is incorrect.
• Hence, the correct answer is They act on two different objects at the same time.

Additional Information

• Origins of the Law: Sir Isaac Newton formulated this law, along with his other two laws of motion, in his monumental work Philosophiae Naturalis Principia Mathematica, published in 1687.
• Mathematical Expression: If F12 represents the force exerted by object 1 on object 2, and F21 represents the force exerted by object 2 on object 1, then the law is expressed as F12 = -F21. The negative sign indicates the opposite direction.
• Common Examples:
• Swimming: A swimmer pushes the water backward (action), and the water pushes the swimmer forward (reaction).
• Bird Flight: A bird's wings push the air downward and backward, while the air pushes the bird upward and forward.
• Rocketry: In a rocket engine, hot exhaust gases are expelled downward, and as a reaction, the gases exert an upward thrust on the rocket.
• Nature of Interaction: The action-reaction pair must be of the same type. For instance, if the action is a magnetic force, the reaction is also a magnetic force; if the action is gravity, the reaction is also gravity.

Important Points

• Mass and Acceleration: While the magnitude of the action and reaction forces is always equal, the resulting acceleration on each object can be very different. According to Newton's Second Law (F = ma), the object with less mass will experience a much greater acceleration. A classic example is a gun recoiling; the bullet and gun feel the same force, but the bullet moves faster due to its small mass.
• System Analysis: When defining a physical system, if both interacting objects are included within the system boundaries, the action and reaction forces are considered internal forces and their vector sum is zero. However, if the system includes only one object, only one force from the pair is considered an external force acting on that system.
• Interaction Requirement: Newton's Third Law highlights that force is not something an object "has," but rather a description of an interaction between two entities.

The correct answer is Pure form of the impure metal.

Key Points

• Electrolytic refining is the process of purifying impure metals using electricity. It is widely used for metals like Copper (Cu), Zinc (Zn), Tin (Sn), Nickel (Ni), Silver (Ag), and Gold (Au).
• In this electrochemical setup, a thick slab of the impure metal is made the Anode (the positive electrode), while a thin strip of pure metal is made the Cathode (the negative electrode).
• When an electric current is passed through the electrolyte (a water-soluble salt of the same metal), metal atoms from the anode lose electrons and enter the solution as positive ions (oxidation).
• These metal ions then migrate toward the Cathode, where they gain electrons and are reduced to form pure metal atoms, which deposit on the cathode surface. Hence, the statement that pure form of the impure metal gets collected at the cathode is correct.
• The organic matter and other volatile substances are typically removed during the roasting or smelting phases, not at the cathode during electrolysis. Hence, option 1 is incorrect.
• The insoluble impurities do not reach the cathode; instead, they settle down under the anode as anode mud. Hence, option 2 is incorrect.
• While the metal atoms form metallic bonds in the new crystal lattice, the most accurate description is the deposition of the pure form of the metal. Hence, option 3 is less accurate than option 4.

Additional Information

• The Electrolyte: The solution used must contain ions of the metal being refined. For example, in copper refining, the electrolyte is usually a mixture of Copper Sulfate (CuSO4) and Sulfuric Acid (H2SO4).
• Anode Mud: This contains insoluble impurities like Antimony, Selenium, Tellurium, Silver, Gold, and Platinum. These are often recovered as valuable by-products, helping to cover the cost of the refining process.
• Oxidation at Anode: The reaction at the positive electrode involves the metal (M) losing n electrons: M → Mⁿ⁺ + ne−.
• Reduction at Cathode: The reaction at the negative electrode involves the metal ions receiving electrons: Mⁿ⁺ + ne− → M (pure).
• Soluble Impurities: Impurities that are more electropositive than the metal being refined remain in the electrolyte solution as ions and do not deposit on the cathode.

Important Points

• Purity Level: Electrolytic refining is highly efficient and can produce metals with a purity of 99.9% to 99.99%. This high level of purity is critical for electrical wires to ensure maximum conductivity and minimum resistance.
• Voltage Control: The voltage applied must be carefully regulated so that only the desired metal ions are reduced at the cathode, preventing other impurity ions from depositing.
• Industrial Application: In the case of Aluminum, a similar but distinct electrolytic process called the Hall-Héroult process is used for extraction, though refining of other metals specifically follows the anode-cathode deposition principle described above.
• Mass Transfer: During the process, the Anode gradually dissolves and becomes thinner, while the Cathode grows thicker due to the continuous accumulation of the pure metal.

The correct answer is 3.33 W.

Key Points

• Power is defined as the rate of doing work or the rate at which energy is transferred or converted.
• The mathematical formula for calculating power is Power (P) = Work Done (W) / Time Taken (t).
• In this specific problem, the Work done is 600 Joules (J) and the Time taken is 3 minutes.
• To find the power in the standard unit of Watts (W), the time must be converted from minutes to seconds. Since 1 minute = 60 seconds, then 3 minutes = 180 seconds.
• Applying the values to the formula: Power = 600 J / 180 s. This simplifies to 60 / 18, which equals 3.333... Watts. Hence, the correct answer is 3.33 W.

Additional Information

• Watt (W): It is the SI unit of power. One Watt is defined as the consumption of one Joule of energy per second. It is named after James Watt, an 18th-century Scottish inventor who significantly improved the steam engine.
• Joule (J): This is the SI unit of work and energy. One Joule is the amount of work done when a force of one Newton is applied over a distance of one meter.
• Scalar Quantity: Power is a scalar quantity because it only has magnitude and no specific direction, unlike velocity or force.
• Units of Time: In physics calculations involving Standard International (SI) units, time is almost always measured in seconds to maintain consistency with Watts and Joules.
• Common Conversions: While the Watt is the standard unit, power is also often measured in Kilowatts (kW), where 1 kW = 1,000 W, or Horsepower (hp), where 1 hp is approximately 746 W.

Important Points

• Average vs. Instantaneous Power: Average power (calculated in this question) is the total work divided by the total time. Instantaneous power is the power measured at a specific, infinitesimal moment in time.
• Relation to Force: Power can also be calculated if the force and velocity are known, using the formula Power = Force x Velocity (P = F x v).
• Electrical Power: In the context of electricity, power is the product of Voltage (V) and Current (I), represented as P = V x I.
• Dimensional Formula: The dimensional representation of power is [ML²T^-3], derived from work [ML²T^-2] divided by time [T].
• Energy Consumption: Power is used to determine energy consumption over time; for instance, Energy = Power x Time. This is why electricity bills are often measured in kilowatt-hours (kWh).

The correct answer is Phenolphthalein.

Key Points

• Phenolphthalein is a synthetic chemical compound often used as an acid-base indicator in laboratories.
• In an acidic medium, phenolphthalein remains completely colourless. It also remains colourless in neutral solutions.
• When the solution becomes basic (alkaline), phenolphthalein molecules undergo a structural change, resulting in a vibrant pink or magenta colour.
• The pH transition range for phenolphthalein is typically between 8.2 and 10.0. Below pH 8.2, it is colourless, and above pH 10.0, it is pink.
• It is widely utilized in analytical chemistry, particularly during titrations involving a strong base and a weak acid, to accurately identify the end point of the reaction.

Additional Information

• Litmus:
• It is a natural indicator extracted from Lichens.
• In an acidic medium, litmus paper or solution turns red, while in a basic medium, it turns blue.
• A neutral litmus solution is purple in colour.
• Methyl Orange:
• This is a synthetic indicator used for titrations involving strong acids.
• It exhibits a red colour in acidic solutions (below pH 3.1) and a yellow colour in basic or neutral solutions (above pH 4.4).
• In the transition range, it appears orange.
• Bromothymol Blue:
• It is a chemical indicator for weak acids and bases. It appears yellow in acidic conditions, green in neutral conditions, and blue in basic conditions.
• Its pKa is approximately 7.1, making it useful for reactions occurring near neutral pH.
• Natural Indicators:
• Turmeric: Remains yellow in acid but turns reddish-brown in a base.
• China Rose (Hibiscus): Turns dark pink (magenta) in acidic solutions and green in basic solutions.
• Olfactory Indicators: These are substances like onion, vanilla essence, and clove oil whose odour changes depending on whether they are mixed with an acid or a base.

The correct answer is The total displacement of the object.

Key Points

• The area under a velocity-time (v-t) graph for an object moving in a straight line represents the total displacement of the object during a given time interval.
• In a velocity-time graph, the velocity is plotted on the y-axis and time is plotted on the x-axis. Since displacement is defined as the product of velocity and time (s = v × t), the geometric area formed between the plot and the time axis signifies the change in position.
• Because velocity is a vector quantity, the area can be positive (above the time axis) or negative (below the time axis). The algebraic sum of these areas gives the net displacement. Hence, option 2 is correct.
• For uniform motion (constant velocity), the graph is a horizontal straight line, and the area under it forms a rectangle, the area of which is length × breadth (velocity × time).
• For uniformly accelerated motion, the graph is a sloping straight line, and the area can be calculated using the formulas for a triangle or a trapezoid to find the displacement.

Additional Information

• Slope of the Graph:
• The slope or gradient of a velocity-time graph represents the acceleration of the object.
• A positive slope indicates positive acceleration (speeding up), while a negative slope indicates retardation or deceleration (slowing down).
• Distance vs. Displacement:
• While the area represents displacement (considering the sign), the total distance is calculated by taking the magnitude of all areas and adding them together, regardless of whether they are above or below the time axis.
• Kinematic Equations:
• The second equation of motion, s = ut + ½at², can be derived by calculating the area under the velocity-time graph for an object with initial velocity (u) and constant acceleration (a).
• Units and Dimensions:
• The unit of area in this context is derived from the product of the units on the axes: (meters/second) × seconds = meters (m). This confirms that the resulting quantity is a length or displacement.

Important Points

• Zero Acceleration: When the velocity-time graph is a straight line parallel to the time axis, the acceleration is 0, and the object is in uniform motion.
• Acceleration-Time Graph: The area under an acceleration-time (a-t) graph represents the change in velocity of the object, not the displacement.
• Position-Time Graph: In a position-time (s-t) graph, the slope represents the velocity, but the area under the curve does not have a standard physical significance in basic mechanics.
• Instantaneous Velocity: At any specific point on the v-t graph, the y-coordinate gives the instantaneous velocity of the object at that particular second or minute.

The correct answer is Tall Plants.

Key Points

• Gregor Johann Mendel, known as the Father of Genetics, conducted his historic experiments on the Garden Pea (Pisum sativum) for seven years between 1856 and 1863.
• He selected seven pairs of contrasting traits to study the inheritance patterns in living organisms.
• Stem Height: Mendel observed that Tall height is the dominant trait, while Dwarf height is the recessive trait. Hence, statement 1 is correct and statement 3 is incorrect.
• Seed Shape: The dominant trait is Round, whereas the Wrinkled shape is recessive. Hence, statement 2 is incorrect.
• Seed Color: In pea plants, Yellow seeds are dominant and Green seeds are recessive. Hence, statement 4 is incorrect.
• Pod Color: Interestingly, for pods, Green is the dominant trait and Yellow is the recessive trait, which is the opposite of the seed color dominance.
• Flower Color and Position: The Violet flower color is dominant over White, and the Axial position of flowers is dominant over the Terminal position.

Additional Information

• Reasons for choosing Pea Plants:
• The Pisum sativum plant is bisexual and naturally undergoes self-pollination, making it easy to produce pure-breeding lines.
• It has a short life cycle, allowing for the study of multiple generations in a short time, and produces a large number of offspring (seeds).
• The plant exhibits distinct and easily detectable contrasting traits, which were crucial for Mendel’s statistical analysis.
• Mendel’s Laws of Inheritance:
• Law of Dominance: In a heterozygous individual (Tt), the dominant allele (T) expresses itself, masking the effect of the recessive allele (t).
• Law of Segregation: During gamete formation, the two alleles for a trait segregate so that each gamete receives only one allele.
• Law of Independent Assortment: Different traits are inherited independently of each other, provided the genes are on different chromosomes.
• Ratios in Genetics:
• In a Monohybrid cross (studying one trait like height), the phenotypic ratio in the F2 generation is 3:1 (3 Tall : 1 Dwarf), and the genotypic ratio is 1:2:1 (1 TT : 2 Tt : 1 tt).
• In a Dihybrid cross (studying two traits), the phenotypic ratio is 9:3:3:1.

Important Points

• Dominant vs. Recessive: A dominant trait is one that appears in the F1 generation after crossing two pure-breeding parents of contrasting traits. A recessive trait is one that remains latent in the F1 generation but reappears in the F2 generation.
• Alleles: These are alternative forms of a gene (e.g., T and t). If both alleles are the same, the organism is homozygous (TT or tt); if they are different, it is heterozygous (Tt).
• Phenotype and Genotype: Phenotype refers to the observable physical characteristics (e.g., Tall), while Genotype refers to the genetic makeup (e.g., Tt).

Shortcut Trick

Ratio of numbers = 4 : 3

LCM of ratio parts = 4 × 3 = 12 units

Given 12 units = 84 ⇒ 1 unit (HCF) = 7

Sum of numbers = (4 + 3) × 7 = 7 × 7 = 49

∴ The correct answer is 49.

Alternate Method

Given:LCM = 84, Ratio = 4 : 3

Formula Used: LCM = HCF × Product of ratio parts (for coprime ratios)

⇒ Let the numbers be 4x and 3x (where x is the HCF).

⇒ LCM of 4x and 3x = 12x

⇒ 12x = 84

⇒ x = 84 ÷ 12 = 7

⇒ First number = 4 × 7 = 28

⇒ Second number = 3 × 7 = 21

⇒ Sum = 28 + 21 = 49

∴ The correct answer is 49.

Additional Information

Fundamental Property of Two Numbers

For any two numbers A and B, Product of numbers (A × B) = HCF(A, B) × LCM(A, B).

Ratio and LCM Relation

If two numbers are in the ratio a : b (where a and b are coprime), then LCM = a × b × HCF.

HCF of Fractions

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

Shortcut Trick

Required percentage = (First value ÷ Second value) × 100

⇒ (1/20 ÷ 10/60) × 100

⇒ (1/20 × 60/10) × 100

⇒ (3/10) × 100 = 30%

∴ The correct answer is 30%.

Alternate Method

Given: First fraction = 1/20, Second fraction = 10/60.

Formula Used: Percentage = (Part/Whole) × 100

Calculation:

⇒ Let 1/20 be x% of 10/60.

⇒ 1/20 = (x/100) × (10/60)

⇒ 1/20 = (x/100) × (1/6)

⇒ 1/20 = x/600

⇒ x = 600/20

⇒ x = 30

∴ The correct answer is 30%.

Additional Information

Fraction to Percentage

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

Percentage Change

Percentage change = (Difference / Original Value) × 100.

Reciprocal Percentages

Common values: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%, 1/20 = 5%.

Shortcut Trick

Convert fractional ratio to integer ratio by multiplying with LCM of 6, 9, and 4 (LCM = 36).

Ratio = (1/6 × 36) : (1/9 × 36) : (1/4 × 36) = 6 : 4 : 9.

Total units = 6 + 4 + 9 = 19 units.

19 units = 646 ⇒ 1 unit = 34.

First monkey (6 units) = 6 × 34 = 204.

∴ The correct answer is 204 bananas.

Alternate Method

Given: Total bananas = 646, Ratio = 1/6 : 1/9 : 1/4

⇒ Simplified Ratio = 1/6 : 1/9 : 1/4

⇒ Multiply by 36: (36/6) : (36/9) : (36/4) = 6 : 4 : 9

⇒ Sum of ratio terms = 6 + 4 + 9 = 19

⇒ Share of first monkey = (6/19) × 646

⇒ Share = 6 × 34 = 204

∴ The correct answer is 204 bananas.

Additional Information

Ratio Simplification

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

Proportional Division

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

Shortcut Trick

Efficiency ratio for Mahi : (Neha + Asha) = 1 : 2 ⇒ Mahi = 1/3 of total.

Efficiency ratio for Asha : (Neha + Mahi) = 1 : 3 ⇒ Asha = 1/4 of total.

Neha efficiency = 1 − (1/3 + 1/4) = 1 − 7/12 = 5/12 of total.

Total time × Total Efficiency = Neha time × Neha efficiency ⇒ 20 × 1 = Days × 5/12.

Days = 20 × 12/5 = 4 × 12 = 48 days.

∴ The correct answer is 48 days.

Alternate Method

Given: Time taken by Mahi = 2 × (Neha + Asha)

⇒ Efficiency ratio (Mahi) : (Neha + Asha) = 1 : 2

⇒ Mahi efficiency = (1/3) × 12 = 4 units/day

Given: Time taken by Asha = 3 × (Neha + Mahi)

⇒ Efficiency ratio (Asha) : (Neha + Mahi) = 1 : 3

⇒ Asha efficiency = (1/4) × 12 = 3 units/day

Calculations:

⇒ Neha efficiency = Total efficiency − Mahi efficiency − Asha efficiency

⇒ Neha efficiency = 12 − 4 − 3 = 5 units/day

⇒ Days taken by Neha = Total Work ÷ Neha efficiency

⇒ Days = 240 ÷ 5 = 48 days

∴ The correct answer is 48 days.

Additional Information

Efficiency and Time Relation

Efficiency is inversely proportional to the time taken to complete a task. If time ratio is a:b, then efficiency ratio is b:a.

Total Work Formula

Total Work = Efficiency × Time. For multiple people, Total Work = (Sum of individual efficiencies) × Combined Time.

Ratio Balancing

When comparing partial groups to a whole, ensure the sum of ratios (parts) matches the total efficiency common to all equations.

Shortcut Trick

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

MP : CP = (100 + 83) : (100 − 77) = 183 : 23.

Given CP = 23 units = ₹92,000.

Multiplier = 92,000 ÷ 23 = 4,000.

MP = 183 × 4,000 = ₹7,32,000.

∴ The correct answer is ₹7,32,000.

Alternate Method

Given → CP = ₹92,000, Discount = 77%, Gain = 83%. Formulas: SP = CP × (100 + Gain%) ÷ 100 | MP = SP × 100 ÷ (100 − Discount%).

⇒ Calculate Selling Price (SP): SP = 92,000 × (100 + 83) ÷ 100 = 92,000 × 1.83. ⇒ SP = ₹1,68,360. ⇒ Since Discount = 77%, SP is 23% of MP: 0.23 × MP = 1,68,360. ⇒ MP = 1,68,360 ÷ 0.23 = ₹7,32,000.

∴ The correct answer is ₹7,32,000.

Additional Information

Relationship between MP and CP

The direct ratio MP/CP = (100 + Profit%)/(100 − Discount%) is the most efficient way to solve problems involving both discount and profit on the same article.

Marked Price and Discount

Discount is always calculated on the Marked Price. SP = MP − Discount. If no discount is offered, MP becomes the SP.

Shortcut Trick

Net filling efficiency = 1/8 − 1/40 = (5 − 1)/40 = 4/40 = 1/10 per minute.

Time to fill full tank = 10 minutes.

Time to fill one-half of the tank = 10 ÷ 2 = 5 minutes.

∴ The correct answer is 5 minutes.

Alternate Method

Given: Filling pipe time = 8 min, Emptying pipe time = 40 min.

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

⇒ Combined efficiency = 5 − 1 = 4 units/min

⇒ One-half tank capacity = 40 ÷ 2 = 20 units

⇒ Time required = 20 ÷ 4 = 5 minutes

∴ The correct answer is 5 minutes.

Additional Information

Net Work Done

When one pipe fills (rate x) and another empties (rate y), the net rate is (1/x − 1/y) per unit time.

Pipes and Cisterns Logic

Inlet pipes have positive efficiency, while outlet (leakage) pipes have negative efficiency.

Partial Filling

If only a fraction k of the tank needs to be filled, the time taken is k × (Total Capacity ÷ Net Efficiency).

Shortcut Trick

Let the Base be b. Then the Altitude (h) = (2/3)b.

Area = 1/2 × Base × Altitude ⇒ 84 = 1/2 × b × (2/3)b.

84 = 1/3 × b² ⇒ b² = 252.

b = √252 = √(36 × 7) = 6√7 cm.

∴ The correct answer is 6√7 cm.

Alternate Method

Given:

⇒ Area of the triangle = 84 cm²

⇒ Altitude (h) = 2/3 of Base (b)

Formula Used:

⇒ Area of a triangle = 1/2 × Base × Altitude

Calculations:

⇒ Area = 1/2 × b × h

⇒ 84 = 1/2 × b × (2/3)b

⇒ 84 = (1/3) × b²

⇒ b² = 84 × 3

⇒ b² = 252

⇒ b = √252

⇒ b = √(36 × 7)

⇒ b = 6√7 cm

∴ The correct answer is 6√7 cm.

Additional Information

Heron's Formula

For a triangle with sides a, b, and c, Area = √[s(s−a)(s−b)(s−c)], where s (semi-perimeter) = (a + b + c) / 2.

Equilateral Triangle Area

The area of an equilateral triangle with side 'a' is given by (√3 / 4) × a².

Right-Angled Triangle

Area = 1/2 × Product of the two sides containing the right angle.

Shortcut Trick

Mean proportional of X and 125 is 25.

By property: X × 125 = 25²

⇒ X × 125 = 625

⇒ X = 625 ÷ 125 = 5

∴ The correct answer is 5.

Alternate Method

Given: First term = X, Third term = 125, Mean Proportional = 25.

Formula Used: If b is the mean proportional between a and c, then b = √(a × c) or b² = a × c.

Calculations:

⇒ 25 = √(X × 125)

Squaring both sides:

⇒ 25² = X × 125

⇒ 625 = X × 125

⇒ X = 625 ÷ 125

⇒ X = 5

∴ The correct answer is 5.

Additional Information

Mean Proportional

The mean proportional between two numbers a and b is the square root of their product, expressed as x = √(ab).

Third Proportional

If a:b :: b:c, then c is called the third proportional to a and b, calculated as c = b² ÷ a.

Fourth Proportional

In the proportion a:b :: c:d, d is the fourth proportional, where d = (b × c) ÷ a.

Shortcut Trick

Assume a base mean of 50 and calculate the deviations: −21, +40, 0, −24, −14, +27, +36, −17, −9.

Net deviation ∑d = (−21 − 24 − 14 − 17 − 9) + (40 + 27 + 36) = −85 + 103 = +18.

Correction in mean = Net deviation ÷ Count = 18 ÷ 9 = +2.

Actual Mean = Base + Correction = 50 + 2 = 52.

∴ The correct answer is 52.

Alternate Method

Given: Observations = 29, 90, 50, 26, 36, 77, 86, 33, 41

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

⇒ Total number of observations (n) = 9

⇒ Sum = 29 + 90 + 50 + 26 + 36 + 77 + 86 + 33 + 41

⇒ Sum = 468

⇒ Mean = 468 ÷ 9

⇒ Mean = 52

∴ The correct answer is 52.

Additional Information

Properties of Arithmetic Mean

The sum of the deviations of all observations from their arithmetic mean is always zero (∑(x_i − x̄) = 0).

Combined Mean Formula

If two groups have sizes n₁, n₂ and means x̄₁, x̄₂, their combined mean is (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).

Effect of Operations

If every observation is increased, decreased, multiplied, or divided by a constant 'k', the mean also changes by the same constant.

Shortcut Trick

Sum of ages after 6 years = 42 + (2 × 6) = 54 years.

Ratio of A : B after 6 years = 2 : 1.

Total units (2 + 1) = 3 units → 54 years.

1 unit (B's age after 6 years) = 54 ÷ 3 = 18 years.

B's present age = 18 − 6 = 12 years.

∴ The correct answer is 12 years.

Alternate Method

Given: Sum of A and B = 42 years. After 6 years, A = 2 × B.

Formula Used: Linear equation in one variable.

Let B's present age be x years.

⇒ A's present age = 42 − x

⇒ After 6 years, A's age = (42 − x) + 6 = 48 − x

⇒ After 6 years, B's age = x + 6

⇒ According to question: 48 − x = 2 × (x + 6)

⇒ 48 − x = 2x + 12

⇒ 3x = 48 − 12

⇒ 3x = 36

⇒ x = 12

∴ The correct answer is 12 years.

Additional Information

Age Ratio Property

When adding years to a sum of ages, multiply the number of years by the number of people: Total Sum_future = Sum_present + (n × years).

Constant Difference

The difference between the ages of two people always remains constant, regardless of the number of years passed.

Shortcut Trick

Walking time = (Distance ÷ Speed) × 60 = (6 ÷ 12) × 60 = 30 minutes.

Rest stops = Distance − 1 = 6 − 1 = 5 stops (No rest after the last km).

Rest time = 5 × 7 = 35 minutes.

Total time = 30 + 35 = 65 minutes.

∴ The correct answer is 65 minutes.

Alternate Method

Given: Speed = 12 km/hr, Distance = 6 km, Rest duration = 7 mins after every 1 km.

Formula Used: Time = Distance ÷ Speed; Total Time = Walking Time + Resting Time.

⇒ Walking time = (6 ÷ 12) hours

⇒ Walking time in minutes = 0.5 × 60 = 30 minutes

⇒ Number of rest intervals = (6 ÷ 1) − 1 = 5

⇒ Total rest duration = 5 × 7 = 35 minutes

⇒ Total time taken = 30 + 35 = 65 minutes

∴ The correct answer is 65 minutes.

Additional Information

Rest Stops Calculation

In problems involving stops, if a person stops every n km, the number of stops is (Total Distance ÷ n), but if the division is exact, subtract 1 because the last stop is at the destination.

Average Speed with Stoppages

Average speed including stoppages = Total Distance ÷ (Walking Time + Stopping Time). Stoppage time per hour = (Fast Speed − Slow Speed) ÷ Fast Speed.

Shortcut Trick

For any point P inside a rectangle ABCD, the British Flag Theorem states:

PA² + PC² = PB² + PD²

⇒ 100² + 85² = 76² + PD²

⇒ 10000 + 7225 = 5776 + PD²

⇒ 17225 − 5776 = PD² ⇒ PD² = 11449

⇒ PD = √11449 = 107 cm

∴ The correct answer is 107 cm.

Alternate Method

Given:

PA = 100 cm, PB = 76 cm, PC = 85 cm

Formula Used:

In a rectangle ABCD, if P is any point in its plane, then PA² + PC² = PB² + PD².

A B C D P 100 cm 76 cm

Shortcut Trick

The total number always represents 100%.

Since adding 20% to 84 gives the full 100%, the value 84 must represent 80% of the number.

We need to find 75% of the same number.

Required Value = (84 ÷ 80) × 75 = 1.05 × 75 = 78.75

∴ The correct answer is 78.75.

Alternate Method

Given:

20% of a number + 84 = The same number

Formula Used:

Total Number = 100%

20% Added Value = 84 (80% part) Total Number (100%) = 105

Calculations:

⇒ Let the number be x.

⇒ 20% of x + 84 = x

⇒ 84 = x − 20% of x

⇒ 84 = 100% of x − 20% of x

⇒ 84 = 80% of x

⇒ x = (84 × 100) ÷ 80

⇒ x = 8400 ÷ 80 = 105

⇒ Now, we need to find 75% of the same number (x).

⇒ 75% of 105 = (75 ÷ 100) × 105

⇒ 75% of 105 = 0.75 × 105 = 78.75

∴ The correct answer is 78.75.

Additional Information

Percentage Base Rule

In any percentage calculation, the original or total quantity is always considered as the 100% base value.

Percentage to Fraction Conversion

To simplify percentage calculations, convert them into fractions by dividing the percentage value by 100 (e.g., 75% = 75 ÷ 100 = 3/4).

Percentage Change

Percentage change is calculated using the formula: ((Final Value − Initial Value) ÷ Initial Value) × 100.

Shortcut Trick Increase in average = (New score − Old average) ÷ Total innings Increase = (231 − 97) ÷ 7 Increase = 134 ÷ 7 ≈ 19.14 Rounding to the nearest whole number, the increase is 19. ∴ The correct answer is 19 runs. Alternate Method Given: Average for 6 innings = 97 Runs in 7th inning = 231

Formula Used: Average = Total runs ÷ Number of innings Calculations: ⇒ Total runs in 6 innings = 97 × 6 = 582 ⇒ Total runs in 7 innings = 582 + 231 = 813 ⇒ New average = 813 ÷ 7 ≈ 116.14 ⇒ Increase in average = 116.14 − 97 = 19.14 ⇒ Rounded to nearest whole number = 19 ∴ The correct answer is 19 runs. Additional Information Basic Average Formula The average (Arithmetic Mean) is calculated by dividing the sum of all observations by the total number of observations. Net Change in Average When a new value is added, the new average increases if the new value > old average, and decreases if new value < old average. Deviation Method Change in average = Total deviation ÷ New total count. This avoids large multiplications during exams.

Shortcut Trick

To find the Square of 2.99, use the Algebraic Identity (a − b)² = a² − 2ab + b².

Let a = 3 and b = 0.01.

(3 − 0.01)² = 3² − (2 × 3 × 0.01) + (0.01)²

= 9 − 0.06 + 0.0001 = 8.9401.

Comparing with the given choices, this matches Option 4.

∴ The correct answer is 8.9401.

Alternate Method

Given: Find the value of (2.99)².

Formula Used: Direct multiplication of decimal numbers.

⇒ (2.99)² = 2.99 × 2.99

⇒ 2.99 = 299/100

⇒ (299/100) × (299/100)

⇒ (299 × 299) / (100 × 100)

⇒ 89401 / 10000

⇒ 8.9401

This result is represented by Option 4.

∴ The correct answer is 8.9401.

Additional Information

Square of a Number (a + b)²

Another useful identity is (a + b)² = a² + 2ab + b², which helps in calculating squares of numbers slightly above a base.

Decimal Multiplication Rule

When multiplying two decimals, the number of decimal places in the product is the sum of the decimal places in the numbers being multiplied.

Difference of Squares

The identity a² − b² = (a − b)(a + b) is helpful for mental math and fast simplification of products.

Shortcut Trick

Expand brackets mentally by distributing the coefficients: a³ + 2a², −3a³ + 9a, and −5a² − 25a.

Combine cubic terms: (1 − 3)a³ = −2a³.

Combine square terms: (2 − 5)a² = −3a².

Combine linear terms: (9 − 25)a = −16a.

∴ The correct answer is −2a³ − 3a² − 16a.

Alternate Method

Given Expression: a²(a + 2) − 3a(a² − 3) − 5a(a + 5)

Formula Used: Distributive Property of multiplication over addition/subtraction: x(y + z) = xy + xz.

Step-by-step Calculation:

⇒ a²(a) + a²(2) − [3a(a²) − 3a(3)] − [5a(a) + 5a(5)]

⇒ (a³ + 2a²) − (3a³ − 9a) − (5a² + 25a)

⇒ a³ + 2a² − 3a³ + 9a − 5a² − 25a

⇒ (a³ − 3a³) + (2a² − 5a²) + (9a − 25a)

⇒ −2a³ − 3a² − 16a

∴ The correct answer is −2a³ − 3a² − 16a.

Additional Information

Like Terms

Terms having the same variable raised to the same power are called like terms. Only like terms can be added or subtracted.

BODMAS in Algebra

Always simplify the brackets first using the distributive law before performing addition or subtraction across terms.

Exponent Rules

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

Shortcut Trick

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

Total Interest % difference for 4 years = 3.5% × 4 = 14%

Required Difference = 14% of ₹4,60,000

⇒ 0.14 × 4,60,000 = ₹64,400

∴ The correct answer is ₹64,400.

Alternate Method

Given: Principal (P) = ₹4,60,000, Time (T) = 4 years, Rates = 3.5% and 7%

Formula Used: Simple Interest (SI) = (P × R × T) / 100

⇒ SI paid to Bank A = (4,60,000 × 3.5 × 4) / 100

⇒ SI_A = 4,600 × 14 = ₹64,400

⇒ SI paid to Bank B = (4,60,000 × 7 × 4) / 100

⇒ SI_B = 4,600 × 28 = ₹1,28,800

⇒ Positive Difference = SI_B − SI_A = 1,28,800 − 64,400

⇒ Difference = ₹64,400

∴ The correct answer is ₹64,400.

Additional Information

Simple Interest Difference Rule

When Principal and Time are constant, the difference in SI is given by: ΔSI = P × (R₂ − R₁) × T / 100.

Total Amount Formula

The total amount (A) to be repaid is the sum of the Principal and the Simple Interest: A = P + SI.

Effect of Rate on Interest

Simple Interest is directly proportional to the Rate. Doubling the rate while keeping P and T constant will double the interest earned.

Shortcut Trick

For equal distances, use the Harmonic Mean formula for average speed.

Average Speed = (2 × x × y) ÷ (x + y)

Average Speed = (2 × 33 × 57) ÷ (33 + 57) = 3762 ÷ 90

Average Speed = 41.8 km/hr

∴ The correct answer is 41.8 km/hr.

Alternate Method

Given:

Speed during onward journey (x) = 33 km/hr

Speed during return journey (y) = 57 km/hr

Formula Used:

Average Speed = Total Distance ÷ Total Time

Let the distance between home and mall be D km.

⇒ Total Distance = D + D = 2D

⇒ Time 1 = D ÷ 33

⇒ Time 2 = D ÷ 57

⇒ Average Speed = 2D ÷ (D/33 + D/57)

⇒ Average Speed = 2 ÷ ((57 + 33) / (33 × 57))

⇒ Average Speed = (2 × 33 × 57) ÷ 90

⇒ Average Speed = 3762 ÷ 90 = 41.8 km/hr

∴ The correct answer is 41.8 km/hr.

Additional Information

Average Speed (Different Distances)

When distances are not equal, use Total Distance (d₁ + d₂ + ...) ÷ Total Time (t₁ + t₂ + ...).

Relative Speed

Same direction: S_relative = S₁ − S₂. Opposite direction: S_relative = 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

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

Given: Side opposite to 60° (Height) = 279 m.

Distance (Side opposite to 30°) = 279 ÷ √3 = (93 × 3) ÷ √3 = 93√3 m.

∴ The correct answer is 93√3 m.

Alternate Method

Given:

Height of lighthouse (h) = 279 m

Angle of depression = 60° ⇒ Angle of elevation from ship = 60°

Formula Used:

tan(θ) = Perpendicular ÷ Base

Calculations:

⇒ tan(60°) = Height ÷ Distance

⇒ √3 = 279 ÷ Distance

⇒ Distance = 279 ÷ √3

⇒ Distance = (93 × 3) ÷ √3

⇒ Distance = 93√3 m

∴ The correct answer is 93√3 m.

Additional Information

Angle of Depression and Elevation

The angle of depression from an observer to an object is numerically equal to the angle of elevation from the object to the observer due to alternate interior angles.

Standard Trigonometric Ratios

For height and distance problems: tan 30° = 1/√3, tan 45° = 1, and tan 60° = √3.

30-60-90 Triangle Ratio

The ratio of sides opposite to 30°, 60°, and 90° is always 1 : √3 : 2 respectively.

Shortcut Trick

Base Income = 100, Savings = 17, Expenditure = 83

Increase in Income = 38% × 100 = 38

Increase in Expenditure = 20% × 83 = 16.6

Increase in Savings (Units) = 38 − 16.6 = 21.4

Actual Increase = (61,500 ÷ 100) × 21.4 = 615 × 21.4 = ₹13,161

∴ The correct answer is increase by ₹13,161.

Alternate Method

Given:

Initial Income = ₹61,500

Initial Savings = 17% of 61,500

Initial Expenditure = 83% of 61,500

⇒ Initial Savings = 61,500 × (17 ÷ 100) = ₹10,455

⇒ Initial Expenditure = 61,500 − 10,455 = ₹51,045

⇒ New Income = 61,500 × 1.38 = ₹84,870

⇒ New Expenditure = 51,045 × 1.20 = ₹61,254

⇒ New Savings = 84,870 − 61,254 = ₹23,616

⇒ Change in Savings = 23,616 − 10,455 = ₹13,161

∴ The correct answer is increase by ₹13,161.

Additional Information

Income-Expenditure Relation

The basic formula is Income = Expenditure + Savings. Any change in two components directly affects the third.

Percentage Change

To find the new value after an x% increase, multiply the original value by (1 + x/100).

Alligation in Income-Exp

Percentage change in Income is the weighted average of percentage changes in Expenditure and Savings based on their initial ratio.

Shortcut Trick

Profit per magazine = Selling Price − Cost Price = 23 − 20 = 3.

Profit percentage = (Profit ÷ Cost Price) × 100.

Profit % = (3 ÷ 20) × 100 = 3 × 5 = 15%.

∴ The correct answer is 15%.

Alternate Method

Given:

Cost Price (CP) of magazine = ₹20

Selling Price (SP) of magazine = ₹23

Formula: Profit% = [(SP − CP) ÷ CP] × 100

⇒ Profit = 23 − 20 = 3

⇒ Profit % = (3 ÷ 20) × 100

⇒ Profit % = 0.15 × 100 = 15%

∴ The correct answer is 15%.

Additional Information

Profit and Loss Calculation

Profit occurs when Selling Price (SP) is greater than Cost Price (CP). Profit = SP − CP.

Percentage Basis

Profit or loss percentage is always calculated on the Cost Price (CP) unless stated otherwise in the problem.

Shortcut Trick

Assume the number is the LCM of denominators (4 and 5) = 20 units.

Three-fourth of 20 = 15 units.

Two-fifth of 20 = 8 units.

Difference (15 − 8) = 7 units → 63.

1 unit = 9 ∴ 20 units = 20 × 9 = 180.

∴ The correct answer is 180.

Alternate Method

Given: Difference between 3/4 and 2/5 of a number is 63.

Formula: Let the number be 'x'.

⇒ (3/4)x − (2/5)x = 63

⇒ [(15 − 8) / 20]x = 63

⇒ (7 / 20)x = 63

⇒ x = (63 × 20) / 7

⇒ x = 9 × 20

⇒ x = 180

∴ The correct answer is 180.

Additional Information

Fraction to Percentage

To solve quickly, convert fractions to percentages: 3/4 = 75% and 2/5 = 40%. The difference 35% of x = 63.

LCM Method for Fractions

Always assume the total quantity as the LCM of denominators to avoid decimal or fractional calculations during steps.

Algebraic Translation

The word 'exceeds' translates to a subtraction equation (Larger Value − Smaller Value = Difference).

Shortcut Trick

Shift vertex B(3, −2) to the origin (0, 0) by subtracting (3, −2) from all vertices.

New coordinates: A'(1, −4), B'(0, 0), and C'(2, 4).

Area of triangle = 1/2 |x₁y₂ − x₂y₁|

Area = 1/2 |(1 × 4) − (2 × −4)| = 1/2 |4 + 8| = 6 sq. units.

∴ The correct answer is 6 sq. units.

Alternate Method

Given: Vertices of the triangle are A(4, −6), B(3, −2), and C(5, 2).

Formula Used: Area = 1/2 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

⇒ Area = 1/2 |4(−2 − 2) + 3(2 − (−6)) + 5(−6 − (−2))|

⇒ Area = 1/2 |4(−4) + 3(2 + 6) + 5(−6 + 2)|

⇒ Area = 1/2 |−16 + 3(8) + 5(−4)|

⇒ Area = 1/2 |−16 + 24 − 20|

⇒ Area = 1/2 |−12|

⇒ Area = 6 sq. units.

∴ The correct answer is 6 sq. units.

Additional Information

Collinearity of Points

Three points are collinear if the area of the triangle formed by them is zero.

Centroid of a Triangle

The centroid G(x, y) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated as ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is given by √((x₂ − x₁)² + (y₂ − y₁)²).

Shortcut Trick

Factor out perfect squares to express all terms as multiples of √3.

√75 = 5√3, √108 = 6√3, √6075 = 45√3

Expression = (5√3 − 6√3 + 45√3)²

Sum inside bracket = 44√3

Square the sum = (44√3)² = 1936 × 3 = 5808

∴ The correct answer is 5808.

Alternate Method

Given:

(√75 − √108 + √6075)²

Calculations:

⇒ Simplifying each surd independently:

⇒ √75 = √(25 × 3) = 5√3

⇒ √108 = √(36 × 3) = 6√3

⇒ √6075 = √(2025 × 3) = 45√3

⇒ Substituting the simplified surds back into the expression:

⇒ (5√3 − 6√3 + 45√3)²

⇒ Grouping the terms together:

⇒ [(5 − 6 + 45)√3]²

⇒ [44√3]²

⇒ Squaring both parts inside the bracket:

⇒ 44² × (√3)²

⇒ 1936 × 3 = 5808

∴ The correct answer is 5808.

Additional Information

Properties of Surds

The multiplication property √(a × b) = √a × √b helps extract perfect squares to simplify complex roots easily.

Perfect Squares

Memorising squares of numbers up to 50 is extremely useful. For example, knowing 45² = 2025 instantly identifies √6075 as 45√3.

BODMAS / PEMDAS Rule

Always simplify the operations inside the Brackets first before applying the exponent (power of 2) to the overall expression.

Shortcut Trick

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

Here, markup a = +29% and discount b = −7%.

Net Gain% = 29 − 7 + (29 × (−7)) ÷ 100

Net Gain% = 22 − 203 ÷ 100 = 22 − 2.03

Net Gain% = 19.97%.

∴ The correct answer is 19.97% gain.

Alternate Method

Given: Markup = 29% above CP, Discount = 7% on MP. Formulas: MP = CP × (100 + Markup%) ÷ 100 | SP = MP × (100 − Discount%) ÷ 100 | Profit% = (SP − CP) ÷ CP × 100.

⇒ Let Cost Price (CP) = 100 units.

⇒ Marked Price (MP) = 100 + 29% of 100 = 129 units.

⇒ Selling Price (SP) = 129 − 7% of 129 = 129 − (7 × 1.29) = 129 − 9.03 = 119.97 units.

⇒ Gain = SP − CP = 119.97 − 100 = 19.97 units.

⇒ Gain % = (19.97 ÷ 100) × 100 = 19.97%.

∴ The correct answer is 19.97% gain.

Additional Information

Relationship between CP, MP, and Discount

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

Successive Percentage Change

When a value is increased by x% and then decreased by y%, the net effect is x − y − (xy ÷ 100)%. If the result is positive, it is a gain; if negative, it is a loss.

The logic followed here is:

For, GMUZ → EJSW

For, YAMN → WXKK

Now, apply the same to SRGE:

So, SRGE → QOEB

Hence, the correct answer is Option 1.

The logic followed here is:

Given number: 178673

Logic:

Add 2 to each odd digit

Subtract 2 from each even digit

Original number: 1 7 8 6 7 3

New number: 3 9 6 4 9 5

Largest digit in the new number = 9

Smallest digit in the new number = 3

Sum = 9 + 3 = 12

Hence, the correct answer is Option 3.

The logic followed here is:

So, 'LKR145' will replace the question mark in the given series.

Hence, the correct answer is "Option 3"

Given: F, G, H, I, M, N and O sit around a circular table facing the centre.

Only two people sit between I and O when counted from the left of O.

G sits third to the left of N.

H sits to the immediate right of N.

H sits second to the left of I.

F is not an immediate neighbour of G.

So, H sits third to the right of M.

Hence, the correct answer is "Option 4".

Family Chart:

Decoding the Symbols:

Given: O - P # T + G @ V

It means: O is the wife of P, P is the father of T, T is the sister of G, and G is the brother of V.

So, O is the wife of P and mother in the family chain leading to V.

Hence, the correct answer is Option 4.

Given: (Left) * + 9 \ / ? > @ 5 5 / ? / 9 * 1 ? # + 6 # (Right)

Now,

Symbols, which are immediately preceded by a number and also immediately followed by a number.

So, Needed pattern is: Number - Symbol - Number

(Left) * + 9 \ / ? > @ 5 5 / ? / 9 * 1 ? # + 6 # (Right)

So, total valid symbols = 1

Hence, the correct answer is Option 3.

The logic followed here is:

For, RY 17 → VW 3

For, IK 11 → MI −3

Now, apply the same to PI 16:

So, PI 16 → TG 2

Hence, the correct answer is Option 2.

The logic followed here is:

Given:

Mr. Akai starts at Point A

18 km South 

25 km West 

9 km North 

11 km East 

5 km South 

4 km East

14 km North 

So, To return from P to A: Move 10 km East

Hence, the correct answer is Option 2.

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

Only two people sit to the right of C.

E sits second to the left of C.

A sits second to the left of D.

B sits to the immediate right of F.

So, there are 2 people between D and B.

Hence, the correct answer is "Option 1".

BODMAS Table: 

Decoding the given information:

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

⇒ 56 ÷ 8 + 2 × 7 − 6 = ?

⇒ 7 + 14 − 6 = ?

⇒ 21 − 6 = ?

⇒ 15

Hence, the correct option is "Option 2".

The logic followed here is:

Given:

Gaurav starts at Point A

11 km South

7 km West (left from South)

14 km North

11 km East

3 km South

So, To return from P to A: Move 4 km East

Hence, the correct answer is Option 2.