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The correct answer is 3X N.

Key Points

• According to Newton's Universal Law of Gravitation, every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The formula for the gravitational force (F) is expressed as F = (G × M × m) / R², where G is the Universal Gravitational Constant, M is the mass of the Earth, m is the mass of the object, and R is the radius of the Earth.
• From the formula, it is evident that the gravitational force is directly proportional to the mass of the object (F ∝ m) when the mass of the Earth and the radius remain constant.
• In the given scenario, the initial force is X for a mass m. If the mass is tripled to 3m, the new force F' becomes (G × M × 3m) / R², which is equivalent to 3 × [(G × M × m) / R²].
• Substituting the value of the initial force, we get F' = 3X. Therefore, tripling the mass results in tripling the gravitational force experienced by the body.

 Additional Information

• Mass vs. Weight:
• Mass is an intrinsic property of matter and remains constant regardless of location, measured in kilograms (kg).
• Weight is the gravitational force acting on an object, calculated as W = m × g, where g is the acceleration due to gravity.
• Acceleration due to Gravity (g):
• The value of g on the surface of Earth is approximately 9.8 m/s².
• Crucially, g is independent of the mass of the falling object; however, the total force (X) depends on the object's mass.
• Universal Gravitational Constant (G):
• Its value is approximately 6.674 × 10⁻¹¹ N⋅m²/kg² and remains the same throughout the universe.

The correct answer is Constant velocity.

Key Points

• In a Velocity-Time (v-t) graph, the velocity is represented on the y-axis (vertical) and the time is represented on the x-axis (horizontal).
• A horizontal straight line parallel to the time axis implies that for any given time interval, the value of velocity remains unchanged.
• When the velocity of an object does not change with respect to time, the object is said to be moving with uniform motion or constant velocity.
• The slope of a velocity-time graph provides the acceleration of the object; since a horizontal line has a zero slope, the acceleration of the object is zero.
• The displacement of the object during a specific time interval can be calculated by finding the area under the horizontal line within that interval.

 Additional Information

• Non-uniform acceleration:
• This is represented by a curved line on a velocity-time graph, indicating that the rate of change of velocity is not constant.
• Zero Velocity:
• If the horizontal line lies exactly on the x-axis (where y = 0), it indicates the object is at rest and has no velocity.
• Uniform Acceleration:
• This is represented by a straight line inclined at a constant angle to the time axis, where the velocity increases or decreases at a steady rate.
• Significance of Area:
• In physics, the integral of velocity with respect to time gives displacement; geometrically, this is the rectangular area formed under the constant velocity line.

The correct answer is Carnivore population will decrease due to less food.

Key Points

• In an ecosystem, the food chain represents a linear sequence of organisms where nutrients and energy are transferred from one trophic level to another.
• Producers, primarily green plants, occupy the first trophic level and convert solar energy into chemical energy through photosynthesis.
• Herbivores are the primary consumers that depend directly on producers for sustenance; if their population decreases, the available energy for the next level is restricted.
• Carnivores act as secondary consumers that rely on herbivores as their primary food source; a sudden drop in the herbivore population results in a scarcity of prey.
• Due to the lack of available biomass and energy from herbivores, the carnivore population will inevitably decline because they cannot directly consume producers to meet their metabolic needs.
• This imbalance creates a trophic cascade where the increase in producers occurs because there are fewer herbivores to consume them, but the higher levels suffer from starvation.

 Additional Information

• Trophic Levels:
• Producers: These are autotrophs like plants and phytoplankton that form the base of the ecological pyramid.
• Herbivores: Also known as primary consumers, these organisms possess specialized digestive systems to process plant cellulose.
• Carnivores: These are secondary or tertiary consumers that regulate the population of lower trophic levels through predation.
• Decomposers: Organisms like bacteria and fungi that break down dead organic matter; their population usually fluctuates based on the total biomass available rather than a specific increase in producers.
• Energy Flow: According to Lindeman's 10 percent law, only 10 percent of the energy is transferred to each higher trophic level, making the survival of carnivores highly sensitive to herbivore density.

The correct answer is 0.1 m/s².

Key Points

• The train starts from rest, which implies that the initial velocity (u) is 0 m/s.
• The final velocity (v) is provided as 216 km/h. To perform calculations in SI units, this must be converted to meters per second (m/s) by multiplying by the factor 5/18. Thus, v = 216 × (5/18) = 60 m/s.
• The time duration (t) is given as 10 minutes. This must be converted into seconds for consistency with SI units: t = 10 × 60 = 600 seconds.
• According to the first equation of motion for uniform acceleration, the relationship between velocity, time, and acceleration is expressed as v = u + at.
• Rearranging the formula to solve for acceleration (a), we get a = (v − u) / t. Substituting the known values: a = (60 − 0) / 600 = 60 / 600 = 0.1 m/s².

 Additional Information

• Acceleration:
• It represents the vector quantity describing the rate of change of velocity over time.
• The standard SI unit for measuring acceleration is m/s² (meters per second squared).
• If the velocity of an object increases, the acceleration is positive; if it decreases, it is termed retardation or deceleration.
• Uniform Motion vs. Non-Uniform Motion:
• In uniform acceleration, the velocity changes by equal amounts in equal time intervals, resulting in a linear velocity-time graph.
• The slope of a velocity-time graph provides the magnitude of the acceleration.
• The area under a velocity-time graph represents the displacement covered by the object.
• Unit Conversions:
• To convert km/h to m/s, multiply by 5/18.
• To convert m/s to km/h, multiply by 18/5.
• Ensuring all variables are in the International System of Units (SI) is crucial for solving physics problems accurately.

The correct answer is 4.4 g.

Key Points

• The chemical reaction described is the thermal decomposition of calcium carbonate (CaCO₃) into calcium oxide (CaO) and carbon dioxide (CO₂).
• This process is governed by the Law of Conservation of Mass, which states that mass is neither created nor destroyed in a chemical reaction. Therefore, the total mass of the reactants must equal the total mass of the products in a closed system.
• The balanced chemical equation for the reaction is: CaCO₃(s) → CaO(s) + CO₂(g).
• According to the data provided: The mass of the reactant (calcium carbonate) is 10 g, and the mass of one product (calcium oxide) is 5.6 g.
• Applying the conservation principle: Mass of CaCO₃ = Mass of CaO + Mass of CO₂.
• By substituting the values: 10 g = 5.6 g + Mass of CO₂.
• To find the mass of the gas produced, we subtract the mass of the solid residue from the initial mass: Mass of CO₂ = 10 g − 5.6 g = 4.4 g.

 Additional Information

• Law of Conservation of Mass:
• Established by Antoine Lavoisier in 1789, this law is a pillar of modern chemistry.
• It dictates that the number of atoms of each element remains constant throughout the reaction, ensuring stoichiometric balance.
• Calcium Carbonate (CaCO₃):
• Commonly found in nature as limestone, chalk, or marble.
• It has a molar mass of approximately 100.1 g/mol.
• Calcium Oxide (CaO):
• Frequently called quicklime, it is used extensively in industrial applications such as water treatment and glass making.
• It has a molar mass of approximately 56.1 g/mol.
• Carbon Dioxide (CO₂):
• A greenhouse gas with a molar mass of approximately 44.01 g/mol.
• In an open container, this gas would escape, causing an apparent loss in mass; however, in a closed container, the total mass remains measurable and constant.

The correct answer is 250 N.

Key Points

• Identify the given physical quantities: The mass (m) of the wooden block is 50 kg. The block starts from rest, which means the initial velocity (u) is 0 m/s.
• The block reaches a final velocity (v) of 10 m/s within a time (t) of 2 s.
• Calculate the acceleration (a) of the block using the first kinematic equation of motion. The formula is defined as a = (v − u) / t.
• By substituting the given values: a = (10 m/s − 0 m/s) / 2 s = 5 m/s². This indicates the rate at which the block's velocity increases per second.
• Apply Newton's Second Law of Motion to find the force. The law states that Force (F) is the product of mass (m) and acceleration (a), represented by the formula F = m × a.
• Substitute the mass and calculated acceleration into the formula to find the value of X: X = 50 kg × 5 m/s² = 250 N.

 Additional Information

• Newton's Second Law of Motion:
• It provides the definition for the magnitude of force. It states that the net force acting on an object is equal to the rate of change of its linear momentum.
• The SI unit of force is the Newton (N). One Newton is the force required to provide an acceleration of 1 m/s² to a mass of 1 kg.
• Smooth Horizontal Surface:
• The term smooth implies that there is zero friction between the block and the surface. Therefore, the applied force X is the net unbalanced force acting on the block.
• On a horizontal surface, the weight (mg) of the block acting downwards is perfectly balanced by the normal reaction force acting upwards, resulting in zero vertical motion.
• Equations of Motion:
• The calculation assumes uniform acceleration, meaning the force X remains constant throughout the 2 s interval.
• Other related equations for uniform acceleration include s = ut + (1/2)at² and v² = u² + 2as.

The correct answer is 7 : 8.

Key Points

• The molecular mass of a compound is calculated by summing the atomic masses of all the individual atoms present in one molecule of that substance.
• A nitrogen molecule (N₂) consists of two nitrogen atoms. Since the atomic mass of nitrogen (N) is approximately 14 atomic mass units (u), the total unit mass of one N₂ molecule is 2 × 14 = 28 u.
• An oxygen molecule (O₂) consists of two oxygen atoms. Given that the atomic mass of oxygen (O) is approximately 16 atomic mass units (u), the total unit mass of one O₂ molecule is 2 × 16 = 32 u.
• To find the required ratio of the mass of a nitrogen molecule to an oxygen molecule, we compare their respective molecular masses: 28 : 32.
• The ratio 28 : 32 can be simplified by dividing both numbers by their greatest common divisor, which is 4. This yields 28/4 : 32/4, resulting in the final ratio of 7 : 8.

 Additional Information

• Atomic Mass Unit (u):
• This is a standard unit of mass that quantifies mass on an atomic or molecular scale.
• It is defined as exactly 1/12^th of the mass of a single unbound neutral atom of carbon-12 in its nuclear and electronic ground state.
• Diatomic Nature:
• Both Nitrogen and Oxygen are classified as homonuclear diatomic molecules under standard temperature and pressure conditions.
• Other elements that naturally exist as diatomic molecules include Hydrogen (H₂), Fluorine (F₂), Chlorine (Cl₂), Bromine (Br₂), and Iodine (I₂).
• Molar Mass vs. Molecular Mass:
• While molecular mass refers to the mass of one single molecule (in u), molar mass refers to the mass of one mole of the substance (6.022 × 10²³ molecules) expressed in grams per mole (g/mol).
• The numerical value remains identical; thus, the molar mass ratio for N₂ to O₂ is also 28 g/mol : 32 g/mol.

The correct answer is Ca3(PO4)2.

Key Points

• Calcium (Ca) is an element belonging to Group 2 of the periodic table, known as the alkaline earth metals. It possesses two electrons in its outermost shell, which it loses to achieve stability, forming a divalent cation with a charge of +2 (Ca²⁺).
• The Phosphate ion (PO₄) is a polyatomic anion consisting of one phosphorus atom and four oxygen atoms. It carries a net negative charge of −3 (PO₄³⁻).
• The chemical formula for an ionic compound is determined by balancing the total positive and negative charges to ensure the molecule is electrically neutral.
• Using the criss-cross method, the magnitude of the charge of the calcium ion (2) becomes the subscript for the phosphate group, and the magnitude of the charge of the phosphate ion (3) becomes the subscript for calcium.
• To balance the total charge, three Ca²⁺ ions (total charge +6) combine with two PO₄³⁻ ions (total charge −6), resulting in the stable chemical formula Ca₃(PO₄)₂.

 Additional Information

• Properties and Occurrences of Calcium Phosphate
• Biological Importance: It is a vital mineral in the human body, serving as the primary structural component of tooth enamel and bone tissue. In the skeletal system, it often exists as hydroxyapatite.
• Industrial Applications: It is extensively used in the production of phosphorus fertilizers. When treated with sulfuric acid, it yields single superphosphate (SSP), which provides essential nutrients to crops.
• Physical State: Pure calcium phosphate is a white solid that is relatively insoluble in water, which is why it remains stable within the biological structures of living organisms.
• Food Industry: It is utilized as an anti-caking agent (E341) in powdered foods and as a calcium supplement in dairy products and cereals to enhance nutritional value.

The correct answer is Turmeric turns red in basic solution.

Key Points

• China rose (Hibiscus) extract is a natural indicator that changes its color to magenta (dark pink) in acidic solutions and green in basic solutions. Hence, statement 1 is correct.
• Methyl orange is a synthetic indicator, not a natural one. It is chemically synthesized and exhibits a red color in acidic media and yellow in basic media. Hence, statement 2 is incorrect.
• Turmeric contains a natural yellow pigment called curcumin. This substance remains yellow in acidic or neutral solutions but undergoes a chemical reaction in the presence of a base, turning red or reddish-brown. Hence, statement 3 is correct.
• Litmus is a natural dye extracted from lichens. Red litmus paper is used to identify bases as it turns blue in basic solutions, while it remains red in acidic solutions. Hence, statement 4 is incorrect.

 Additional Information

• Acid-Base Indicators are substances that change color based on the hydrogen ion concentration (pH) of a solution.
• Natural Indicators:
• Red Cabbage: Turns red in acids and green/yellow in bases.
• Onion/Vanilla: Known as olfactory indicators, their characteristic smell disappears in basic solutions but persists in acids.
• Synthetic Indicators:
• Phenolphthalein: A colorless organic compound that remains colorless in acidic solutions but turns deep pink in basic environments.

The correct answer is To optimize soil fertility and crop yield.

Key Points

• Balanced Nutrient Management (BNM) involves the application of essential plant nutrients, including Nitrogen (N), Phosphorus (P), and Potassium (K), along with secondary and micronutrients, in the correct proportions required by specific crops.
• The primary objective is to enhance soil fertility by replenishing nutrients extracted by previous harvests, thereby preventing soil degradation and ensuring long-term agricultural sustainability.
• By providing a steady and balanced supply of nutrients, BNM significantly improves crop yield, quality, and the plants overall resistance to biotic and abiotic stresses.
• It improves Nutrient Use Efficiency (NUE), ensuring that the maximum amount of applied fertilizer is actually absorbed by the plant rather than being lost to the environment through leaching or volatilization.
• Scientific BNM is often guided by soil testing, which helps in determining the exact nutrient deficit of a specific land parcel, leading to precision farming.

 Additional Information

• Water Requirements and Pesticides:
• Nutrient management cannot eliminate water requirements; in fact, nutrients are transported into the plant through osmosis and transpiration, which require adequate soil moisture.
• While healthy plants are more resistant to pests, the goal is never to maximize pesticide use, as excessive chemicals lead to soil toxicity and environmental pollution.
• NPK Ratios:
• In many regions like India, the ideal NPK ratio for cereal crops is often recommended as 4:2:1, but imbalances often occur due to the over-application of Urea.
• The Soil Health Card Scheme, launched in 2015, is a major initiative to promote balanced nutrient use based on 12 key soil parameters.
• Ecological Impact:
• Imbalanced fertilization, particularly the excess use of nitrogen, can lead to eutrophication of nearby water bodies and the emission of Nitrous Oxide (N₂O), a potent greenhouse gas.