National Institute of Technology Raipur Entrance Exam Set

The question probes the understanding of the fundamental principles of sustainable urban development, a key area of focus for institutions like the National Institute of Technology Raipur, which emphasizes resource efficiency and community well-being. The scenario presented involves a city grappling with increased vehicular emissions and a growing demand for public transportation. To address this, a multi-pronged approach is necessary, integrating technological advancements with policy interventions. The core of the solution lies in a holistic strategy that prioritizes the reduction of private vehicle dependency and the enhancement of public transit infrastructure. This includes investing in electric vehicle charging networks to encourage adoption of cleaner private transport, expanding and modernizing the public bus fleet with low-emission or zero-emission vehicles, and developing dedicated lanes for buses and bicycles to improve efficiency and safety. Furthermore, implementing congestion pricing in high-traffic zones can disincentivize private car use during peak hours. Smart traffic management systems, utilizing real-time data, can optimize traffic flow and reduce idling times, thereby lowering emissions. Promoting mixed-use development, where residential, commercial, and recreational spaces are integrated, reduces the need for long commutes. Finally, public awareness campaigns about the benefits of sustainable transportation and the availability of alternative modes are crucial for behavioral change. Considering these elements, the most comprehensive and effective strategy would involve a combination of infrastructure upgrades, technological integration, and policy measures that directly address the root causes of increased emissions and traffic congestion. This aligns with the National Institute of Technology Raipur’s commitment to fostering innovative solutions for societal challenges through engineering and planning.

The question probes the understanding of the fundamental principles of sustainable urban development and resource management, particularly relevant to the context of a growing city like Raipur, which is a hub for engineering and technology. The core concept being tested is the integration of ecological considerations with infrastructural planning. A key aspect of sustainable urban planning is the efficient management of water resources, especially in regions that might experience seasonal variations in rainfall. The concept of a “closed-loop water system” or “circular water economy” is central to minimizing external water dependency and reducing wastewater discharge. This involves treating and reusing water for various purposes, such as irrigation, industrial processes, and even potable water after advanced treatment. Furthermore, incorporating green infrastructure, like bioswales and permeable pavements, plays a crucial role in managing stormwater runoff, recharging groundwater, and mitigating the urban heat island effect. The question requires candidates to synthesize knowledge of environmental engineering, urban planning, and resource economics to identify the most comprehensive approach. The correct answer emphasizes a multi-faceted strategy that includes water conservation, advanced wastewater treatment and reuse, and the implementation of green infrastructure. This holistic approach aligns with the forward-thinking educational philosophy of institutions like the National Institute of Technology Raipur, which encourages interdisciplinary problem-solving and innovation in addressing societal challenges. The other options, while containing elements of good practice, are less comprehensive. Focusing solely on rainwater harvesting, while important, does not address the full spectrum of water management. Similarly, prioritizing industrial water efficiency without considering domestic use and wastewater treatment, or concentrating only on green spaces without integrating water management, presents an incomplete solution. The correct answer represents the most robust and integrated strategy for long-term urban sustainability.

The question probes the understanding of the fundamental principles of sustainable urban development, a core area of study within engineering and planning disciplines at institutions like the National Institute of Technology Raipur. The scenario involves a hypothetical city, “Nirvana Nagar,” facing challenges related to resource management and environmental impact. To determine the most effective strategy, one must analyze the interconnectedness of urban systems and the long-term implications of various interventions. The core concept here is the integration of ecological principles into urban planning. A truly sustainable approach prioritizes minimizing environmental footprint while ensuring social equity and economic viability. Let’s break down why the correct option is superior. Option A, focusing on a comprehensive, multi-sectoral approach that integrates green infrastructure, circular economy principles, and community engagement, directly addresses the multifaceted nature of sustainability. Green infrastructure, such as urban forests, permeable pavements, and green roofs, helps manage stormwater, reduce the urban heat island effect, and improve air quality. Circular economy principles, by contrast to linear “take-make-dispose” models, aim to keep resources in use for as long as possible, extracting maximum value from them and then recovering and regenerating products and materials at the end of each service life. This minimizes waste and resource depletion. Community engagement is crucial for ensuring that development plans are socially inclusive and meet the needs of residents, fostering a sense of ownership and long-term commitment to sustainability initiatives. This holistic strategy aligns with the advanced research and educational goals of NIT Raipur, which often emphasizes interdisciplinary problem-solving. Option B, while addressing renewable energy, is too narrow. Renewable energy is a vital component of sustainability, but it doesn’t encompass the full spectrum of challenges, such as waste management, water conservation, or social equity. Option C, emphasizing strict zoning regulations and industrial relocation, might offer some environmental benefits by separating polluting industries from residential areas, but it can also lead to economic disruption and social displacement, and it doesn’t inherently promote resource efficiency or ecological restoration. It’s a more traditional, often less effective, approach to environmental management. Option D, prioritizing technological solutions like advanced waste-to-energy plants, is also limited. While technology plays a role, it often overlooks the importance of behavioral change, community participation, and the fundamental redesign of urban systems to be inherently more sustainable. Furthermore, waste-to-energy plants can still have environmental impacts and may not address the root causes of waste generation. Therefore, the most effective strategy for Nirvana Nagar, reflecting the sophisticated understanding expected at NIT Raipur, is the integrated, multi-sectoral approach that tackles environmental, social, and economic dimensions simultaneously.

The question probes the understanding of the fundamental principles governing the design and operation of a basic electrical circuit, specifically focusing on Ohm’s Law and its implications for power dissipation. The scenario describes a scenario where a resistor’s resistance is doubled while the voltage across it remains constant. First, let’s consider the initial state of the circuit. Let the initial resistance be \(R_1\) and the voltage be \(V\). According to Ohm’s Law, the initial current \(I_1\) flowing through the resistor is given by \(I_1 = \frac{V}{R_1}\). The initial power dissipated by the resistor, \(P_1\), can be calculated using the formula \(P_1 = V \times I_1\) or \(P_1 = \frac{V^2}{R_1}\). Now, in the second scenario, the resistance is doubled, so the new resistance is \(R_2 = 2R_1\). The voltage \(V\) across the resistor remains constant. The new current \(I_2\) will be \(I_2 = \frac{V}{R_2} = \frac{V}{2R_1}\). Notice that the new current is half of the original current, \(I_2 = \frac{1}{2}I_1\). The new power dissipated, \(P_2\), is calculated as \(P_2 = V \times I_2 = V \times \frac{V}{2R_1} = \frac{V^2}{2R_1}\). Comparing this to the initial power \(P_1 = \frac{V^2}{R_1}\), we can see that \(P_2 = \frac{1}{2} \times \frac{V^2}{R_1} = \frac{1}{2}P_1\). Therefore, the power dissipated by the resistor is halved. This understanding is crucial for electrical engineering students at NIT Raipur, as it relates to thermal management in electronic components and efficient energy utilization, core concepts in circuit analysis and design. The ability to predict how changes in circuit parameters affect power consumption is fundamental for designing reliable and efficient systems.

The question probes the understanding of fundamental principles in materials science and engineering, particularly concerning the behavior of crystalline structures under stress, a core area relevant to many disciplines at the National Institute of Technology Raipur. The scenario describes a metallic alloy exhibiting anisotropic elastic properties, meaning its stiffness varies with direction. This anisotropy is a direct consequence of the underlying crystal lattice structure and the nature of atomic bonding. When subjected to a tensile stress along a specific crystallographic direction, the strain experienced by the material will be dependent on the elastic constants associated with that direction. In a cubic crystal system, the Young’s modulus \(E\) along a general direction specified by direction cosines \(\lambda_1, \lambda_2, \lambda_3\) is related to the elastic stiffness coefficients \(C_{ijkl}\) by the equation: \[ \frac{1}{E} = s_{1111} – 2(s_{1111} – s_{1212})\left(\lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2\right) + 4(s_{1111} – s_{1212} – 2s_{1212})\lambda_1^2\lambda_2^2\lambda_3^2 \] where \(s_{ijkl}\) are the elastic compliance coefficients. For cubic crystals, these can be expressed in terms of three independent compliance constants: \(s_{11}\), \(s_{12}\), and \(s_{44}\). The relationship between Young’s modulus \(E\) along a direction \([hkl]\) and the principal moduli \(E_{100}\), \(E_{110}\), \(E_{111}\) is often simplified using the anisotropy factor \(A = \frac{2s_{44}}{s_{11} – s_{12}}\). A value of \(A > 1\) indicates that the material is stiffer along close-packed directions like \([111]\) compared to directions like \([100]\). The question asks about the *maximum* strain. For a given tensile stress, the strain will be maximum when the Young’s modulus is minimum. The Young’s modulus is typically minimum along directions that are less densely packed with atoms or where the bonding is weaker. In many cubic metals, the \([100]\) direction exhibits a lower Young’s modulus compared to close-packed directions like \([111]\). This is because the \([100]\) direction involves fewer atomic bonds per unit length compared to the \([111]\) direction in a face-centered cubic (FCC) or body-centered cubic (BCC) lattice. Therefore, applying stress along the \([100]\) direction would result in the largest elastic deformation (strain) for a given stress value, assuming the material’s elastic behavior is governed by these directional dependencies. This concept is crucial for understanding material selection in structural applications where specific orientations might experience different stress states, and predicting failure or deformation requires knowledge of directional elastic properties, a key consideration in advanced materials engineering programs at NIT Raipur.

The question probes the understanding of the fundamental principles of structural integrity and material behavior under stress, particularly relevant to civil engineering disciplines at NIT Raipur. The scenario involves a cantilever beam supporting a uniformly distributed load. To determine the maximum bending stress, we first need to calculate the maximum bending moment. For a cantilever beam with a uniformly distributed load \(w\) over its entire length \(L\), the maximum bending moment occurs at the fixed support and is given by \(M_{max} = \frac{wL^2}{2}\). In this case, \(w = 10 \, \text{kN/m}\) and \(L = 5 \, \text{m}\). Therefore, \(M_{max} = \frac{(10 \, \text{kN/m})(5 \, \text{m})^2}{2} = \frac{10 \times 25}{2} = 125 \, \text{kN-m}\). The bending stress (\(\sigma\)) at any point in the beam is given by the flexure formula: \(\sigma = \frac{My}{I}\), where \(M\) is the bending moment at that section, \(y\) is the distance from the neutral axis to the extreme fiber, and \(I\) is the moment of inertia of the beam’s cross-section about the neutral axis. The maximum bending stress occurs where both \(M\) and \(y\) are maximum. For a rectangular cross-section of width \(b\) and depth \(h\), the moment of inertia about the neutral axis is \(I = \frac{bh^3}{12}\). The distance to the extreme fiber is \(y_{max} = \frac{h}{2}\). Given a rectangular cross-section of width \(b = 0.2 \, \text{m}\) and depth \(h = 0.4 \, \text{m}\), the moment of inertia is \(I = \frac{(0.2 \, \text{m})(0.4 \, \text{m})^3}{12} = \frac{0.2 \times 0.064}{12} = \frac{0.0128}{12} \approx 0.001067 \, \text{m}^4\). The distance to the extreme fiber is \(y_{max} = \frac{0.4 \, \text{m}}{2} = 0.2 \, \text{m}\). The maximum bending stress is then \(\sigma_{max} = \frac{M_{max} y_{max}}{I} = \frac{(125 \, \text{kN-m})(0.2 \, \text{m})}{0.001067 \, \text{m}^4}\). To ensure consistency in units, we convert kN-m to N-mm: \(125 \, \text{kN-m} = 125 \times 10^3 \, \text{N} \times 10^3 \, \text{mm} = 125 \times 10^6 \, \text{N-mm}\). And \(y_{max} = 0.2 \, \text{m} = 200 \, \text{mm}\). And \(I = 0.001067 \, \text{m}^4 = 0.001067 \times (1000 \, \text{mm})^4 = 0.001067 \times 10^{12} \, \text{mm}^4 = 1.067 \times 10^9 \, \text{mm}^4\). So, \(\sigma_{max} = \frac{(125 \times 10^6 \, \text{N-mm})(200 \, \text{mm})}{1.067 \times 10^9 \, \text{mm}^4} = \frac{25 \times 10^9}{1.067 \times 10^9} \, \text{N/mm}^2 \approx 23.43 \, \text{N/mm}^2\) or \(23.43 \, \text{MPa}\). This calculation is fundamental to understanding how structural elements like beams resist bending loads, a core concept in civil engineering at NIT Raipur. The ability to accurately predict maximum stress is crucial for ensuring safety and efficiency in design, preventing material failure, and selecting appropriate materials. This involves not only applying the correct formulas but also understanding the underlying principles of stress distribution within a material under load, which is a key learning objective for aspiring engineers at the institute. The analysis highlights the interplay between applied load, beam geometry, and material properties in determining structural performance.

The question probes the understanding of fundamental principles in material science and engineering, particularly concerning the behavior of crystalline structures under stress, a core area of study at institutions like the National Institute of Technology Raipur. The scenario involves a cubic crystal lattice, a common model for many metals and ceramics. The concept of slip systems, which are crystallographic planes and directions along which plastic deformation occurs most easily, is central to this question. For a face-centered cubic (FCC) lattice, the most densely packed planes are the {111} planes, and within these planes, the close-packed directions are the directions. Therefore, the slip systems in FCC structures are of the type {111}. This means that slip can occur on any of the four {111} planes, in any of the three directions within each of those planes, resulting in a total of 12 primary slip systems. The question asks about the *most common* slip systems, which are indeed the {111} type due to the high atomic density on these planes and the close packing of atoms along these directions, minimizing the energy required for dislocation movement. Other crystal structures, like body-centered cubic (BCC), have different slip systems (e.g., {110}, {211}), and hexagonal close-packed (HCP) structures have specific basal and non-basal slip systems. The question specifically refers to a cubic lattice without specifying FCC or BCC, but the phrasing “most common slip systems” in the context of general materials science often defaults to the highly symmetric and widely studied FCC structure, or it implies a general understanding of how crystal structure dictates slip. Given the options, the {111} designation is the most universally recognized and prevalent slip system in materials exhibiting high ductility and ease of plastic deformation, aligning with the foundational knowledge expected of students entering engineering programs at NIT Raipur.

The question probes understanding of the fundamental principles of structural integrity and material science as applied in civil engineering, a core discipline at NIT Raipur. The scenario describes a bridge designed with a specific load-bearing capacity and a known material property (Young’s Modulus). The core concept to evaluate is how changes in geometric parameters, specifically the cross-sectional area, affect the structural response under load, particularly stiffness and deflection. A bridge’s stiffness is directly proportional to its cross-sectional moment of inertia (I) and Young’s Modulus (E), and inversely proportional to its length (L). For a rectangular cross-section of width \(b\) and height \(h\), the moment of inertia about the neutral axis is \(I = \frac{bh^3}{12}\). If the cross-sectional area \(A = bh\) is kept constant, and the height \(h\) is increased while the width \(b\) is decreased proportionally (so \(b = A/h\)), the moment of inertia changes. Specifically, if the height is doubled and the width halved, the new moment of inertia \(I’ = \frac{(A/2)(2h)^3}{12} = \frac{(A/2)(8h^3)}{12} = \frac{4Ah^3}{12} = 4 \times \frac{Ah^3}{12}\). However, the question states the *cross-sectional area* is doubled, not that the dimensions are changed while keeping the area constant. Let the original cross-sectional area be \(A_1\) and the new area be \(A_2 = 2A_1\). For a given shape, the moment of inertia is generally related to the area. For a simple rectangular beam of width \(b\) and height \(h\), \(A = bh\) and \(I = \frac{bh^3}{12}\). If we assume the shape of the cross-section remains similar (e.g., a square or a rectangle with a constant aspect ratio), and the area is doubled, then the linear dimensions (like height and width) are scaled by a factor of \(\sqrt{2}\). If the height is \(h_1\) and width is \(b_1\), then \(A_1 = b_1 h_1\). If \(A_2 = 2A_1\), and the aspect ratio \(h/b\) is constant, then \(h_2 = \sqrt{2} h_1\) and \(b_2 = \sqrt{2} b_1\). The new moment of inertia would be \(I_2 = \frac{b_2 h_2^3}{12} = \frac{(\sqrt{2}b_1)(\sqrt{2}h_1)^3}{12} = \frac{\sqrt{2}b_1 (2\sqrt{2}h_1^3)}{12} = \frac{4b_1 h_1^3}{12} = 4I_1\). The deflection (\(\delta\)) of a beam under a given load is inversely proportional to the product of Young’s Modulus (E) and the moment of inertia (I), i.e., \(\delta \propto \frac{1}{EI}\). Since the material (E) and the load remain constant, doubling the cross-sectional area, assuming a similar geometric scaling of the cross-section, leads to a quadrupling of the moment of inertia. Therefore, the deflection will be reduced by a factor of 4. The question asks about the *reduction* in deflection. If the original deflection was \(\delta_1\), the new deflection \(\delta_2 = \delta_1 / 4\). The reduction in deflection is \(\delta_1 – \delta_2 = \delta_1 – \delta_1/4 = \frac{3}{4}\delta_1\). This means the deflection is reduced *to* one-fourth of its original value, or by three-fourths. The question asks for the factor by which the deflection is reduced, which implies the new deflection relative to the old. A reduction *by* a factor of 4 means the new deflection is 1/4 of the original. The correct answer is that the deflection is reduced to one-fourth of its original magnitude. This is a critical concept in structural design, as minimizing deflection is crucial for serviceability and preventing aesthetic issues or damage to non-structural elements. Understanding how geometric properties influence stiffness is fundamental for civil engineers graduating from NIT Raipur, enabling them to design efficient and safe structures. This principle is directly applicable to bridge design, building frames, and various other civil engineering applications where load-carrying capacity and deformation control are paramount. The ability to predict and control deflection through appropriate material selection and geometric design is a hallmark of sound engineering practice.

The question probes understanding of the fundamental principles of sustainable urban development, a key area of focus for engineering and planning programs at institutions like the National Institute of Technology Raipur. Specifically, it tests the ability to identify the most impactful strategy for mitigating the urban heat island (UHI) effect, a phenomenon directly relevant to environmental engineering and civil infrastructure design. The UHI effect is caused by the replacement of natural landscapes with impervious surfaces like concrete and asphalt, which absorb and re-emit more solar radiation. This leads to higher ambient temperatures in urban areas compared to their rural surroundings. To effectively combat the UHI effect, strategies must address the root causes: increased heat absorption and reduced evapotranspiration. While increasing green spaces (like parks and rooftop gardens) is a well-known mitigation technique, its primary mechanism is through evapotranspiration, which cools the surrounding air. However, the question asks for the *most* impactful strategy in terms of direct heat reduction and energy efficiency. Reflective surfaces, often termed “cool pavements” or “cool roofs,” directly reduce the amount of solar radiation absorbed by urban infrastructure. By reflecting a higher percentage of sunlight back into the atmosphere, these materials significantly lower surface temperatures and, consequently, ambient air temperatures. This approach is particularly effective in densely built urban environments where green space may be limited. The reduction in absorbed heat also translates to lower energy consumption for cooling buildings, a critical aspect of sustainable urban planning and a direct concern for the National Institute of Technology Raipur’s focus on resource efficiency. Other options, such as improving public transportation or enhancing waste management, are important for overall sustainability but do not directly address the thermal properties of the urban fabric in the same way as reflective surfaces.

The question probes the understanding of material science principles relevant to structural integrity, a core concern in engineering disciplines at NIT Raipur. The scenario describes a bridge experiencing fatigue under cyclic loading. Fatigue failure in metals is primarily characterized by crack initiation and propagation, driven by stress concentrations and material microstructure. The key to preventing such failures lies in understanding the material’s response to repeated stress cycles. A material’s resistance to fatigue is quantified by its fatigue strength or endurance limit, which is the stress level below which the material can withstand an infinite number of stress cycles without failure. However, in practical engineering, especially for advanced materials and complex loading scenarios, the focus shifts to understanding the mechanisms of fatigue crack growth. This involves parameters like the stress intensity factor range, \( \Delta K \), and the material’s Paris law constants, \( C \) and \( m \), which describe the rate of crack growth per cycle. The scenario implies that the bridge material is susceptible to fatigue. To mitigate this, engineers would analyze the material’s properties and the applied stresses. Options related to increasing tensile strength or hardness might offer some improvement, but they don’t directly address the *mechanism* of fatigue crack propagation. Similarly, improving ductility might delay crack initiation but not necessarily arrest propagation once a crack has formed. The most effective approach to combat fatigue in a structural component like a bridge, especially when dealing with cyclic loading, is to select materials with a high fatigue limit or to design the structure to minimize stress concentrations and ensure that operating stresses remain below this limit. For a given material, understanding its crack growth resistance characteristics is paramount. Therefore, focusing on the material’s ability to resist crack propagation under cyclic stress, often related to its fracture toughness and microstructure, is the most pertinent consideration for preventing fatigue failure. This involves selecting materials with a favorable combination of strength, toughness, and a high endurance limit, and ensuring that the design minimizes stress risers. The explanation emphasizes the fundamental material science principles that underpin the design and maintenance of robust structures, aligning with the rigorous engineering education at NIT Raipur.

The question probes the understanding of the fundamental principles of sustainable urban development, a key area of focus for institutions like the National Institute of Technology Raipur, which emphasizes holistic engineering and societal impact. The scenario describes a city grappling with increased vehicular emissions and a growing population, directly impacting air quality and public health. To address this, the city council is considering various interventions. The core concept being tested is the identification of a strategy that promotes long-term environmental and social well-being, aligning with the principles of smart city planning and ecological responsibility. Option (a) proposes the development of a comprehensive, integrated public transportation network, coupled with incentives for non-motorized transport and strict emission controls for existing vehicles. This approach tackles the root cause of increased emissions by reducing reliance on private vehicles and promoting cleaner alternatives. It fosters a healthier urban environment, enhances accessibility for all citizens, and contributes to energy efficiency, all critical components of sustainable development. This aligns with the National Institute of Technology Raipur’s commitment to innovative solutions for societal challenges. Option (b), focusing solely on expanding road infrastructure, would likely exacerbate the problem by encouraging more private vehicle usage, leading to increased congestion and emissions, thus contradicting sustainability goals. Option (c), while addressing pollution through technological upgrades, fails to tackle the underlying issue of vehicle volume and might not be universally accessible or affordable. Option (d), concentrating on green spaces without addressing transportation, offers partial benefits but doesn’t resolve the primary source of air pollution in this scenario. Therefore, the integrated public transport and non-motorized transport strategy is the most effective and sustainable solution.

The question probes the understanding of the fundamental principles of sustainable urban development, a key area of focus for institutions like the National Institute of Technology Raipur, which emphasizes holistic engineering solutions. The scenario involves a hypothetical city, “Nirvana Nagar,” facing typical urban challenges. To determine the most effective strategy for sustainable growth, one must evaluate the core tenets of sustainability: environmental protection, social equity, and economic viability. Option (d) proposes a multi-pronged approach that integrates green infrastructure development, community engagement in planning, and the promotion of circular economy principles. Green infrastructure (e.g., urban forests, permeable pavements) directly addresses environmental concerns by mitigating pollution, managing stormwater, and enhancing biodiversity. Community engagement ensures social equity by giving residents a voice in development decisions, fostering a sense of ownership, and addressing local needs. Circular economy principles (e.g., waste reduction, resource reuse) bolster economic viability by creating new industries, reducing reliance on virgin materials, and minimizing waste disposal costs. This integrated strategy aligns with the comprehensive vision of sustainable development that the National Institute of Technology Raipur would champion, as it addresses the interconnectedness of ecological, social, and economic factors. Option (a) focuses solely on technological solutions, which, while important, can overlook crucial social and economic dimensions. For instance, advanced waste management systems might be economically prohibitive for certain communities or may not be adopted without public buy-in. Option (b) emphasizes economic growth above all else, which can lead to environmental degradation and social disparities if not balanced with other considerations. Rapid industrialization without adequate environmental safeguards or social impact assessments is a common pitfall. Option (c) prioritizes environmental conservation but might neglect the economic realities and social needs of the population, potentially leading to resistance or unfulfilled basic requirements. A truly sustainable approach, as expected in the rigorous academic environment of NIT Raipur, requires a balanced and integrated strategy.

The question probes the understanding of the fundamental principles of sustainable urban development, a core area of study within engineering and planning disciplines at institutions like the National Institute of Technology Raipur. The scenario involves a city facing increased water scarcity due to climate change and population growth, requiring a multifaceted approach to resource management. The correct answer, focusing on integrated water resource management (IWRM) that combines conservation, efficient infrastructure, and decentralized supply, directly addresses the interconnectedness of these challenges. IWRM emphasizes a holistic view, considering all water users and environmental needs, which is crucial for long-term sustainability. This approach aligns with the National Institute of Technology Raipur’s commitment to fostering innovative solutions for societal challenges through interdisciplinary research and education. The other options, while potentially contributing to water management, are less comprehensive. Focusing solely on technological solutions without addressing demand management or community involvement, or prioritizing short-term fixes over systemic changes, would not achieve the long-term resilience required. The emphasis on community participation and policy reform within IWRM also reflects the socio-technical aspects of engineering and planning, encouraging a broader understanding of problem-solving beyond purely technical fixes.

The question probes the understanding of the fundamental principles of **material science and engineering design**, specifically concerning the selection of materials for components subjected to cyclic loading and potential fatigue failure. The scenario describes a critical structural element within a new research facility at the National Institute of Technology Raipur, requiring a material that can withstand repeated stress cycles without fracturing. The core concept tested here is **fatigue resistance**, which is the ability of a material to endure repeated cyclic loading. Materials with high fatigue strength, characterized by a high endurance limit (the stress level below which a material can theoretically withstand an infinite number of cycles), are preferred for such applications. Furthermore, **ductility** is also a crucial factor. While high strength is desirable, a material that is too brittle might fail catastrophically with little warning. Ductile materials tend to deform plastically before fracturing, providing some indication of impending failure. Considering the context of a research facility at NIT Raipur, which likely involves advanced instrumentation and potentially sensitive experiments, **reliability and safety** are paramount. A material that exhibits a good combination of fatigue strength and ductility, along with reasonable stiffness and corrosion resistance (though not explicitly stated as the primary concern, it’s a common consideration in structural design), would be the most suitable choice. Option a) focuses on a material known for its excellent fatigue resistance and good ductility, making it a strong candidate for cyclic loading applications. This aligns with the need to prevent fatigue failure in a critical structural component. Option b) describes a material that might have high tensile strength but could be prone to brittle fracture under cyclic stress, especially if not properly heat-treated or if the stress concentrations are significant. Its lack of inherent fatigue resistance makes it less ideal. Option c) suggests a material that, while strong, might have limited ductility and a lower endurance limit compared to other options, making it susceptible to fatigue crack initiation and propagation under repeated stress. Option d) presents a material that might be suitable for static loads but lacks the specific properties required for components undergoing significant cyclic stress, potentially failing prematurely due to fatigue. Therefore, the material exhibiting superior fatigue resistance and adequate ductility is the most appropriate selection for the described structural component at the National Institute of Technology Raipur.

The question probes the understanding of fundamental principles of structural integrity and material science as applied in civil engineering, a core discipline at NIT Raipur. The scenario involves a cantilever beam subjected to a uniformly distributed load. The maximum bending moment in a cantilever beam with a uniformly distributed load \(w\) over its entire length \(L\) occurs at the fixed support and is given by the formula \(M_{max} = \frac{wL^2}{2}\). The maximum shear force also occurs at the fixed support and is equal to the total load, which is \(V_{max} = wL\). The question asks about the critical factor influencing the beam’s ability to withstand this load without failure. While all listed factors are important in structural design, the most *critical* for a cantilever beam under a distributed load, in terms of determining its capacity to resist bending, is the beam’s section modulus. The section modulus (\(Z\)) is a geometric property of a cross-section that relates to its resistance to bending. The maximum bending stress (\(\sigma_{max}\)) in the beam is directly proportional to the maximum bending moment (\(M_{max}\)) and inversely proportional to the section modulus (\(Z\)): \(\sigma_{max} = \frac{M_{max}}{Z}\). A higher section modulus means the beam can withstand a larger bending moment before the stress exceeds the material’s yield strength. While the material’s yield strength is crucial for determining the stress limit, the section modulus dictates how effectively the beam’s geometry distributes that stress. The length of the beam directly influences the magnitude of the bending moment (\(M_{max} \propto L^2\)), but once the moment is determined, the section modulus is the primary geometric factor resisting it. The density of the material affects the self-weight of the beam, which contributes to the load, but the question implies an external distributed load is the primary concern, and the material’s strength-to-weight ratio is a broader consideration than just density. Therefore, the section modulus is the most direct geometric property that governs the beam’s resistance to the bending stresses induced by the load.

The question probes the understanding of the fundamental principles of sustainable urban development, a key area of focus for institutions like the National Institute of Technology Raipur, which emphasizes responsible engineering and societal impact. The core concept being tested is the integration of ecological preservation with urban growth. Option A, focusing on the synergistic integration of green infrastructure and smart technology for resource efficiency and citizen well-being, directly addresses this by proposing a holistic approach that balances environmental concerns with technological advancement and human needs. This aligns with the institute’s commitment to innovation that serves societal goals. Option B, while mentioning environmental regulations, is too narrow and reactive, failing to capture the proactive and integrated nature of sustainable development. Option C, concentrating solely on economic incentives for developers, overlooks the crucial ecological and social dimensions. Option D, emphasizing the preservation of historical architectural styles, is a valid aspect of urban planning but not the overarching principle of sustainable development in its entirety, which encompasses broader ecological and resource management strategies. Therefore, the most comprehensive and forward-thinking approach, reflecting the ethos of NIT Raipur, is the integration of green infrastructure and smart technologies.

The question probes the understanding of fundamental principles in materials science and engineering, particularly concerning the behavior of crystalline structures under stress. Specifically, it addresses the concept of slip systems in face-centered cubic (FCC) and body-centered cubic (BCC) metals, which are crucial for plastic deformation. In FCC structures, the most densely packed planes are the {111} planes, and within these planes, the close-packed directions are the directions. Therefore, the slip systems in FCC metals are of the type {111}. There are 4 unique {111} planes and 6 unique directions within these planes, resulting in a total of 24 possible slip systems. However, some of these are crystallographically equivalent. The primary slip systems are typically considered to be the ones with the highest resolved shear stress. In BCC structures, the situation is more complex. While {110} planes are often considered the most important slip planes due to their high atomic density, slip can also occur on {112} and, less commonly, {123} planes. The slip directions are generally directions. Therefore, BCC metals have slip systems of the type {110}, {112}, and {123}. The number of slip systems in BCC metals is significantly higher than in FCC metals, contributing to their generally higher ductility and ability to deform at room temperature. For example, there are 12 {110} systems, 24 {112} systems, and 24 {123} systems, totaling 60 potential slip systems. The question asks about the *minimum* number of slip systems required for general plastic deformation according to the von Mises criterion. The von Mises criterion states that plastic deformation occurs when the distortion energy reaches a critical value. For arbitrary plastic deformation to occur in a polycrystalline material, at least five independent slip systems must be operative. This is because there are five independent components of strain that need to be accommodated. If fewer than five slip systems are available, the material may not be able to deform plastically in an arbitrary manner, potentially leading to fracture or brittle behavior. Both FCC and BCC metals possess more than five slip systems, which explains their general ductility. Therefore, the fundamental requirement for general plastic deformation, as dictated by the von Mises criterion, is the availability of at least five independent slip systems.

The question probes the understanding of fundamental principles in materials science and engineering, particularly concerning the behavior of crystalline solids under stress, a core area of study at NIT Raipur. The scenario describes a metallic alloy exhibiting a specific stress-strain curve. The key to answering lies in recognizing that the elastic limit represents the point beyond which permanent deformation occurs. In the provided stress-strain behavior, the initial linear portion of the curve signifies elastic deformation, where the material returns to its original shape upon removal of the load. The point where this linearity ceases, and the curve begins to deviate, is the elastic limit. Beyond this point, the material undergoes plastic deformation. The question asks to identify the characteristic that is *not* directly represented by the elastic limit. The elastic limit is fundamentally about the transition from reversible to irreversible deformation. It is intrinsically linked to the material’s ability to withstand stress without permanent change. Therefore, the elastic limit directly implies that the material will return to its original dimensions if the applied stress is removed below this point. It also signifies the maximum stress the material can endure elastically. Furthermore, it is a critical parameter in determining the material’s suitability for applications requiring dimensional stability under load. However, the elastic limit does not, in itself, dictate the total amount of plastic deformation the material can undergo before fracture. While it marks the *onset* of plastic deformation, the extent of this subsequent plastic deformation is governed by other material properties, such as ductility and strain hardening, which are distinct from the elastic limit itself. The elastic limit is a threshold, not a measure of subsequent deformation capacity. Therefore, the total strain at fracture is not directly determined by the elastic limit alone.

The question probes the understanding of the fundamental principles governing the operation of a synchronous generator, specifically focusing on the relationship between excitation current, terminal voltage, and power factor under varying load conditions, a core concept in electrical engineering relevant to the curriculum at National Institute of Technology Raipur. When a synchronous generator operates at a constant terminal voltage and frequency, and its load is varied from no-load to full-load, the excitation current (field current) must be adjusted to maintain the terminal voltage. At no load, the terminal voltage is primarily determined by the induced EMF, which is proportional to the excitation current. As a load is applied, the armature reaction (demagnetizing or magnetizing effect) and the synchronous impedance drop (which includes armature resistance and synchronous reactance) cause the terminal voltage to decrease. To counteract this voltage drop and maintain a constant terminal voltage, the excitation current must be increased. Consider the power factor. At unity power factor, the armature reaction is largely demagnetizing, requiring a higher excitation current compared to the no-load condition to maintain the same terminal voltage. At lagging power factor, the armature reaction is more strongly demagnetizing, necessitating an even greater increase in excitation current. Conversely, at leading power factor, the armature reaction is magnetizing, which helps to boost the terminal voltage. Therefore, less excitation current is needed to maintain the same terminal voltage compared to the no-load condition. The question asks for the condition where the excitation current is *least* for a constant terminal voltage. This occurs when the magnetizing effect of the armature current is at its maximum, which happens at a leading power factor. Specifically, a purely capacitive load (leading power factor of 1.0) would provide the maximum magnetizing effect from the armature reaction, thus requiring the least excitation current to maintain a constant terminal voltage.

The core principle being tested here is the understanding of how the fundamental theorem of calculus relates to the concept of instantaneous rate of change and accumulation. The question asks about the rate at which the volume of water in a reservoir is changing at a specific moment, given the net flow rate into the reservoir. The net flow rate, \(F_{net}(t)\), represents the rate of change of the volume of water, \(V(t)\), with respect to time. Mathematically, this is expressed as \(F_{net}(t) = \frac{dV}{dt}\). The question asks for the rate of change of volume at \(t = 5\) hours. Therefore, we need to evaluate \(F_{net}(5)\). Given the net flow rate function \(F_{net}(t) = 150 – 10\sqrt{t}\) liters per hour, we substitute \(t=5\) into the function: \(F_{net}(5) = 150 – 10\sqrt{5}\) To provide a numerical approximation for clarity in understanding the concept, we can approximate \(\sqrt{5} \approx 2.236\). \(F_{net}(5) \approx 150 – 10(2.236)\) \(F_{net}(5) \approx 150 – 22.36\) \(F_{net}(5) \approx 127.64\) liters per hour. The explanation should focus on the direct application of the derivative concept. The net flow rate is the derivative of the volume function. Therefore, to find the rate of change of volume at a specific time, one directly evaluates the net flow rate function at that time. This is a direct application of the definition of the derivative in a real-world context, relevant to civil engineering and environmental management disciplines often studied at NIT Raipur. Understanding this relationship is crucial for analyzing dynamic systems where rates of change are paramount. It highlights how a function representing a rate can be directly used to determine the instantaneous rate of change of the accumulated quantity. This concept is foundational for understanding concepts like mass balance, fluid dynamics, and environmental transport models, all of which are areas of study at institutions like NIT Raipur. The question tests the ability to connect a given rate function to the instantaneous rate of change of an accumulated quantity without needing to perform integration or further differentiation.

The question probes the understanding of the fundamental principles governing the behavior of semiconductor materials under varying environmental conditions, a core concept in materials science and electrical engineering programs at National Institute of Technology Raipur. The intrinsic carrier concentration \(n_i\) in a semiconductor is highly dependent on temperature. As temperature increases, more covalent bonds break, releasing electron-hole pairs, thus increasing \(n_i\). This relationship is generally described by an exponential function. For silicon, a common semiconductor, the intrinsic carrier concentration at room temperature (approximately 300 K) is around \(1.5 \times 10^{10} \text{ cm}^{-3}\). When a semiconductor is doped with an impurity atom that has more valence electrons than the host material (n-type doping, e.g., phosphorus in silicon), the majority carriers become electrons. Conversely, doping with an atom having fewer valence electrons (p-type doping, e.g., boron in silicon) makes holes the majority carriers. The question asks about the effect of increasing temperature on a p-type semiconductor. While increasing temperature increases the intrinsic carrier concentration, it also increases the thermal generation of electron-hole pairs. In a p-type semiconductor, the majority carriers are holes, and their concentration is primarily determined by the doping concentration. However, at sufficiently high temperatures, the intrinsic carrier concentration can become comparable to or even exceed the doping concentration. When this happens, the semiconductor starts behaving more like an intrinsic semiconductor, meaning the concentration of electrons and holes becomes nearly equal. The question specifically asks about the *net effect* on the majority carrier concentration. While the doping sets the baseline for majority carriers, the increased thermal generation at higher temperatures leads to a significant increase in minority carriers (electrons in a p-type material) and a proportional increase in intrinsic carriers. This phenomenon, where the intrinsic carrier concentration becomes dominant over the doping concentration at high temperatures, is known as the “intrinsic region” of operation for a doped semiconductor. Therefore, the majority carrier concentration, while still predominantly holes, will increase significantly due to thermal generation, approaching the intrinsic carrier concentration. The question is designed to test the understanding that temperature’s effect on intrinsic carrier generation is a dominant factor at elevated temperatures, impacting the overall carrier concentrations, including the majority carriers, even in a doped material. The correct answer reflects this understanding of the interplay between doping and thermal effects on carrier concentration.

The question probes understanding of the fundamental principles of sustainable urban development, a core area of study within engineering and planning disciplines at institutions like the National Institute of Technology Raipur. The scenario involves a hypothetical city, “Nirvana Nagar,” facing common urban challenges. The correct answer, focusing on integrated resource management and community participation, directly addresses the multifaceted nature of sustainability. This approach acknowledges that environmental protection, economic viability, and social equity are interconnected and require holistic solutions. For instance, implementing a closed-loop water system (resource management) alongside incentivizing local businesses that use recycled materials (economic viability) and establishing community gardens in underutilized urban spaces (social equity) exemplifies this integrated approach. Such strategies are crucial for developing resilient and livable cities, aligning with the forward-thinking educational ethos of NIT Raipur, which emphasizes practical application of knowledge to real-world problems. Other options, while touching on aspects of urban improvement, lack the comprehensive, systemic perspective essential for true sustainability. Focusing solely on technological upgrades without considering social impact, or prioritizing economic growth at the expense of environmental preservation, represents incomplete or potentially detrimental approaches to urban planning.

The question probes the understanding of fundamental principles in materials science and engineering, specifically concerning the behavior of crystalline solids under stress, a core area for aspiring engineers at NIT Raipur. The scenario involves a metallic alloy exhibiting a specific stress-strain relationship. The key is to identify the material property that directly quantifies the material’s resistance to permanent deformation. 1. **Elastic Limit:** This is the maximum stress a material can withstand without undergoing permanent deformation. Beyond this point, the material will not return to its original shape upon removal of the load. 2. **Yield Strength:** This is often used interchangeably with the elastic limit, especially in engineering contexts, and represents the stress at which a material begins to deform plastically. For many materials, the elastic limit and yield strength are very close. 3. **Ultimate Tensile Strength (UTS):** This is the maximum stress a material can withstand while being stretched or pulled before necking (local reduction in cross-sectional area) begins. It is the peak of the stress-strain curve. 4. **Modulus of Elasticity (Young’s Modulus):** This measures the stiffness of a material, defined as the ratio of stress to strain in the elastic region. It describes how much a material will deform elastically under a given load, not its resistance to permanent deformation. The scenario describes a material that, when subjected to increasing stress, initially deforms elastically. Upon unloading from a stress level of 350 MPa, it returns to its original shape, indicating that this stress is within its elastic limit. However, when the stress is increased to 450 MPa and then removed, a permanent deformation (residual strain) of 0.002 is observed. This residual strain signifies that the stress of 450 MPa has exceeded the material’s capacity to deform elastically and has caused plastic deformation. Therefore, the stress at which permanent deformation begins, the yield strength (or elastic limit), must be between 350 MPa and 450 MPa. The question asks for the property that defines the *onset* of permanent deformation. This is precisely the definition of yield strength.

The question probes the understanding of the fundamental principles of sustainable urban development, a core area of study in engineering and planning disciplines at institutions like the National Institute of Technology Raipur. The scenario involves a hypothetical city, “Vikas Nagar,” facing challenges related to resource management and environmental impact. To determine the most effective approach for Vikas Nagar, one must consider the interconnectedness of various urban systems and the long-term viability of proposed solutions. The core concept here is the integration of ecological principles with urban planning. This involves minimizing waste, optimizing resource utilization (water, energy), and fostering biodiversity within the urban fabric. A truly sustainable approach would not merely address one aspect in isolation but would seek synergistic solutions. Let’s analyze the options: 1. **Focusing solely on advanced waste-to-energy conversion:** While beneficial, this addresses only the waste management aspect and might not tackle water scarcity or energy efficiency comprehensively. It’s a partial solution. 2. **Implementing a comprehensive smart grid for energy distribution:** This is excellent for energy efficiency but doesn’t directly address water conservation or waste reduction strategies. It’s also focused on a single utility. 3. **Developing a closed-loop water management system with integrated green infrastructure:** This option directly tackles water scarcity through recycling and reuse, and simultaneously addresses environmental concerns by incorporating green infrastructure. Green infrastructure, such as bioswales, permeable pavements, and urban forests, plays a crucial role in stormwater management, reducing runoff pollution, improving air quality, and enhancing urban biodiversity. These elements contribute to a more resilient and ecologically sound urban environment, aligning with the holistic approach to sustainability emphasized in engineering and planning education. This integrated approach offers a more systemic solution to the multifaceted challenges faced by Vikas Nagar. 4. **Mandating strict building codes for energy efficiency in new constructions:** This is a vital step but primarily affects future development and doesn’t address the existing infrastructure’s sustainability challenges or the immediate need for resource management. Therefore, the most effective and holistic approach for Vikas Nagar, considering the principles of sustainable urban development and the need for integrated solutions, is the development of a closed-loop water management system with integrated green infrastructure. This strategy addresses multiple environmental and resource challenges simultaneously, promoting long-term resilience and ecological balance.

The question probes the understanding of the fundamental principles governing the behavior of semiconductor devices, specifically focusing on the concept of doping and its impact on electrical conductivity. In intrinsic semiconductors, the number of electrons in the conduction band is equal to the number of holes in the valence band, denoted by \(n_i\). When a semiconductor is doped with a pentavalent impurity (like Phosphorus in Silicon), it becomes an n-type semiconductor. In an n-type semiconductor, the majority charge carriers are electrons, and their concentration is approximately equal to the doping concentration, \(N_D\). The minority charge carriers are holes, and their concentration, \(p\), can be determined using the mass action law, which states that the product of electron and hole concentrations in thermal equilibrium is constant and equal to the square of the intrinsic carrier concentration: \(np = n_i^2\). Given that the semiconductor is doped with \(N_D = 5 \times 10^{22} \text{ m}^{-3}\) and the intrinsic carrier concentration is \(n_i = 1.5 \times 10^{16} \text{ m}^{-3}\), and assuming the doping concentration is significantly higher than the intrinsic carrier concentration (\(N_D \gg n_i\)), the electron concentration \(n\) in the n-type semiconductor is approximately equal to the doping concentration, \(n \approx N_D = 5 \times 10^{22} \text{ m}^{-3}\). Now, we can calculate the hole concentration \(p\) using the mass action law: \[p = \frac{n_i^2}{n}\] \[p = \frac{(1.5 \times 10^{16} \text{ m}^{-3})^2}{5 \times 10^{22} \text{ m}^{-3}}\] \[p = \frac{2.25 \times 10^{32} \text{ m}^{-6}}{5 \times 10^{22} \text{ m}^{-3}}\] \[p = 0.45 \times 10^{10} \text{ m}^{-3}\] \[p = 4.5 \times 10^9 \text{ m}^{-3}\] This calculation demonstrates that the concentration of holes, the minority charge carriers, is significantly reduced compared to the intrinsic carrier concentration due to the high concentration of majority electrons introduced by doping. This reduction in minority carrier concentration is a fundamental consequence of doping and is crucial for understanding the electrical characteristics of semiconductor devices, such as diodes and transistors, which are core components in many fields of engineering studied at institutions like the National Institute of Technology Raipur. The ability to predict and control these carrier concentrations is vital for designing and optimizing electronic circuits and systems.

The question probes the understanding of the fundamental principles of sustainable urban development and resource management, a key area of focus for engineering and planning programs at institutions like the National Institute of Technology Raipur. The scenario describes a city grappling with increasing water scarcity due to population growth and climate change, necessitating a shift towards more resilient water management strategies. The core of the problem lies in identifying the most effective approach to mitigate this crisis. Option A, focusing on integrated water resource management (IWRM) that incorporates rainwater harvesting, wastewater recycling, and efficient distribution networks, directly addresses the multifaceted nature of water scarcity. IWRM emphasizes a holistic approach, viewing water as a finite resource that requires careful planning and management across all sectors. Rainwater harvesting captures a natural, albeit variable, source. Wastewater recycling, when treated to appropriate standards, can supplement supply for non-potable uses or even potable uses after advanced treatment, reducing reliance on traditional sources. Efficient distribution networks minimize losses due to leakage, which can be substantial in aging infrastructure. This comprehensive strategy aligns with the principles of sustainability and resilience, crucial for long-term urban planning. Option B, while beneficial, is a partial solution. Increasing the capacity of conventional water treatment plants primarily addresses the purification of existing sources but does not inherently expand the available water supply or reduce demand. It’s a reactive measure rather than a proactive, systemic one. Option C, focusing solely on public awareness campaigns for water conservation, is important but often insufficient on its own to overcome deep-seated behavioral patterns and infrastructural inefficiencies. While behavioral change is a component of IWRM, it’s not the sole or most impactful solution when faced with severe scarcity and infrastructure issues. Option D, while technologically advanced, represents a high-cost, energy-intensive solution that may not be universally applicable or sustainable in the long term, especially for a developing urban center. Desalination is typically employed in coastal regions with limited freshwater access and significant energy resources, and its environmental impact (brine disposal) needs careful consideration. Therefore, the integrated approach of IWRM, encompassing multiple strategies, offers the most robust and sustainable solution to the described water scarcity challenge, reflecting the forward-thinking approach expected in advanced technical education at NIT Raipur.

The question probes the understanding of the fundamental principles of **digital signal processing (DSP)**, specifically concerning the properties of the **Discrete Fourier Transform (DFT)** and its implications for signal analysis. The scenario involves analyzing a finite-duration discrete-time signal \(x[n]\) of length \(N\). The DFT of this signal is denoted as \(X[k]\), where \(k = 0, 1, \dots, N-1\). The core concept being tested is the **periodicity of the DFT**. The DFT of a finite-length sequence of length \(N\) is inherently periodic with period \(N\). This means that \(X[k] = X[k+N]\) for all integer values of \(k\). This property arises from the definition of the DFT, which involves complex exponentials \(e^{-j \frac{2\pi}{N} nk}\). When \(k\) is replaced by \(k+N\), the term \(e^{-j \frac{2\pi}{N} n(k+N)}\) becomes \(e^{-j \frac{2\pi}{N} nk} e^{-j \frac{2\pi}{N} nN} = e^{-j \frac{2\pi}{N} nk} e^{-j 2\pi n}\). Since \(n\) is an integer, \(e^{-j 2\pi n} = 1\), thus preserving the original value of \(X[k]\). Therefore, \(X[N] = X[0]\), \(X[N+1] = X[1]\), and so on. This periodicity is crucial for understanding how the DFT represents the frequency content of a signal and how it is used in applications like spectral analysis and filtering. The National Institute of Technology Raipur’s curriculum in Electrical Engineering and Computer Science emphasizes these foundational DSP concepts for students to effectively analyze and manipulate signals. Understanding this periodicity is a prerequisite for grasping more advanced topics such as the Fast Fourier Transform (FFT) and its efficient implementation.

The question probes the understanding of fundamental principles in material science and structural integrity, particularly relevant to engineering disciplines at NIT Raipur. The scenario involves a beam subjected to a bending moment, and the core concept tested is the relationship between stress, strain, and material properties in a non-uniform stress distribution. In a beam under bending, the neutral axis experiences zero strain and zero stress. Points above the neutral axis are in compression, and points below are in tension, with the stress magnitude increasing linearly with the distance from the neutral axis. The maximum tensile stress occurs at the extreme fiber furthest from the neutral axis on the tension side, and similarly, maximum compressive stress occurs at the extreme fiber on the compression side. The question asks about the state of stress at a point *within* the beam’s cross-section, specifically at a distance \(y\) from the neutral axis. The bending stress (\(\sigma\)) in a beam is given by the flexure formula: \(\sigma = \frac{My}{I}\), where \(M\) is the bending moment, \(y\) is the distance from the neutral axis, and \(I\) is the moment of inertia of the cross-section about the neutral axis. The key here is that the stress is not uniform across the cross-section. It varies linearly from zero at the neutral axis to a maximum at the outer fibers. Therefore, at a distance \(y\) from the neutral axis, the stress will be non-zero and will depend on the magnitude of the bending moment, the geometry of the cross-section (through \(I\)), and the specific location \(y\). The stress will be either tensile or compressive depending on whether the point is below or above the neutral axis, respectively, and its magnitude will be proportional to \(y\). The correct answer must reflect this linear variation and the presence of stress, not zero stress (unless \(y=0\)) or a uniform stress. The concept of shear stress also exists in beams, but the question specifically asks about the state of stress related to bending, implying normal stress. The stress is directly proportional to the distance from the neutral axis.

The question probes the understanding of the fundamental principles of sustainable development, a core tenet in engineering and societal planning, which is highly relevant to the interdisciplinary approach at the National Institute of Technology Raipur. The scenario presented involves a proposed industrial expansion near a vital river ecosystem. To assess the sustainability of this expansion, one must consider the three pillars of sustainable development: economic viability, social equity, and environmental protection. Economic viability would necessitate that the expansion generates sufficient revenue and employment to justify the investment and contribute to regional economic growth. Social equity requires that the benefits and burdens of the expansion are distributed fairly among the local population, considering factors like community engagement, displacement, and access to resources. Environmental protection demands that the expansion minimizes its ecological footprint, particularly concerning the river ecosystem, by controlling pollution, conserving biodiversity, and ensuring responsible resource management. The most comprehensive approach to evaluating the sustainability of such a project, especially in the context of a prestigious institution like NIT Raipur which emphasizes responsible innovation, involves a holistic assessment that integrates all three pillars. This means not just focusing on economic gains or environmental regulations in isolation, but understanding how they interrelate and impact the long-term well-being of the community and the environment. Therefore, a thorough Environmental Impact Assessment (EIA) coupled with a Social Impact Assessment (SIA) and a robust economic feasibility study, all integrated into a comprehensive sustainability framework, would provide the most accurate and responsible evaluation. This approach aligns with the ethical considerations and forward-thinking research fostered at NIT Raipur, preparing students to tackle complex real-world challenges with a balanced perspective.

The question probes the understanding of the fundamental principles governing the behavior of semiconductor devices under varying environmental conditions, specifically focusing on the impact of temperature on carrier concentration and conductivity. For an intrinsic semiconductor, the number of electrons in the conduction band (\(n_i\)) is equal to the number of holes in the valence band (\(p_i\)), and this concentration is highly temperature-dependent. The relationship is generally exponential, approximated by \(n_i \approx AT^{3/2} e^{-E_g/(2kT)}\), where \(A\) is a material-dependent constant, \(T\) is the absolute temperature, \(E_g\) is the band gap energy, and \(k\) is the Boltzmann constant. As temperature increases, the thermal energy available to electrons also increases, leading to more electrons being excited from the valence band to the conduction band. This results in a significant increase in both the electron concentration (\(n\)) and hole concentration (\(p\)) in an intrinsic semiconductor. Consequently, the conductivity (\(\sigma\)), which is proportional to the product of carrier concentration and mobility (\(\sigma = q(n\mu_n + p\mu_p)\)), also increases. In intrinsic semiconductors, \(n = p = n_i\), so \(\sigma = qn_i(\mu_n + \mu_p)\). While carrier mobility (\(\mu\)) generally decreases with increasing temperature due to increased lattice scattering (typically proportional to \(T^{-3/2}\) for acoustic phonon scattering), the exponential increase in carrier concentration dominates the conductivity change. Therefore, the conductivity of an intrinsic semiconductor increases significantly with temperature. For extrinsic semiconductors (doped semiconductors), the behavior is more complex. At low temperatures, ionization of dopant atoms is the primary factor. As temperature increases, dopant ionization saturates, and the carrier concentration becomes relatively constant (saturation region). At very high temperatures, intrinsic behavior starts to dominate, and the carrier concentration increases exponentially with temperature, similar to intrinsic semiconductors. However, the question implicitly refers to the general behavior of semiconductors, and the most pronounced and universally understood effect of increasing temperature on conductivity is the significant rise in intrinsic carrier concentration, which is a fundamental concept tested in semiconductor physics, relevant to materials science and electrical engineering programs at NIT Raipur. The question tests the understanding of the dominant mechanism of conductivity change with temperature in semiconductors, emphasizing the exponential increase in intrinsic carrier generation.