Past Questions

Thevenin's Theorem is a fundamental principle in network analysis used to simplify a complex linear circuit into a simple equivalent circuit. According to the theorem, any combination of voltage sources, current sources, and resistors with two terminals can be replaced by a single voltage source (called the Thevenin Voltage, Vth) connected in series with a single resistor (called the Thevenin Resistance, Rth). Breakdown of options: - Option (a) is correct: This is the definition of a Thevenin equivalent circuit. - Option (b) is incorrect: A current source in parallel with a resistor describes Norton's Theorem, not Thevenin's. - Option (c) is incorrect: Thevenin's theorem deals with resistive networks (or impedance in AC), but the standard definition for basic network theory refers to a resistor, not a capacitor. - Option (d) is incorrect: A resistor only would imply the network has no active sources, which is not the general case for Thevenin's theorem.

Step-by-step solution: 1. **Identify the given parameters:** - Input Voltage ($V_{in}$) = $12\text{ V}$ - Pulse width per unit voltage ($\text{width/V}$) = $5\text{ μs/V}$ - Clock Frequency ($f$) = $20\text{ MHz} = 20 \times 10^6\text{ Hz}$ 2. **Calculate the total pulse width ($T$):** The voltage-controlled generator produces a pulse width proportional to the input voltage. $T = V_{in} \times (\text{width/V})$ $T = 12\text{ V} \times 5\text{ μs/V} = 60\text{ μs} = 60 \times 10^{-6}\text{ s}$ 3. **Calculate the pulse count ($N$):** The pulse count is the number of clock cycles that occur during the duration of the generated pulse width. $N = T \times f$ $N = (60 \times 10^{-6}\text{ s}) \times (20 \times 10^6\text{ Hz})$ $N = 60 \times 20 = 1200$ Conclusion: The pulse count corresponding to an input signal of $12\text{ V}$ is $1200$. Therefore, option (b) is correct.

A two-port network is reciprocal if the ratio of the response at one port to the excitation at the other port remains the same when the positions of the excitation and response are interchanged. In terms of network parameters, the conditions for reciprocity are: 1. Z-parameters: Z12 = Z21 2. Y-parameters: Y12 = Y21 3. ABCD-parameters: AD - BC = 1 4. h-parameters: h12 = -h21 (which is equivalent to h21 = -h12) Evaluating the options: - Option (a) Z11 = Z22: This is the condition for symmetry, not reciprocity. - Option (b) Y21 = Y22: This is not a standard condition for reciprocity or symmetry. - Option (c) h21 = -h12: This is the correct condition for a reciprocal network in terms of hybrid parameters. - Option (d) AD - BC = 0: For a reciprocal network, AD - BC must equal 1. A value of 0 would imply a non-reciprocal or singular network. Therefore, option (c) is the correct condition.

The Superposition Theorem states that in a linear bilateral network containing multiple independent sources, the current through (or voltage across) any element is the algebraic sum of the currents (or voltages) produced by each source acting alone, while all other independent sources are replaced by their internal impedances. 1. **Linearity**: The theorem relies on the principle of linearity. Voltage (V) and Current (I) are linear responses in a linear circuit. Therefore, options (a) and (b) are valid applications of the theorem. 2. **Non-linearity of Power**: Power (P) is calculated as P = I²R or P = V²/R or P = VI. Because power depends on the square of the current or voltage, it is a non-linear (quadratic) function. 3. **Mathematical Proof**: If two sources produce currents I1 and I2 respectively, the total current is I = I1 + I2. The total power is P_total = (I1 + I2)²R = (I1² + I2² + 2I1I2)R. However, the sum of individual powers would be P1 + P2 = I1²R + I2²R. Since (I1 + I2)²R ≠ I1²R + I2²R (unless one current is zero), the superposition theorem cannot be applied to power.

The intrinsic impedance of a medium (denoted as η) is the ratio of the electric field (E) to the magnetic field (H) for a plane wave. For free space (vacuum), it is calculated using the formula: η₀ = √(μ₀ / ε₀), where μ₀ is the permeability of free space (4π × 10⁻⁷ H/m) and ε₀ is the permittivity of free space (approximately 8.854 × 10⁻¹² F/m). Plugging in these values: η₀ = √( (4π × 10⁻⁷) / (8.854 × 10⁻¹²) ) ≈ 376.73 Ω, which is commonly rounded to 377 Ω. Options a, c, and d are incorrect because they provide values in Volts (V), which is a unit of electric potential, not impedance (which is measured in Ohms Ω).

A covalent bond is a type of chemical bond that occurs when two atoms share one or more pairs of valence electrons to achieve a stable electron configuration (usually an octet). This typically occurs between two non-metal atoms with similar electronegativities. Analysis of options: - Option (a) 'Transfer of electrons' describes an **Ionic Bond**, where one atom gives up an electron to another, resulting in electrostatic attraction between a cation and an anion. - Option (b) 'Sharing of electrons' is the defining characteristic of a **Covalent Bond**. - Option (c) 'Sharing of neutrons' is scientifically incorrect; neutrons are located in the nucleus and are not involved in chemical bonding. - Option (d) is incorrect because only option (b) is accurate.

Step-by-step reasoning: 1. The resonant frequency (f) of a parallel LC resonant circuit is given by the formula: f = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. 2. From this formula, we can see that the resonant frequency is inversely proportional to the square root of the capacitance (f ∝ 1/√C). 3. To tune the circuit to a lower frequency (decrease f), the denominator of the fraction must increase. 4. Therefore, the value of the capacitance (C) must be increased to lower the resonant frequency. Analysis of options: - Option (a) 'Be increased': Correct, as increasing C decreases f. - Option (b) 'Be decreased': Incorrect, as decreasing C would increase the resonant frequency. - Option (c) 'Be zero': Incorrect, if C = 0, the circuit is no longer a resonant circuit and the frequency would theoretically tend toward infinity. - Option (d) 'Remain the same': Incorrect, as changing the frequency requires a change in the component values.

Superconductivity is a quantum mechanical phenomenon where a material exhibits zero electrical resistance and the expulsion of magnetic fields (Meissner effect). However, this state is fragile and can be destroyed by three primary factors: 1. Temperature: If the temperature rises above the Critical Temperature (Tc), the material returns to a normal conducting state. 2. Magnetic Field: If an external magnetic field exceeds the Critical Magnetic Field (Hc), the superconducting state is destroyed. 3. Current Density: If the current flowing through the superconductor exceeds the Critical Current Density (Jc), it reverts to a normal state. Analyzing the options: - Option (a): Adding impurities can affect the critical temperature or the type of superconductor (Type I vs Type II), but it does not inherently 'destroy' the phenomenon of superconductivity across the board. - Option (b): Reducing temperatures actually helps a material enter the superconducting state; it does not destroy it. - Option (c): Application of a magnetic field (above the critical threshold) directly destroys the superconducting state by penetrating the material and breaking the Cooper pairs. Therefore, this is the correct answer.

Maxwell's equations are a set of four fundamental partial differential equations that form the foundation of classical electromagnetism. They are: 1. Gauss's Law for Electricity: Relates electric fields to charge densities. 2. Gauss's Law for Magnetism: States that there are no magnetic monopoles. 3. Faraday's Law of Induction: Describes how a changing magnetic field creates an electric field. 4. Ampere's Law (with Maxwell's addition of displacement current): Describes how electric currents and changing electric fields create magnetic fields. Comparing this to the options: - Option b (Gauss's Law), Option c (Faraday's Law), and Option d (Ampere's Law) are all integral components of Maxwell's equations. - Option a (Planck's Law) describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium. It is a fundamental law of Quantum Mechanics, not classical electromagnetism, and is not part of Maxwell's equations.

To determine the poorest conductor of electricity among the given options, we must compare their electrical conductivities. 1. **Silver (c):** Silver is the most conductive element known to man. It has the highest electrical conductivity of all metals. 2. **Aluminium (d):** Aluminium is an excellent conductor, though less so than silver or copper. It is widely used in power transmission lines due to its conductivity and light weight. 3. **Steel (b):** Steel is an alloy primarily composed of iron. While it is a conductor, its conductivity is significantly lower than that of pure metals like silver or aluminium due to the presence of alloying elements (like carbon) which scatter electrons. 4. **Carbon (a):** Carbon exists in several allotropes. While graphite is a conductor, carbon in general (especially in forms like diamond or amorphous carbon) is a poor conductor compared to metals. Even graphite's conductivity is orders of magnitude lower than that of silver, aluminium, or steel. Comparing the four, carbon has the highest electrical resistivity (and thus the lowest conductivity) among the listed materials.

In a lossless dielectric medium, the conductivity (σ) is zero. According to Maxwell's equations for a uniform plane wave, the relationship between the Electric field (E) and the Magnetic field (H) is governed by the intrinsic impedance of the medium (η). For a lossless medium, the intrinsic impedance η = sqrt(μ/ε) is a purely real quantity. The phase difference between the E and H fields is given by the phase angle of the intrinsic impedance. Since η is real, the phase angle is 0°, meaning the electric and magnetic fields reach their maximum and minimum values at the same time and position. Therefore, they are 'in phase'. In contrast, in a lossy medium (where σ > 0), the intrinsic impedance becomes complex, causing the magnetic field to lag behind the electric field.

1. **Ideal Current Source**: An ideal current source is a theoretical circuit element that provides a constant current regardless of the voltage across its terminals. Its internal resistance is considered to be infinite (open circuit). 2. **Practical Current Source**: In reality, no source is ideal. A practical current source has a finite internal resistance. As the load resistance changes, some of the current is diverted through this internal resistance, meaning the current delivered to the load decreases as the load resistance increases. 3. **Circuit Modeling**: To model this behavior, the internal resistance must be placed in a way that it can 'shunt' or divert current away from the load. If the resistance were in series (Option a), the total current flowing through the circuit would remain constant (determined by the ideal source), which is the definition of an ideal source. Therefore, to represent the loss of current delivery, the internal resistance must be placed in **parallel** with the ideal current source. 4. **Conclusion**: A practical current source is represented by an ideal current source in parallel with a resistance (often called the Norton equivalent circuit).

In an analog measuring instrument, three torques are typically involved to ensure accurate measurement: 1. Deflecting Torque (Td): This torque is produced by the electrical quantity being measured (current, voltage, etc.). Its primary function is to move the pointer away from the zero position to a point proportional to the magnitude of the quantity being measured. 2. Controlling Torque (Tc): This torque opposes the deflecting torque and ensures that the pointer stops at a unique position where Td = Tc. It also returns the pointer to zero when the measurement is removed. 3. Damping Torque: This torque is used to stop the oscillations of the pointer quickly so that a steady reading can be taken. Analyzing the options: - Option (a) is incorrect because holding the pointer at zero is a function of the controlling torque in the absence of a signal. - Option (b) is correct as it describes the fundamental purpose of the deflecting torque. - Option (c) is incorrect as this is the function of the damping torque. - Option (d) is incorrect as reducing error involves calibration and design precision, not the basic function of the deflecting torque.

A Permanent Magnet Moving Coil (PMMC) instrument operates on the principle of the Lorentz Force Law. When a current-carrying conductor (the moving coil) is placed within a magnetic field produced by a permanent magnet, it experiences a mechanical force. The deflecting torque (Td) is given by the formula Td = BINA, where B is the magnetic flux density, I is the current, N is the number of turns, and A is the area of the coil. This force is a direct result of the electromagnetic interaction between the magnetic field of the permanent magnet and the magnetic field generated by the current flowing through the coil. Why other options are incorrect: - a. Eddy currents: These are circulating currents induced in conductors by changing magnetic fields, usually causing losses or used in specific types of meters (like induction meters), but not for deflection in PMMC. - c. Spring control: This provides the 'controlling torque' (Tc) to oppose the deflecting torque and return the pointer to zero; it does not produce the deflection. - d. Air friction: This is a form of 'damping torque' used to prevent the pointer from oscillating, not to produce the deflection.

The Reciprocity Theorem states that in any linear, bilateral network, the ratio of an excitation (voltage or current) at one point to the response (current or voltage) at another point remains the same if the positions of the excitation and response are interchanged. 1. **Linearity**: The theorem relies on the principle of superposition and the fact that the relationship between voltage and current is a linear function (V = IZ). If the network were non-linear (e.g., containing diodes or transistors), the proportionality would change with the magnitude of the signal, making the theorem invalid. 2. **Bilateralism**: A bilateral element is one whose properties are the same regardless of the direction of current flow (e.g., resistors, inductors, capacitors). If a network contains unilateral elements (like diodes), the current cannot flow in the reverse direction with the same characteristics, thus violating the reciprocity condition. Therefore: - Option (b) is incorrect because non-linear networks do not satisfy the proportionality required. - Option (c) is incorrect because unilateral networks change behavior based on direction. - Option (d) is incorrect because it combines both non-linearity and unilateralism.

The velocity of an electromagnetic (EM) wave in a medium is derived from Maxwell's equations. For a non-conducting, linear, isotropic medium, the wave equation for the electric field (E) or magnetic field (H) leads to the expression for the phase velocity (v). The formula is given by v = 1 / √(με), where μ is the magnetic permeability of the medium and ε is the electric permittivity of the medium. 1. In a vacuum, μ = μ₀ (permeability of free space) and ε = ε₀ (permittivity of free space), so the speed of light c = 1 / √(μ₀ε₀) ≈ 3 × 10⁸ m/s. 2. In any other medium, the velocity decreases as μ and ε increase. Analysis of options: - Option (a) v = 1/√(με) is the mathematically correct derivation from the wave equation. - Option (b) v = √(με) is dimensionally incorrect. - Option (c) v = μ/ε is dimensionally incorrect. - Option (d) v = ε/μ is dimensionally incorrect.

A Current Transformer (CT) is designed to step down high primary currents to a lower, measurable value in the secondary. In a CT, the primary is connected in series with the load. 1. **The Danger of Open-Circuiting**: If the secondary of a CT is open-circuited while the primary is energized, the primary current (which is essentially the full load current) becomes the magnetizing current. Since there is no secondary current to oppose the primary flux, the core becomes heavily saturated. This leads to an extremely high voltage buildup across the secondary terminals, which can cause insulation failure, damage the transformer, and pose a severe safety risk (electric shock) to personnel. 2. **The Solution**: To prevent this, the secondary must always be closed. When a measuring instrument (like an ammeter) is removed or not in use, the secondary terminals must be short-circuited. A short circuit in the secondary ensures that the secondary current opposes the primary current, keeping the net flux in the core low and safe. 3. **Analysis of other options**: - **Option a**: Open-circuiting a CT is dangerous and can lead to catastrophic failure. - **Option c**: A voltmeter measures voltage; connecting it would create a high-impedance path, effectively open-circuiting the CT. - **Option d**: While a wattmeter can be used, the requirement is not limited to wattmeters; any low-impedance load or a direct short is required for safety.

Step-by-step reasoning: 1. **Definition of Disinfection**: Disinfection is the process of killing pathogenic microorganisms in water to make it safe for human consumption. 2. **Evaluating Option (a) Chlorine**: Chlorine is the most widely used disinfectant globally due to its effectiveness, low cost, ease of application, and most importantly, its ability to provide 'residual disinfection.' This means chlorine remains in the water as it travels through the distribution pipes, preventing re-contamination. 3. **Evaluating Option (b) Ozone**: While ozone is a very powerful oxidant and disinfectant (more effective than chlorine against certain cysts), it is expensive to produce on-site and provides no residual protection once the water leaves the treatment plant. 4. **Evaluating Option (c) Ultraviolet (UV) rays**: UV radiation is a physical process that disrupts the DNA of microorganisms. Like ozone, it provides no residual effect and requires the water to be very clear (low turbidity) to be effective. 5. **Evaluating Option (d) RO (Reverse Osmosis)**: RO is a membrane filtration process used for desalination and removing dissolved solids. While it removes microbes physically, it is a filtration method, not a chemical disinfectant. Conclusion: Because of its cost-effectiveness and residual protection, chlorine is the most common disinfectant used in municipal water treatment.

The question asks for a device that measures fluid flow velocity (or discharge) based on pressure difference. 1. **Venturimeter (Correct):** A Venturimeter works on Bernoulli's principle. It consists of a converging section, a throat, and a diverging section. As fluid flows through the constricted throat, its velocity increases and its pressure decreases. By measuring the pressure difference between the inlet and the throat using a manometer, the flow velocity and discharge rate can be calculated. 2. **Manometer (Incorrect):** A manometer is a device used to measure the static pressure of a fluid or the pressure difference between two points, but it does not measure velocity on its own; it is often a component used *within* a Venturimeter. 3. **Hydrometer (Incorrect):** A hydrometer is used to measure the relative density (specific gravity) of a liquid based on the principle of buoyancy. 4. **Barometer (Incorrect):** A barometer is used to measure atmospheric pressure.

In groundwater hydrology and well hydraulics, the 'Specific Capacity' (often referred to in the context of a well's specific yield) is defined as the discharge (Q) obtained per unit drawdown (s). Mathematically, it is expressed as Specific Capacity = Q/s. - Option (a) 'Static water level' refers to the level of water in the well when no pumping is occurring. - Option (b) 'Maximum discharge per hour' is simply the peak flow rate, not a 'specific' property relative to the energy loss (drawdown). - Option (c) 'Yield per unit drawdown' correctly describes the efficiency of the well in delivering water for a given drop in the water table/piezometric surface. - Option (d) 'Water lost to surroundings' refers to seepage or leakage, not the yield of the well. Note: While 'Specific Yield' in a general aquifer context refers to the volume of water that drains by gravity, when applied specifically to the performance of a 'well' (as phrased in the question), it refers to the Specific Capacity.

Turbidity is a measure of the cloudiness or haziness of a fluid caused by individual particles (suspended solids). According to the World Health Organization (WHO) Guidelines for Drinking-water Quality, for an effective disinfection process and for aesthetic acceptability, turbidity should ideally be less than 1 NTU (Nephelometric Turbidity Unit). While WHO notes that turbidity levels up to 5 NTU may be acceptable in some contexts, the gold standard for treated drinking water to ensure the removal of pathogens (like Cryptosporidium and Giardia) and to ensure the efficiency of chlorination is 1 NTU. In many competitive engineering exams, 1 NTU is recognized as the permissible limit for treated drinking water to ensure safety and clarity.

The Isohyetal method is one of the primary techniques used in hydrology to calculate the average depth of precipitation over a specific catchment area. An 'isohyet' is a line connecting points of equal rainfall depth (similar to isotherms for temperature or contours for elevation). By mapping these lines and calculating the area between successive isohyets, the weighted average rainfall for the entire basin can be determined. This method is generally considered the most accurate because it accounts for the spatial distribution of rainfall and topographic influences. Why other options are incorrect: - Option (a): Evaporation is measured using evaporation pans or lysimeters. - Option (b): The runoff coefficient is typically determined using the Rational Method (Q=CiA) or based on land-use characteristics, not isohyetal maps. - Option (c): Groundwater levels are determined using observation wells or piezometers.

The selection of a hydraulic turbine depends primarily on the available head (the vertical distance the water falls) and the discharge (the volume of water flowing per unit time). 1. **Pelton Turbine (Impulse Turbine):** These are designed specifically for **high head** and **low discharge** conditions. They use high-velocity jets of water that strike buckets on the periphery of the wheel to transfer kinetic energy into mechanical energy. 2. **Francis Turbine (Reaction Turbine):** These are versatile and are typically used for **medium head** and **medium discharge** applications. 3. **Kaplan and Propeller Turbines (Reaction Turbines):** These are axial-flow turbines designed for **low head** and **high discharge** conditions. Kaplan turbines have adjustable blades to maintain efficiency over a range of flows, while propeller turbines have fixed blades. Since the question specifies 'high head and low discharge', the Pelton turbine is the most suitable choice.

Waterlogging occurs when the water table rises to a level where the root zone of crops becomes saturated, displacing air and hindering plant growth. To control waterlogging, the excess groundwater must be removed to lower the water table. 1. **Installing tube wells (Correct):** This is a method of 'vertical drainage'. By pumping out groundwater through tube wells, the water table is artificially lowered, which drains the saturated root zone and mitigates waterlogging. 2. **Increasing irrigation (Incorrect):** Adding more water to an already waterlogged field will exacerbate the problem, raising the water table further. 3. **Deforestation (Incorrect):** Trees help in reducing waterlogging through transpiration (biological pumping). Removing trees reduces the amount of water removed from the soil, potentially worsening the condition. 4. **Reducing crop rotation (Incorrect):** Crop rotation, specifically including deep-rooted crops, can help lower the water table. Reducing this practice does not provide a mechanism to remove excess water.

In irrigation engineering, 'Intensity of Irrigation' is defined as the percentage of the total cultivable area of a region that is irrigated annually. It is calculated as: (Area Irrigated / Total Cultivable Area) × 100. Analysis of other options: - Option (b) 'Depth of water supplied' refers to the 'Duty' or 'Delta' concepts (specifically Delta is the total depth of water required by a crop). - Option (c) 'Number of waterings given to a crop' refers to the 'Frequency of Irrigation'. - Option (d) 'Duration of irrigation season' refers to the 'Base Period' (the time between the first watering at sowing and the last watering before harvest).