Past Questions

Young's Modulus (E) is a measure of the stiffness of a solid material. It is defined as the ratio of tensile or compressive stress to the corresponding longitudinal strain. 1. Formula: E = Stress / Strain 2. Stress is defined as Force per unit Area. Its SI unit is Pascal (Pa), which is equivalent to Newton per square meter (N/m²) or Newton per square millimeter (N/mm²). 3. Strain is the ratio of change in length to original length (ΔL/L), making it a dimensionless quantity (it has no units). 4. Therefore, the unit of Young's Modulus is the same as the unit of stress. Evaluating the options: - (a) Newton: This is the unit of Force, not modulus. - (b) Newton/mm²: This is a standard unit of pressure/stress (1 N/mm² = 1 MPa), which is the correct unit for Young's Modulus. - (c) Newton/cm²: While technically a unit of stress, in the context of engineering and technical exams for Strength of Materials, N/mm² (MPa) is the standard convention used for material properties. - (d) Newton-cm²: This is a product of force and area, which does not represent stress or modulus.

CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique) are the two primary network analysis techniques used in project management. 1. CPM (Critical Path Method) is an 'activity-oriented' technique. In CPM, the focus is on the activity itself (the work being performed), and the arrows in the network diagram represent the activities. It is typically used for deterministic projects where activity durations are known with certainty (e.g., construction projects). 2. PERT (Program Evaluation and Review Technique) is an 'event-oriented' technique. In PERT, the focus is on milestones or events (the start or completion of an activity), and the nodes represent these events. It is typically used for probabilistic projects where durations are uncertain (e.g., R&D projects). Therefore, option (c) is correct. Option (b) describes PERT, and options (a) and (d) are not standard classifications for these network techniques.

In thick cylinder hoop stress varies hyperbolically through thickness, longitudinal stress remains uniform.

In structural engineering, members are classified based on the type of internal forces they are primarily designed to resist: 1. **Column**: A vertical structural member designed to support an axial compressive load. While 'strut' also carries compression, a column is the general term for a primary structural member subjected to axial compression. 2. **Beam**: A structural member primarily subjected to transverse loads, which causes bending and shear stresses, rather than axial compression. 3. **Strut**: A strut is also a member subjected to axial compression, but it is typically a shorter, secondary member used in bracing or trusses. In the context of general structural members subjected to axial compressive forces, 'column' is the standard technical term. Therefore, a structural member subjected to an axial compressive force is most accurately defined as a column.

A blast furnace is used for the smelting of iron ore to produce iron. In this process, iron oxides (ore) are reduced by carbon monoxide (derived from coke) at very high temperatures. The resulting molten metal is saturated with carbon (typically 3.5% to 4.5%) and other impurities like silicon and phosphorus. This crude, high-carbon iron is called 'pig iron' because it is cast into molds (called pigs and sows) as it cools. Why other options are incorrect: - Cast iron (a): While pig iron is a type of cast iron, 'cast iron' usually refers to pig iron that has been remelted and refined for specific casting purposes. - Wrought iron (c): This is the purest form of iron (very low carbon), produced by removing carbon from pig iron through processes like puddling. - White iron (d): This is a specific type of cast iron where all carbon is present as cementite, usually achieved through rapid cooling or specific alloying, not as the primary direct output of a blast furnace.

Step-by-step reasoning: 1. Elasticity refers to the ability of a material to return to its original shape after the removal of an applied load. 2. For most metals and ceramics, the relationship between stress and strain is linear (follows Hooke's Law) within the elastic limit. 3. Rubber is an elastomer. Elastomers consist of long-chain polymers that are cross-linked. When stretched, these chains uncoil, allowing for very large deformations. 4. The stress-strain curve for rubber is characterized by a non-linear relationship; it does not follow a straight line (Hooke's Law) even though the deformation is reversible (elastic). 5. Therefore, while rubber is elastic, its stress-strain behavior is non-linear. Option (a) is incorrect because rubber does not follow a linear relationship. Option (c) is incorrect because plastic behavior refers to permanent deformation, whereas rubber is primarily known for its elasticity.

1. Definition: A point of contra-flexure (or point of inflection) is a point in a structural element where the bending moment changes sign (from positive to negative or vice versa). At this point, the curvature of the beam changes, and the bending moment is zero. 2. Analysis of Options: - Cantilever Beams: In a standard cantilever beam loaded from the free end, the bending moment is typically negative (hogging) throughout the entire length. Since the sign does not change, there is no point of contra-flexure. - Simply Supported Beams: For a simply supported beam with downward loads, the bending moment is typically positive (sagging) throughout the span. Since the sign does not change, there is no point of contra-flexure. - Overhanging Beams: An overhanging beam is a beam that extends beyond its supports. In such beams, the portion between the supports typically experiences sagging (positive moment), while the overhanging portion experiences hogging (negative moment). Therefore, there must be a point between the support and the span where the bending moment is zero and changes sign. This is the point of contra-flexure. 3. Conclusion: Among the given options, only overhanging beams (and fixed beams or continuous beams, which are not listed) can exhibit a point of contra-flexure.

To determine which factor is independent of a sales forecast, we must analyze the relationship between demand forecasting and operational management: 1. **Sales Forecast** is an estimate of future demand. It serves as the primary input for planning how much to produce and what resources are needed. 2. **Inventory Control (Option b):** This is directly dependent on sales forecasts. To avoid stockouts or overstocking, a company must know the expected demand to determine safety stock levels and reorder points. 3. **Production Planning (Option c):** This is the process of deciding how to allocate resources to meet demand. It is fundamentally driven by the sales forecast (e.g., Aggregate Production Planning). 4. **Production Control (Option d):** This involves monitoring the actual production process to ensure it aligns with the plan. Since the plan itself is based on the sales forecast, production control is indirectly dependent on it to ensure the right quantity is produced at the right time. 5. **Productivity (Option a):** Productivity is a measure of efficiency (Output / Input). It describes how well a system converts resources into products regardless of the volume of demand. For example, if a machine produces 10 units per hour, its productivity is 10 units/hr whether the sales forecast is for 1,000 units or 10,000 units. While a company may strive to improve productivity to meet a high forecast, the *metric* of productivity itself is an internal efficiency measure and is independent of the external sales forecast.

To determine which alloy does not contain copper, let's analyze the composition of each option: 1. **Delta metal**: This is a high-strength alloy primarily composed of Copper (Cu), Silver (Ag), and Gold (Au). It contains copper. 2. **Monel metal**: This is a group of nickel-copper alloys. The most common form is approximately 67% Nickel (Ni) and 30% Copper (Cu). It contains copper. 3. **Nichrome**: This is an alloy consisting primarily of Nickel (Ni) and Chromium (Cr), often with small amounts of iron. It is widely used in heating elements due to its high electrical resistance and oxidation resistance. It does NOT contain copper. 4. **Silicon bronze**: As the name suggests, it is a type of bronze. Bronze is an alloy of Copper (Cu) and Tin (Sn), and silicon bronze specifically adds silicon to improve corrosion resistance and strength. It contains copper. Therefore, Nichrome is the only alloy among the choices that does not have copper as a constituent.

In arc welding (specifically Shielded Metal Arc Welding - SMAW), the electrode consists of a metal core wire coated with a chemical material known as 'flux'. 1. Why 'flux' is correct: The flux coating serves several critical purposes: it melts to create a gaseous shield (protecting the molten weld pool from atmospheric contamination like oxygen and nitrogen), it stabilizes the electric arc, and it forms a layer of slag on top of the weld to slow the cooling rate and prevent oxidation. 2. Why other options are incorrect: - 'Protective layer' is a general term, but the specific technical name for the coating material in welding is flux. - 'Binder' is a component within the flux coating that holds the flux chemicals to the core wire, but it is not the name of the coating itself. - 'Slag' is the byproduct (the solidified waste) that forms after the flux has reacted and cooled; it is the result of the coating, not the coating material itself.

To understand the range of speed, we must compare the Porter governor and the Proell governor. 1. **Porter Governor**: This is a modification of the Watt governor where a central dead weight is added to the sleeve. This increases the force acting downwards, allowing the governor to operate at higher speeds for the same lift of the sleeve. 2. **Proell Governor**: This is a further modification of the Porter governor where the balls are attached to the arms by extensions (extensions of the arms). This increases the effective length of the arms and changes the geometry of the mechanism. 3. **Comparison**: For a given lift of the sleeve, the Proell governor is designed to be more sensitive than the Porter governor. Sensitivity in a governor is defined as the ratio of the change in speed to the change in lift of the sleeve (dS/dh). A more sensitive governor requires a smaller change in speed to achieve the same lift of the sleeve. Therefore, for the same lift of the sleeve, the range of speed (the difference between the maximum and minimum speed) for a Proell governor is less than that of a Porter governor. This makes the Proell governor more efficient in maintaining a constant speed.

In the study of Strength of Materials, strain energy is the energy stored in a body due to deformation. 1. **Resilience**: This is the capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. It is the strain energy stored up to any point within the elastic limit. 2. **Proof Resilience**: This is the maximum energy that can be absorbed by a body per unit volume without permanent deformation. In other words, it is the strain energy stored when the material is stressed up to its elastic limit (or proportional limit). Since it is the 'maximum' energy stored within the elastic range, it is the correct answer. 3. **Toughness**: This is the total energy absorbed by a material up to the point of fracture. It includes both elastic and plastic deformation. While it is a 'maximum' energy, it refers to the energy until failure, not the stored energy typically associated with resilience. 4. **Impact Energy**: This refers to the energy absorbed by a material during a high-strain rate loading (like a Charpy or Izod test), which is a measure of toughness rather than a specific property of stored strain energy. Therefore, the maximum strain energy that can be stored (specifically within the elastic limit) is known as proof resilience.

In PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method), 'Slack' (or Float) is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks or the overall project completion date. 1. **Slack Calculation**: Slack is calculated as the difference between the Latest Start time (LS) and the Earliest Start time (ES), or the Latest Finish time (LF) and the Earliest Finish time (EF). Formula: Slack = LS - ES or LF - EF. 2. **Positive Slack**: When the slack is positive, it means the activity has a 'buffer'. The activity can be delayed by the amount of the slack without affecting the project deadline. If an activity is completed earlier than its latest allowable finish time, it indicates that the specific path is 'ahead of schedule' relative to the critical path. 3. **Zero Slack**: Activities with zero slack are on the 'Critical Path'. Any delay in these activities will directly delay the project completion. 4. **Negative Slack**: This occurs when the planned completion date is earlier than the calculated earliest completion date, meaning the project is 'beyond schedule' (behind schedule). Therefore, positive slack indicates that there is extra time available, meaning the project/activity is ahead of the critical schedule.

Creep in a belt drive is the relative motion between the belt and the pulley surface. It occurs because the belt is subjected to different tensions at the tight side and the slack side of the drive. As the belt moves from the tight side to the slack side, it undergoes a slight contraction; conversely, as it moves from the slack side to the tight side, it undergoes a slight extension. This continuous cycle of extension and contraction causes the belt to 'slip' slightly relative to the pulley, even if there is no gross slippage. - Option (a) and (b) are incorrect because while the material properties (elasticity) allow creep to happen, the 'cause' of the motion is the tension differential, not the material itself. - Option (c) is incorrect because simple expansion (like thermal expansion) does not describe the cyclic process of creep during operation.

The Moment of Inertia (I) of an area about any axis is governed by the Parallel Axis Theorem, which is expressed as: I = I_centroid + Ad², where 'I_centroid' is the moment of inertia about the axis passing through the centroid (central axis), 'A' is the area, and 'd' is the perpendicular distance between the centroidal axis and the parallel axis in question. Since A and d² are always non-negative, the value of I is minimized when d = 0. This occurs when the axis of rotation is the centroidal axis (central axis). Therefore, the moment of inertia is always least with respect to the central axis compared to any other parallel axis (such as the bottom-most axis or other horizontal/vertical axes not passing through the centroid).

In discrete mathematics, there is a fundamental one-to-one correspondence between the number of equivalence relations on a set and the number of partitions of that set. The number of ways to partition a set of n elements is given by the Bell number, denoted as B_n. For a set with n = 4 elements {1, 2, 3, 4}, we can calculate the Bell numbers using the Bell triangle or the recurrence relation B_{n+1} = Σ (k=0 to n) [C(n, k) * B_k], where B_0 = 1. Step-by-step calculation: 1. B_0 = 1 2. B_1 = C(0,0) * B_0 = 1 * 1 = 1 3. B_2 = C(1,0)*B_0 + C(1,1)*B_1 = (1*1) + (1*1) = 2 4. B_3 = C(2,0)*B_0 + C(2,1)*B_1 + C(2,2)*B_2 = (1*1) + (2*1) + (1*2) = 1 + 2 + 2 = 5 5. B_4 = C(3,0)*B_0 + C(3,1)*B_1 + C(3,2)*B_2 + C(3,3)*B_3 = (1*1) + (3*1) + (3*2) + (1*5) = 1 + 3 + 6 + 5 = 15. Alternatively, we can list the partition types for n=4: - {4}: 1 way (all in one block) - {3, 1}: 4 ways (choosing 1 element to be alone) - {2, 2}: 3 ways (pairing elements: {12,34}, {13,24}, {14,23}) - {2, 1, 1}: 6 ways (choosing 2 elements to be together) - {1, 1, 1, 1}: 1 way (all elements separate) Total = 1 + 4 + 3 + 6 + 1 = 15. Therefore, the number of equivalence relations is 15.

Thevenin's and Norton's theorems are used to simplify a linear, bilateral network into a single source and a single impedance. For these theorems to be applicable, the network must be 'self-contained'—meaning all sources and dependencies must exist within the network being simplified. 1. Options (a) and (b) are incorrect because Thevenin/Norton equivalents are specifically designed to handle independent voltage and current sources. 2. Option (c) is incorrect because dependent sources (controlled sources) located *within* the network can be handled using the test-source method (applying a voltage or current source at the terminals to find the equivalent resistance). 3. Option (d) is correct because if a source or a component within the network depends on a variable (voltage or current) located *outside* the network (in another network or the load), the network is no longer linear or independent. The equivalent circuit would change depending on what is connected to the terminals, violating the fundamental premise of Thevenin's theorem, which states that the equivalent circuit remains constant regardless of the load connected.

Even-zero count: an=2*an-1 (end 1/2) + (3^{n-1}-an-1) (end 0 makes odd even) = an-1+3^{n-1}.

Thevenin's and Norton's theorems are fundamental tools in network analysis used to simplify a linear circuit into a single source and a single impedance. 1. **Linearity Requirement**: The primary requirement for these theorems to apply is that the network must be 'linear'. A linear network is one where the components (resistors, capacitors, inductors) have constant values and the sources are independent or dependent linear sources. 2. **Independent Sources**: The theorems obviously apply to networks containing only independent voltage or current sources. 3. **Dependent Sources**: The theorems also apply to networks containing dependent (controlled) sources. The only difference is in the method of calculating the equivalent resistance (Thevenin resistance, Rth). While for independent sources we can simply 'kill' the sources (short voltage, open current), for dependent sources, we must either: - Use the Open Circuit Voltage (Voc) and Short Circuit Current (Isc) method: Rth = Voc / Isc. - Apply an external test voltage or current source at the terminals and calculate the resulting response. Therefore, the presence of dependent sources does not invalidate the theorem; it only changes the procedure for finding the equivalent resistance. Options a, b, and c are too restrictive, as they suggest the theorems only work under specific subsets of linear components.

In network theory, 'steady-state condition' for a DC circuit refers to the state reached after a long period of time where the voltages and currents have stabilized (transients have decayed). 1. Inductor (L): The voltage across an inductor is given by V = L(di/dt). In DC steady state, the current (i) becomes constant, meaning the rate of change of current (di/dt) is zero. Therefore, V = 0. A component with zero voltage drop across it while carrying current behaves as a short circuit. 2. Capacitor (C): The current through a capacitor is given by I = C(dv/dt). In DC steady state, the voltage (v) across the capacitor becomes constant, meaning the rate of change of voltage (dv/dt) is zero. Therefore, I = 0. A component that allows no current to flow through it behaves as an open circuit. Thus, during DC steady-state, the inductor behaves as a short circuit and the capacitor behaves as an open circuit.

In mathematics and digital electronics, when performing a subtraction operation (regardless of whether it is done using direct subtraction or 2's complement addition), the terms are defined as follows: 1. Minuend: The number from which another number is subtracted. 2. Subtrahend: The number that is being subtracted from the minuend. 3. Difference: The result of the subtraction. In the context of 2's complement subtraction, the operation 'A - B' is performed by adding the 2's complement of B to A. In this scenario, 'A' is the minuend and 'B' is the subtrahend. Therefore, the number which is subtracted from another number is known as the subtrahend. Analysis of other options: - Option (a) Carry: This refers to the bit that 'carries over' to the next higher column during addition. - Option (c) Minuend: This is the number that is being subtracted FROM, not the number being subtracted.

In filter design, a 'prototype filter' (usually a low-pass prototype) is a normalized filter where the component values are chosen such that the characteristic impedance (Z0) is normalized to a specific value, typically 1 Ohm or a standard resistive value (like 50 or 600 Ohms). The purpose of the prototype is to define the mathematical transfer function and attenuation characteristics. For the filter to effectively pass signals in the passband without introducing phase distortion or reflection, the image impedance (Z0) must be purely resistive. If Z0 were inductive or capacitive in the passband, it would cause impedance mismatching and frequency-dependent phase shifts, which contradicts the goal of a normalized prototype filter. Therefore, Z0 is designed to be purely resistive in the passband to ensure maximum power transfer and a flat response.

A flip-flop is a fundamental building block of digital electronics used to store one bit of data. It is called a 'bistable multivibrator' because it has two stable states: logic '0' (low) and logic '1' (high). It will remain in one of these states indefinitely until an external trigger (like a clock pulse or input change) forces it to switch to the other state. Analysis of options: - (a) Stable: This is too general; while it is stable in its states, the term 'stable' does not describe the circuit's behavior of switching between two states. - (b) Bistable: Correct. 'Bi' means two, and 'stable' refers to the states. A flip-flop is a bistable device. - (c) Tri stable: This is not a standard term in digital electronics. Multivibrators are typically categorized as astable (no stable states), monostable (one stable state), or bistable (two stable states). - (d) Not stable: Incorrect, as flip-flops are specifically designed to hold a state reliably.

To determine which statement is true, we must check if the given sets satisfy the four group axioms under the specified operation: Closure, Associativity, Identity, and Inverse. 1. **Analysis of Option (a):** The set of all rational negative numbers under multiplication. - **Closure:** If we multiply two negative rational numbers (e.g., -1 * -1), the result is a positive rational number (1). Since the result is not a negative number, the set is not closed under multiplication. Therefore, it cannot be a group. 2. **Analysis of Option (b):** The set of all non-singular matrices (matrices with a non-zero determinant) under multiplication. - **Closure:** The product of two non-singular matrices $A$ and $B$ is also non-singular because $\det(AB) = \det(A)\det(B)$. Since $\det(A) \neq 0$ and $\det(B) \neq 0$, their product is non-zero. - **Associativity:** Matrix multiplication is inherently associative. - **Identity:** The identity matrix $I$ is non-singular ($\det(I) = 1$) and acts as the identity element. - **Inverse:** By definition, a non-singular matrix has a multiplicative inverse $A^{-1}$ such that $AA^{-1} = I$, and $A^{-1}$ is also non-singular. Since all four axioms are satisfied, this is a group (known as the General Linear Group $GL(n, F)$). 3. **Analysis of Option (c):** The set of all matrices under multiplication. - **Inverse:** For a set to be a group, every element must have an inverse. Singular matrices (matrices with determinant 0) do not have multiplicative inverses. Therefore, the set of all matrices cannot form a group under multiplication. 4. **Analysis of Option (d):** Since (c) is false, (d) must also be false. Conclusion: Only option (b) is mathematically correct.

y=ABCD+ABC'=ABC(D+D')=ABC*1=ABC.