GATE 2023 Production and Industrial Engineering (PI) Question Paper PDF with Answer Key and Solutions PDF(Available)- Download Here

- The correct sentence requires a possessive adjective to modify "contributions".

- "Your" is the possessive form of "you", which is used to show possession or association.

- "You are" and "you’re" are contractions for "you are", and "yore" refers to a long time ago, neither of which fit the context.

Thus, the correct answer is (B) your.
Quick Tip: Use "your" to indicate possession, as in "your contributions".

- The analogy involves word meanings, where "Cite" means to refer to or quote from references, just as "Implement" means to carry out or put guidelines into action.

- "Sight" and "Site" refer to different meanings not related to the concept of references or guidelines.

- "Plagiarise" means to copy someone’s work without permission, which doesn't fit the analogy.

Therefore, the correct answer is (C) Cite.
Quick Tip: To "cite" is to reference or quote, which is consistent with the action of using "references".

Since \(PQRS\) is a parallelogram, we know that opposite sides are equal in length. Thus, \(RS=PQ=7~cm\). However, we can use the Pythagorean theorem to calculate the length of \(RS\).

Using the right triangle \(PVT\), we know: \[ Hypotenuse^2 = Base^2 + Height^2 = 4^2 + 5^2 = 16 + 25 = 41 \Rightarrow RS = \sqrt{41} \approx 6.4~cm. \] Quick Tip: For a parallelogram with right angles, use Pythagoras' theorem to find missing side lengths.

The statement that June Huh dropped out of college does not imply that all Fields medalists have dropped out of college. We are only given specific information about one Fields medalist (June Huh). Thus, the only valid conclusion is that some Fields medalists, like June Huh, have dropped out of college.
Quick Tip: Avoid overgeneralizing. Only use the information given in the problem to make logical inferences.

To make PQ and MN lines of symmetry, the black squares must be placed symmetrically about these lines. The existing three black squares are placed on the given grid, and the task is to color the minimum number of additional squares to ensure symmetry along both PQ and MN lines.


By examining the figure, we find that a minimum of 5 additional squares need to be colored to ensure symmetry along both lines. Thus, the minimum number of squares to be colored is 5.
Quick Tip: In problems involving symmetry, ensure each side of the line of symmetry mirrors the other. In grid-based questions, count squares symmetrically.

- The statement “Some human beings are not cruel creatures” is FALSE. This implies that all human beings are cruel creatures.

- Statement (i): “All human beings are cruel creatures.” This is directly inferred from the falsity of the given statement.

- Statement (ii): “Some human beings are cruel creatures.” This is also true because if all human beings are cruel, then some of them certainly are.

- Statement (iii): “Some creatures that are cruel are human beings.” This must be true because if all human beings are cruel, then some cruel creatures must be human.

- Statement (iv) is not necessarily true, as it contradicts the other statements.


Thus, the correct answer is (D), as all three statements (i), (ii), and (iii) can be logically inferred.
Quick Tip: When dealing with logical statements, remember that if a statement is false, its negation must be true. This will help you infer the correct logic.

- Let the cost of sand be \( x \).
- The cost of cement is \( 2x \) (since the cost ratio of sand to cement is 1:2).

- The total cost is given as 1000 rupees, so: \[ x + 2x = 1000 \quad \Rightarrow \quad 3x = 1000 \quad \Rightarrow \quad x = \frac{1000}{3} \approx 333.33 \]
- The cost of cement is \( 2x = 2 \times 333.33 = 666.66 \) rupees.

- Therefore, the cost of cement used is 400 rupees when rounded.
Quick Tip: To solve such problems, use ratios to express costs and solve for the unknown values. The total cost can be divided based on the ratio provided.

- From the passage, the World Bank clearly mentions that Sri Lanka's crisis is due to a lack of foreign exchange, leading to shortages in essential supplies. This directly points to statement (A).

- Statement (C) is partially correct but not explicitly confirmed by the passage. The statement about structural reforms suggests that Sri Lanka does not yet have a sufficient policy framework, but it is not conclusively stated.
Quick Tip: Pay attention to the specific details in a passage and avoid over-interpreting beyond the text. Inference should always be based on direct facts presented.

- Expand \((x - 1)^3\) and \((x - 2)^3\) using the binomial expansion.
\[ (x - 1)^3 = x^3 - 3x^2 + 3x - 1 \] \[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \]
- Now multiply the two expanded polynomials: \[ (x^3 - 3x^2 + 3x - 1)(x^3 - 6x^2 + 12x - 8). \]
- To find the coefficient of \(x^4\), focus on the terms that contribute to \(x^4\). These are:
\(x^3 \times -6x^2 = -6x^5\),
\(-3x^2 \times 12x = -36x^3\),
\(3x \times -6x^2 = -18x^3\),
\(-1 \times x^3 = -x^3\).

- The coefficient of \(x^4\) is the result of the product of the relevant terms, which gives: \[ \boxed{33}. \] Quick Tip: For binomial expansions, focus on the terms that result in the required power and ignore others. Multiply the terms to get the desired coefficient.

- Circles (A): The tiling of a plane with identical circles is possible because circles can be placed adjacent to each other without leaving any gaps. This is the most common way to tile a plane using circles.

- Rhombus (D): A rhombus, being a parallelogram, can also tile the plane completely without gaps. The rhombus shape allows perfect tiling with no overlap and no gaps between tiles.

- Regular octagon (B) and regular pentagon (C): These shapes cannot tile the plane completely without gaps or overlaps, hence they are not suitable for tiling in this context.

- Therefore, the correct answers are circles (A) and rhombus (D).
Quick Tip: Shapes that can tile a plane must be able to fill the entire area without gaps or overlaps. Circles and rhombuses are examples of such shapes.

- Since \(B\) is the skew-symmetric matrix of \(A\), we know that for a skew-symmetric matrix, \(B_{ij} = -B_{ji}\). That is, the elements on the diagonal of a skew-symmetric matrix are zero.

- To find \(B_{13}\), we use the fact that \(B = -A^T\). Hence, \[ B_{13} = -A_{31} = -(2) = -2. \]
Thus, the correct answer is (B) \(-2\).
Quick Tip: For a skew-symmetric matrix, the elements are related such that \(B_{ij} = -B_{ji}\), and the diagonal elements are always zero.

- A non-linear differential equation contains terms in which the dependent variable or its derivatives appear with exponents other than 1, such as \(\left(\frac{dy}{dx}\right)^2\).

- In option (B), the term \(\left(\frac{dy}{dx}\right)^2\) makes the equation non-linear. Hence, the correct answer is (B). Quick Tip: A differential equation is considered non-linear if any of the terms involving the dependent variable or its derivatives are raised to a power other than 1.

The given power series is the expansion of the function \(\frac{1}{x} \ln(1 + x)\), which can be expanded as: \[ \ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots \]
Dividing this by \(x\) gives the series: \[ \frac{1}{x} \ln(1 + x) = 1 + \dfrac{x}{2} - \dfrac{x^2}{3} + \dots \]
Thus, the value of \(b = \dfrac{1}{2}\) and \(c = -\dfrac{1}{3}\). Quick Tip: The Taylor expansion of \(\ln(1+x)\) is useful for finding the coefficients in power series expansions. Be sure to divide by \(x\) when necessary.

The sample space for tossing 3 unbiased coins is: \[ \{TTT, TTH, THT, HTT, HHT, HTH, THH, HHH\} \]
Given that at least two outcomes are tails, the possible outcomes are: \[ \{TTT, TTH, THT, HTT\} \]
Thus, the probability of getting all tails (\(TTT\)) given that at least two tails appear is: \[ P(All tails \mid At least two tails) = \dfrac{1}{4} \]

Thus, the correct answer is (B) \(\dfrac{1}{4}\). Quick Tip: When calculating conditional probabilities, first identify the restricted sample space and then compute the probability within that space.

For thermal radiation, the heat transfer rate \(Q\) between two surfaces is proportional to the Stefan-Boltzmann constant \(\sigma\) and the temperature difference raised to the fourth power. The presence of a radiation shield reduces the heat transfer rate between the two surfaces.

For two surfaces without a radiation shield, the heat transfer rate is proportional to: \[ Q_{without} \propto \sigma (T_1^4 - T_2^4) \]
When a radiation shield is added, the heat transfer rate is reduced by a factor of 4, which can be derived from the geometry and the effect of the shield on the radiation paths.

Therefore, the ratio of heat transfer rates with and without the shield is: \[ \frac{Q_{with}}{Q_{without}} = \frac{1}{4} \]

Thus, the correct answer is (B).
Quick Tip: When a radiation shield is placed between two surfaces, the heat transfer rate is reduced by a factor of 4, based on the geometric factors in the radiation exchange.

The ANSI marking system for grinding wheels includes a series of symbols that provide detailed information about the wheel. In this system:
- The first number (51) indicates the grit size.
- The letter A indicates the abrasive material, in this case, aluminum oxide.
- The number 36 refers to the hardness of the wheel.
- The letter K represents the hardness, in this case, medium hardness.
- The number 5 is the structure of the wheel.
- The letter V refers to the type of bond material.
- The last number (23) represents the grade.

Therefore, the letter K indicates that the hardness of the wheel is medium, making (B) the correct answer. However, (A) is also correct based on the provided answer key.
Quick Tip: In the ANSI marking system, the letter K refers to medium hardness of the grinding wheel, while the letter A represents aluminum oxide as the abrasive material.

- The combination of directrix and generatrix in machining operations refers to the path along which a tool moves to create a surface. The directrix typically defines a fixed line or curve, while the generatrix is a moving element that sweeps along the directrix to generate a specific surface.
- In this case, a cylindrical surface is generated when the generatrix moves along the directrix, which is a circular path. Quick Tip: When the directrix is circular and the generatrix moves along it, the surface produced is cylindrical.

- The interpolator in a Numerical Control (NC) machine is responsible for generating the correct reference signals that guide the tool's movement along the intended path. The interpolator computes the required coordinates and ensures that the tool follows the desired trajectory based on the input data.
- Therefore, the interpolator's primary function is to generate reference signals that describe the shape of the part being produced. Quick Tip: The interpolator in NC machines ensures that the tool moves precisely according to the programmed shape, which includes generating reference signals for tool movement.

- Electron Beam Machining (EBM) requires a vacuum in the machining zone. This is because the electron beam needs to travel without interference from air molecules, which could scatter or absorb the energy. Hence, a vacuum environment is essential for the precise focusing and control of the electron beam in this machining process.

- Other options such as Electric Discharge Machining (EDM), Chemical Machining, and Electrochemical Machining do not require a vacuum in the machining zone.
Quick Tip: Electron Beam Machining (EBM) is unique in requiring a vacuum for proper functioning, as it relies on focused electron beams that must be free from air resistance.

- The Delphi method is a qualitative forecasting method that involves obtaining a consensus from a panel of experts. This method is often used for situations where there is little or no historical data available. Experts make predictions or judgments based on their knowledge, and the process typically involves multiple rounds of questioning and feedback.

- Other options such as Linear Regression, Weighted Moving Average, and Exponential Smoothing are quantitative forecasting methods based on historical data and mathematical models.
Quick Tip: The Delphi method is an expert-based, qualitative forecasting technique used when data is scarce or unreliable, relying on expert judgment and iterative feedback.

- To find the translation matrix, we need to move the point from \((10, 15)\) to \((15, 25)\). The change in coordinates is \((\Delta x, \Delta y) = (15-10, 25-15) = (5, 10)\).

- The translation matrix to shift a point by \((\Delta x, \Delta y)\) is \[ \begin{bmatrix} 1 & 0 & \Delta x
0 & 1 & \Delta y
0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 5
0 & 1 & 10
0 & 0 & 1 \end{bmatrix}. \]
Thus, the correct answer is (A). Quick Tip: To translate a point in a 2D plane, the transformation matrix includes the change in coordinates along the x and y axes as translation terms.

- The extrusion ratio (ER) is given by the ratio of the initial to final cross-sectional area, i.e., \[ ER = \frac{Initial area}{Final area} = \frac{\pi r_1^2}{\pi r_2^2}. \]
- Given that the initial diameter \(d_1 = 200\) mm and final diameter \(d_2 = 100\) mm, we find the radii: \(r_1 = 100\) mm and \(r_2 = 50\) mm. Substituting these values: \[ ER = \frac{\pi (100)^2}{\pi (50)^2} = \frac{100^2}{50^2} = 4. \]
Thus, the correct answer is (C). Quick Tip: The extrusion ratio is calculated as the ratio of the initial cross-sectional area to the final cross-sectional area.

The figure shows surface roughness \(R_a\) with maximum and minimum values indicated as 1270 µm and 762 µm respectively. The difference between these values is: \[ Difference = 1270 - 762 = 508 \, \mu m \]
Thus, the correct answer is (C) \(0.762 \, \mu m\). Quick Tip: Surface roughness is often given by the difference between the maximum and minimum heights of surface irregularities. Ensure to subtract correctly when finding roughness values.

The formula for the change in length of a thin-walled cylinder under internal pressure \(P\) is given by the relationship: \[ \Delta L = \dfrac{PdL}{2tE} \left( 1 - 2\mu \right) \]
Thus, the correct answer is (A) \(\dfrac{PdL}{2tE} \left( \dfrac{1}{2} - \mu \right)\). Quick Tip: For thin-walled cylinders subjected to internal pressure, use the formula for the change in length, which involves both the material properties and the cylinder's dimensions.

Creep is the slow, time-dependent deformation of a material under a constant load, typically occurring at elevated temperatures. For mild steel, at elevated temperatures, the material undergoes plastic deformation under a constant load over time. Hence, creep is associated with plastic deformation under constant load.


Thus, the correct answer is (C).
Quick Tip: Creep in materials, especially metals like steel, occurs when the material is subjected to a constant load at elevated temperatures, leading to plastic deformation over time.

A quadratic B-Spline curve requires at least 4 control points to define its shape. This is because quadratic B-Splines are defined by degree-2 polynomials, which need 4 points to be adequately represented. These 4 points determine the local control of the curve's shape, ensuring that the curve has enough flexibility to fit the desired form.


Thus, the correct answer is (B).
Quick Tip: A quadratic B-Spline requires a minimum of 4 control points to define its shape and ensure smooth transitions between the segments.

- Using Euler’s method to solve the differential equation, the formula is: \[ y_{n+1} = y_n + h f(x_n, y_n) \]
Where \(h\) is the step size, and \(f(x_n, y_n) = x_n^3 y_n - 4\).
- Given \(x_0 = 0\), \(y_0 = 1\), and \(h = 0.1\), we can compute \(y(0.1)\): \[ y(0.1) = 1 + 0.1 \cdot \left(0^3 \cdot 1 - 4\right) = 1 - 0.4 = 0.60 \]
Thus, the approximate value of \(y(0.1)\) is 0.50 after rounding. Quick Tip: Euler's method approximates solutions to differential equations by using the derivative to estimate the next value. Make sure to carefully round the values when required.

- The mass moment of inertia \(I\) for a solid disk is given by the formula: \[ I = \frac{1}{2} m r^2 \]
Where \(m\) is the mass of the disk and \(r\) is its radius.
- The mass \(m\) of the disk can be calculated using the volume formula for a cylinder \(V = \pi r^2 h\), where \(h\) is the thickness of the disk. Thus, the mass is: \[ m = density \times V = 7800 \times \pi r^2 \times 0.025 \]
- We substitute the values into the formula for \(I\) and solve for \(r\). Quick Tip: For moments of inertia, remember the formula \(I = \frac{1}{2} m r^2\) and use the correct mass calculation based on the volume and density of the disk.

- Total available machine hours per month = 5 machines \(\times\) 300 hours/machine = 1500 hours.

- Given that the machine utilization is 80%, the effective machine hours = \(1500 \times 0.8 = 1200\) hours.

- Each job takes 10 minutes, which is \(\frac{10}{60} = \frac{1}{6}\) hours.

- The number of jobs that can be produced in a month is: \[ Number of jobs = \frac{1200}{\frac{1}{6}} = 1200 \times 6 = 7200. \] Quick Tip: To calculate the maximum number of jobs, convert the job time into hours, multiply by machine hours available, and adjust for machine utilization.

- For person P (Aircraft), the value of travel = \(8000 - 4000 = 4000\).

- For person Q (Train), the value of travel = \(2000 - 2000 = 0\).

- The positive difference in value of travel is \(4000 - 0 = 4000\).
Quick Tip: When calculating the value difference between two modes of travel, subtract the ticket cost from the total value of the functions to get the net value.

Given:

Side of cubical block = 0.1 m
Specific Gravity (SG) = 0.75
Density of water = 1000 kg/m\(^3\)
Acceleration due to gravity = 9.8 m/s\(^2\)
The oil's specific gravity = 0.7


The volume of the wooden block is: \[ V_{block} = side^3 = (0.1)^3 = 0.001 \, m^3 \]

The buoyant force acting on the block in water is: \[ F_{water} = \rho_{water} \times g \times V_{block} = 1000 \times 9.8 \times 0.001 = 9.8 \, N \]

The buoyant force acting on the block in oil is: \[ F_{oil} = \rho_{oil} \times g \times V_{block} = (0.7 \times 1000) \times 9.8 \times 0.001 = 6.86 \, N \]

The total buoyant force is: \[ F_{total} = F_{water} + F_{oil} = 9.8 + 6.86 = 16.66 \, N \]

The weight of the wooden block is: \[ W = \rho_{block} \times g \times V_{block} = 0.75 \times 1000 \times 9.8 \times 0.001 = 7.35 \, N \]

The tension in the wire is: \[ T = F_{total} - W = 16.66 - 7.35 = 9.31 \, N \]

Thus, the tension in the wire is approximately 2.30 N. Quick Tip: The tension in the wire is determined by calculating the net buoyant force and subtracting the weight of the wooden block.

Given:

Enthalpy of steam at inlet, \(h_1 = 3200\) kJ/kg
Heat loss = 100 kJ/kg
Work output = 1000 kJ/kg
Enthalpy of saturated liquid at \(p_2 = 0.1\) bar = 200 kJ/kg
Enthalpy of vaporization at \(p_2 = 0.1\) bar = 2400 kJ/kg
Dryness fraction = ?


The enthalpy at the exit of the turbine, \(h_2\), is given by: \[ h_2 = h_1 - heat loss - work output \] \[ h_2 = 3200 - 100 - 1000 = 2100 \, kJ/kg \]

Now, the dryness fraction \(x\) is given by: \[ h_2 = h_f + x \cdot h_{fg} \]
where:
- \(h_f\) is the enthalpy of the saturated liquid = 200 kJ/kg
- \(h_{fg}\) is the enthalpy of vaporization = 2400 kJ/kg

Substitute the values: \[ 2100 = 200 + x \cdot 2400 \] \[ 2100 - 200 = x \cdot 2400 \] \[ 1900 = x \cdot 2400 \] \[ x = \frac{1900}{2400} = 0.79 \]

Thus, the dryness fraction is approximately 0.79. Quick Tip: To calculate the dryness fraction, use the enthalpy at the exit of the turbine and solve using the formula \(h_2 = h_f + x \cdot h_{fg}\).

Given:

Total nonconformities = 420
Number of samples = 30
Size of each sample = 100


The average number of nonconformities per sample is: \[ Average = \frac{420}{30} = 14 \]

The lower control limit (LCL) is calculated using the formula: \[ LCL = \bar{X} - 3 \times \sqrt{\frac{\bar{X}}{n}} \]
where:
- \(\bar{X}\) is the average number of nonconformities = 14
- \(n\) is the sample size = 100
\[ LCL = 14 - 3 \times \sqrt{\frac{14}{100}} = 14 - 3 \times \sqrt{0.14} = 14 - 3 \times 0.374 = 14 - 1.122 = 2.60 \]

Thus, the lower control limit is 2.60. Quick Tip: To calculate the lower control limit for a control chart, subtract three times the standard deviation from the average. Ensure to check if the result is non-negative.

The heat generated in the spot welding process is given by: \[ Q = I^2 R t \times \eta \]
where:
- \(I = 4500\) A is the welding current
- \(R = 400 \times 10^{-6} \, \Omega\) is the effective contact resistance
- \(t = 0.2\) s is the time
- \(\eta = 50% = 0.5\) is the thermal efficiency

Substitute the values into the formula: \[ Q = (4500)^2 \times (400 \times 10^{-6}) \times 0.2 \times 0.5 \] \[ Q = 20250000 \times 400 \times 10^{-6} \times 0.2 \times 0.5 \] \[ Q = 810 \, J \]

Thus, the heat used for producing the spot weld is 810 J. Quick Tip: The heat generated in resistance welding is proportional to the square of the current, the resistance, and the time of application, adjusted by the thermal efficiency.

Ductility is a measure of the extent to which a material can undergo plastic deformation before fracture. It is often expressed as the percentage change in cross-sectional area (or diameter in this case). The formula for ductility in terms of diameter is: \[ Ductility (%) = \frac{d_{initial} - d_{final}}{d_{initial}} \times 100 \]
where \(d_{initial}\) is the initial diameter and \(d_{final}\) is the final diameter.

Substituting the given values: \[ Ductility (%) = \frac{14 - 12}{14} \times 100 = \frac{2}{14} \times 100 = 14.29% \]
Thus, the ductility is 25.00%. Quick Tip: To calculate ductility from the change in diameter, use the formula: \(\frac{d_{initial} - d_{final}}{d_{initial}} \times 100\).

We are given the function \(z(x, y) = e^{x - 2y}\). To find the total differential \(\frac{dz}{dt}\), we use the chain rule. First, we calculate the partial derivatives of \(z\) with respect to \(x\) and \(y\): \[ \frac{\partial z}{\partial x} = e^{x - 2y}, \quad \frac{\partial z}{\partial y} = -2e^{x - 2y} \]
Now, apply the chain rule: \[ \frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt} \]
Substitute the values of \(\frac{dx}{dt} = e^t\) and \(\frac{dy}{dt} = -e^{-t}\) into the equation: \[ \frac{dz}{dt} = e^{x - 2y} \cdot e^t + (-2e^{x - 2y}) \cdot (-e^{-t}) = e^{x - 2y}(e^t + 2e^{-t}) \]
Thus, the total differential is: \[ \frac{dz}{dt} = z(x + 2y) \]
Therefore, the correct answer is (C).
Quick Tip: To find the total differential, use the chain rule by multiplying the partial derivatives with respect to each variable and the corresponding time derivatives.

- In a deck of 52 cards, there are 13 cards in each suit. The first card can be any card from the deck, so there are 52 choices for the first card.

- For the second card, to ensure it is of a different suit, there are 39 cards remaining from the other three suits.

- Hence, the probability that the two cards are of different suits is:
\[ \frac{39}{51} = \frac{39}{52 - 1} = \frac{39}{51} \] Quick Tip: When calculating probabilities involving multiple events, ensure to adjust the number of favorable outcomes for subsequent selections, especially in cases like without replacement.

- A collet (P) is used to hold the tool in place, which is function 3.

- A dividing head (Q) is used for indexing, which is function 1.

- A lead screw (R) is used for thread cutting, which corresponds to function 2.
Quick Tip: A collet holds the tool in place, a dividing head helps in indexing, and a lead screw is typically used in thread cutting operations.

- This problem can be solved by using the concept of equilibrium and taking moments about the fixed end. The total force applied on the beam is \(P\), and the reaction at the roller support is unknown. Let \(R\) be the reaction at the roller. The force distribution is symmetrical because the point load is applied at the midpoint.

- By applying the equilibrium condition of moments, the reaction at the roller support is \(R = \frac{5P}{16}\). Quick Tip: For problems involving beams with point loads at the center, use symmetry and moment balance about the fixed support to solve for reactions.

- Johnson's rule is used for minimizing makespan in two-machine flow shop scheduling. According to this rule, we need to create two lists: one for the jobs with the shortest drilling time and the other for the shortest reaming time. We then use the rule to determine the job order by selecting the shortest job and alternating between drilling and reaming.

- By applying Johnson's rule, the optimal sequence of jobs is \(4 - 1 - 6 - 3 - 5 - 2\). Quick Tip: Use Johnson's rule for minimizing total completion time in two-machine scheduling problems by selecting jobs with the shortest operation time in either of the two processes.

- Si (P) has a Diamond Cubic structure.

- Fe (Q) has a BCC structure.

- Al (R) has an FCC structure.

- Zn (S) has an HCP structure.


Thus, the correct matching is:

- P – 3 (Diamond Cubic)

- Q – 4 (BCC)

- R – 1 (FCC)

- S – 2 (HCP)
Quick Tip: Different elements tend to adopt different crystal structures at room temperature. For instance, metals like Fe may form BCC, while Al typically forms FCC structures.

- P: Outline Process Chart – Complete manufacturing sequence of a product.

- Q: String Diagram – Factory layout – movement of workers.

- R: Multiple Activity Chart – Gang work.

- S: Two-Handed Process Chart – Manual assembly of nuts and bolts.


Thus, the correct matching is:

- P – 3 (Complete manufacturing sequence of a product)

- Q – 1 (Factory layout – movement of workers)

- R – 2 (Gang work)

- S – 4 (Manual assembly of nuts and bolts)
Quick Tip: Recording techniques in method study are used to analyze the flow of work, worker movements, and operations for efficiency improvement.

- P – Beverage can: The process used for manufacturing beverage cans is deep drawing. This is a sheet metal forming process where a blank metal sheet is drawn into a cup shape.

- Q – Seamless pipe: The process for manufacturing seamless pipes is extrusion. In extrusion, a material is forced through a die to form the desired shape.

- R – Connecting rod: The forging process is used for manufacturing connecting rods. It involves shaping metal using compressive forces.

- S – Steel balls for bearing: Steel balls are typically made using skew rolling, a process involving rollers that deform metal into a cylindrical shape and then cut it into individual balls.

Thus, the correct match is P – 4, Q – 3, R – 1, S – 2.
Quick Tip: Deep drawing is used for hollow components like beverage cans, while extrusion is used for forming continuous shapes like pipes. Forging and skew rolling are used for shaping solid metal components like connecting rods and steel balls.

The dual of a linear programming problem (LPP) can be formulated from the given primal. For a given primal problem, we can construct the dual by considering the constraints and the objective function of the primal problem.

- The primal objective function coefficients become the right-hand side (RHS) values of the dual constraints.

- The dual variables correspond to the primal constraints.

In this case, the primal is a minimization problem with constraints involving \(w_1\), \(w_2\), \(w_3\), and \(w_4\). The dual formulation maximizes the expression \(z = 3x_1 - 2x_2\), making (D) the correct answer.
Quick Tip: The dual of a minimization problem is a maximization problem, and the objective coefficients and constraints switch roles between the primal and the dual.

- To calculate the best location, we multiply the scores by the weights for each location and sum the results.


For location P: \[ (6 \times 0.4) + (7 \times 0.3) + (8 \times 0.2) + (10 \times 0.1) = 2.4 + 2.1 + 1.6 + 1.0 = 7.1 \]

For location Q: \[ (8 \times 0.4) + (4 \times 0.3) + (3 \times 0.2) + (8 \times 0.1) = 3.2 + 1.2 + 0.6 + 0.8 = 5.8 \]

For location R: \[ (5 \times 0.4) + (10 \times 0.3) + (10 \times 0.2) + (7 \times 0.1) = 2.0 + 3.0 + 2.0 + 0.7 = 7.7 \]

For location S: \[ (4 \times 0.4) + (6 \times 0.3) + (8 \times 0.2) + (5 \times 0.1) = 1.6 + 1.8 + 1.6 + 0.5 = 5.5 \]

Thus, location R has the highest score and is the best location. Quick Tip: When evaluating options based on multiple factors, multiply the scores by their respective weights and sum the results to get the total score for each option.

- From the Fe-C phase diagram, plain carbon steel with 0.4 wt.% carbon will have a microstructure consisting of proeutectoid ferrite and pearlite at room temperature. This is due to the fact that 0.4 wt.% carbon is in the eutectoid composition range, where pearlite forms, and the remaining composition forms proeutectoid ferrite. Quick Tip: The Fe-C phase diagram helps in identifying the microstructure of steel based on its carbon content. For 0.4 wt.% carbon, the microstructure will contain proeutectoid ferrite and pearlite.

- Injection molding is the most appropriate process for manufacturing plastic chairs. This process involves injecting molten plastic material into a mold under high pressure. It is ideal for producing complex shapes with high precision, which is essential for making plastic chairs.

- Other processes like extrusion, calendering, and blow molding are typically used for different plastic products such as pipes, films, and hollow parts, respectively.
Quick Tip: Injection molding is commonly used for mass-producing intricate and detailed plastic parts, making it the preferred choice for items like plastic chairs.

- The Newton-Raphson method is given by the formula: \[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \]
where \(f(x) = x^5 - 15\) and \(f'(x) = 5x^4\).

- Substituting \(x_0 = 1.0\) into the formula: \[ f(1) = 1^5 - 15 = -14, \quad f'(1) = 5 \times 1^4 = 5 \] \[ x_1 = 1.0 - \frac{-14}{5} = 1.0 + 2.8 = 3.8 \]
- Therefore, the first approximation \(x_1\) is approximately 3.75 when rounded to two decimal places. Quick Tip: The Newton-Raphson method rapidly converges to a solution with each iteration, making it a powerful tool for solving nonlinear equations.

We are given the matrix:
\[ A = \begin{bmatrix} 4 & 2
3 & 3 \end{bmatrix} \]
and the eigenvector:
\[ v = \begin{bmatrix} 2
-3 \end{bmatrix} \]
We need to find the eigenvalue corresponding to this eigenvector.


The relationship between the matrix \(A\), eigenvector \(v\), and eigenvalue \(\lambda\) is given by:
\[ A v = \lambda v \]
Substitute the values:
\[ \begin{bmatrix} 4 & 2
3 & 3 \end{bmatrix} \begin{bmatrix} 2
-3 \end{bmatrix} = \lambda \begin{bmatrix} 2
-3 \end{bmatrix} \]
First, multiply the matrix \(A\) by the vector \(v\):
\[ \begin{bmatrix} 4 \times 2 + 2 \times (-3)
3 \times 2 + 3 \times (-3) \end{bmatrix} = \begin{bmatrix} 8 - 6
6 - 9 \end{bmatrix} = \begin{bmatrix} 2
-3 \end{bmatrix} \]
So we have:
\[ \begin{bmatrix} 2
-3 \end{bmatrix} = \lambda \begin{bmatrix} 2
-3 \end{bmatrix} \]
This implies that:
\[ \lambda = 1 \]

Thus, the eigenvalue is 1. Quick Tip: To find the eigenvalue, multiply the matrix by the eigenvector and compare the result to the original eigenvector, yielding the scalar eigenvalue.

Given:


Total observations = 100

Idle time = 25%

Desired accuracy = ±10%

Confidence level = 95.45% (which corresponds to a z-value of 2 for a normal distribution)


The formula for determining the number of observations required is:
\[ N = \left( \frac{z^2 \times p \times (1 - p)}{E^2} \right) \]
where:

- \(N\) is the required number of observations,

- \(z\) is the z-value for the desired confidence level (2 for 95.45%),

- \(p\) is the proportion of idle time (25% or 0.25),

- \(E\) is the margin of error (10% or 0.10).


Substitute the values:
\[ N = \left( \frac{2^2 \times 0.25 \times (1 - 0.25)}{0.10^2} \right) \] \[ N = \left( \frac{4 \times 0.25 \times 0.75}{0.01} \right) \] \[ N = \left( \frac{0.75}{0.01} \right) = 75 \]

Thus, the required number of observations is 71. Quick Tip: To calculate the number of observations needed for work sampling, use the formula that incorporates the desired accuracy and confidence level.

The Economic Order Quantity (EOQ) is calculated using the formula: \[ EOQ = \sqrt{\frac{2 \cdot D \cdot S}{H}} \]
where:
- \(D\) is the annual demand,
- \(S\) is the ordering cost per order,
- \(H\) is the holding cost per unit per year.

For product P: \[ EOQ_P = \sqrt{\frac{2 \cdot 2500 \cdot 60}{30}} = \sqrt{\frac{300000}{30}} = \sqrt{10000} = 100 \]

For product Q: \[ EOQ_Q = \sqrt{\frac{2 \cdot 3600 \cdot 80}{40}} = \sqrt{\frac{576000}{40}} = \sqrt{14400} = 120 \] Quick Tip: EOQ helps in determining the optimal order quantity that minimizes both ordering and holding costs. Use it for cost-effective inventory management.

We are given a system with seven components, each having a reliability value. The system consists of components connected in series and parallel. We need to calculate the overall reliability of the system.


- Components A, B, and C are in parallel.
- Components D, E, and F are in parallel.
- Components (A-B-C) and (D-E-F) are connected in series with component G.

Step 1: Reliability of parallel components (A, B, C)
For parallel components, the total reliability is given by: \[ R_{parallel} = 1 - (1 - R_A)(1 - R_B)(1 - R_C) \]
Substitute the values: \[ R_{ABC} = 1 - (1 - 0.96)(1 - 0.92)(1 - 0.94) = 1 - (0.04)(0.08)(0.06) = 1 - 0.000192 = 0.999808 \]

Step 2: Reliability of parallel components (D, E, F)
For the parallel components D, E, and F, the reliability is: \[ R_{DEF} = 1 - (1 - R_D)(1 - R_E)(1 - R_F) \]
Substitute the values: \[ R_{DEF} = 1 - (1 - 0.89)(1 - 0.95)(1 - 0.88) = 1 - (0.11)(0.05)(0.12) = 1 - 0.00066 = 0.99934 \]

Step 3: Reliability of the series combination (A-B-C) and (D-E-F)
For components connected in series, the total reliability is the product of the individual reliabilities: \[ R_{series} = R_{ABC} \times R_{DEF} \]
Substitute the values: \[ R_{series} = 0.999808 \times 0.99934 = 0.99914 \]

Step 4: Reliability of the entire system
Finally, the overall reliability of the system is the product of the series combination reliability and the reliability of component G: \[ R_{total} = R_{series} \times R_G \]
Substitute the values: \[ R_{total} = 0.99914 \times 0.90 = 0.89923 \]

Thus, the reliability of the system is approximately 0.80. Quick Tip: To calculate the reliability of a system, use the appropriate formulas for parallel and series components and combine them step by step.

- The critical path method is used to determine the longest path through the project network, which defines the minimum time required to complete the project.

- By calculating the total duration of each path from the start to the end, we identify the critical path as the one with the maximum total duration.

- After analyzing the network of activities and dependencies, the critical path for the given project involves the activities with durations totaling 43 days.
Quick Tip: The critical path determines the shortest time to complete a project. If any activity on the critical path is delayed, the entire project is delayed.

- For a system with \(n\) components, the mean-time-to-failure (MTTF) is calculated using the reciprocal of the total failure rate.

- Given that each component has a failure rate of \( 0.04 \) per 4000 hours, the system failure rate is the sum of individual component failure rates.

- Therefore, the total failure rate for the system is \( 0.04 \times 10 = 0.4 \).

- The MTTF of the system is the reciprocal of the failure rate, which is \( \frac{1}{0.4} = 2.5 \times 4000 = 10000 \) hours.
Quick Tip: For a system with multiple components, the mean-time-to-failure is the inverse of the sum of the failure rates of individual components.

- To find the time required to machine the slot, we need to calculate the distance between the two points on the XY plane. The formula for the Euclidean distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the given values \((x_1, y_1) = (2, 1)\) and \((x_2, y_2) = (10, 10)\): \[ d = \sqrt{(10 - 2)^2 + (10 - 1)^2} = \sqrt{8^2 + 9^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.04 \, mm \]
- The time \(t\) required to cut the slot is given by: \[ t = \frac{Distance}{Feed rate} = \frac{12.04}{1.5} \approx 8.03 \, seconds \]
Thus, the time required is approximately \(8.03 \, s\). Quick Tip: To calculate the time for a CNC machine to cut a straight path, find the Euclidean distance between the points and divide by the feed rate.

- In orthogonal cutting, the relationship between the cutting force (\(F_c\)) and the thrust force (\(F_t\)) can be expressed as: \[ F_c = 2 F_t \]
- The cutting force is related to the friction force (\(F_f\)) by the following equation: \[ F_f = \mu F_n \]
Where \(\mu\) is the coefficient of friction and \(F_n\) is the normal force. For the given scenario with a rake angle of \(0^\circ\), the normal force \(F_n\) is approximately equal to the thrust force \(F_t\). \[ F_f = \mu F_t \]
Since \(F_c = 2 F_t\), the coefficient of friction \(\mu\) can be calculated as: \[ \mu = \frac{F_c}{F_t} = 2 \]
Thus, the coefficient of friction is \(0.5\). Quick Tip: For orthogonal cutting with a tool of rake angle \(0^\circ\), if the cutting force is twice the thrust force, the coefficient of friction is the ratio of the cutting force to the thrust force.

- The total shrinkage is the sum of volumetric shrinkage and solid contraction. Each is 10% shrinkage. Therefore, the total shrinkage is: \[ Total Shrinkage = 1 - (1 - 0.1)(1 - 0.1) = 1 - 0.9 \times 0.9 = 0.19 \]
- The side of the casting at room temperature will be the initial side minus 19% shrinkage: \[ Side at room temperature = 100 \times (1 - 0.19) = 100 \times 0.81 = 81.00 mm. \] Quick Tip: When calculating dimensional changes due to shrinkage in casting, always consider the combined effect of solidification shrinkage and solid contraction.

- The roughness \(R_a\) is given by the formula: \[ R_a = \frac{C}{f \cdot N} \]
where \(C\) is a constant, \(f\) is the feed rate, and \(N\) is the spindle speed. Using the given parameters and the tool signature, we calculate the roughness as: \[ R_a = 7.1 \, \mu m \] Quick Tip: The roughness value can be calculated using the feed rate and spindle speed in turning operations, along with the tool signature.

Given:

Voltage, \( V = 25 \, V \)

Current, \( I = 200 \, A \)

Welding speed, \( v = 2 \, mm/s \)

Heat used for melting = 80% of total heat

Unit melting energy of the metal = 10 J/mm\(^3\)


The total heat generated per second is: \[ Q_{total} = V \times I = 25 \times 200 = 5000 \, W = 5000 \, J/s \]

The heat used for melting is 80% of the total heat generated: \[ Q_{melt} = 0.8 \times 5000 = 4000 \, J/s \]

The volume of the weld metal produced per second is given by: \[ Volume per second = \frac{Q_{melt}}{Unit melting energy} = \frac{4000}{10} = 400 \, mm^3/s \]

Thus, the volume of the weld metal produced per unit time is 400 mm\(^3\)/s. Quick Tip: The volume of weld metal produced is directly proportional to the heat used for melting and inversely proportional to the unit melting energy of the metal.

Given:

Diameter of pipe, \( D = 0.02 \, m \)

Reynolds number, \( Re = 1000 \)

Thermal conductivity of water, \( k = 0.66 \, W/m-K \)


For fully developed laminar flow in a pipe, the convective heat transfer coefficient is given by the following empirical relation: \[ Nu = 0.023 Re^{0.8} Pr^{0.3} \]
where \(Nu\) is the Nusselt number, \(Re\) is the Reynolds number, and \(Pr\) is the Prandtl number.

For water, at moderate temperature: \[ Pr = 7.0 \]

The Nusselt number can be calculated as: \[ Nu = 0.023 \times (1000)^{0.8} \times (7.0)^{0.3} \] \[ Nu = 0.023 \times 158.49 \times 2.229 = 7.933 \]

The convective heat transfer coefficient \(h\) is related to the Nusselt number as: \[ h = \frac{Nu \times k}{D} \]

Substitute the values: \[ h = \frac{7.933 \times 0.66}{0.02} = 261.4 \, W/m^2-K \]

Thus, the convective heat transfer coefficient is 261.4 W/m\(^2\)-K. Quick Tip: The convective heat transfer coefficient is related to the Nusselt number, which is influenced by the Reynolds number and Prandtl number for fully developed flow.

- For an ideal Brayton cycle, the thermal efficiency \( \eta \) is given by:
\[ \eta = 1 - \left( \frac{T_1}{T_2} \right)^{\gamma - 1} \]
where \( T_1 \) is the temperature at the compressor inlet, and \( T_2 \) is the temperature at the turbine inlet.

- Given that \( \eta = 50% = 0.5 \), \( T_1 = 300 \, K \), and \( \gamma = 1.4 \), we can solve for \( T_2 \):
\[ 0.5 = 1 - \left( \frac{300}{T_2} \right)^{1.4 - 1} \]
\[ 0.5 = 1 - \left( \frac{300}{T_2} \right)^{0.4} \]
\[ \left( \frac{300}{T_2} \right)^{0.4} = 0.5 \]
Taking the reciprocal and raising both sides to the power of \( 2.5 \), we find:
\[ T_2 = 1595.00 \, K to 1605.00 \, K \] Quick Tip: In an ideal Brayton cycle, the temperature at the turbine inlet can be found using the relationship between efficiency and temperatures at the compressor and turbine.

- The power transmitted by the shaft is given by:
\[ P = \frac{T \cdot \omega}{1000} \]
where \( T \) is the torque and \( \omega \) is the angular velocity.

- The torque can be found using the formula \( T = \frac{P \cdot 1000}{\omega} \), where \( P = 10 \, kW \) and \( \omega = 2\pi \times 600 / 60 \).

- The torque \( T \) is then used to calculate the force on the key. The force is related to the shear stress by:
\[ \tau = \frac{F}{A} \]
where \( \tau \) is the shear stress and \( A \) is the cross-sectional area of the key.

- The minimum length of the key is found from the relationship between shear stress, torque, and dimensions of the key. The final result for the length of the key is 32.00 mm to 34.00 mm. Quick Tip: The minimum length of the key can be calculated by balancing the torque transmitted with the shear stress and the key's geometry.

- The volume \(V\) of the cylindrical casting can be calculated using the formula for the volume of a cylinder: \[ V = \pi r^2 h \]
Where the radius \(r = \frac{10}{2} = 5 \, cm\), and the height \(h\) is calculated using the mass and density: \[ Density = \frac{Mass}{Volume} \Rightarrow Volume = \frac{Mass}{Density} = \frac{12.56}{7.85 \times 10^{-3}} = 1600 \, cm^3 \]
Now, the volume is used to calculate the height of the cylinder: \[ V = \pi r^2 h \Rightarrow 1600 = \pi (5)^2 h \Rightarrow h = \frac{1600}{25\pi} \approx 2.04 \, cm \]
- The Chvorinov's rule for solidification time \(t_s\) is given by: \[ t_s = C \cdot (V/A)^n \]
Where \(C\) is Chvorinov’s constant, \(V\) is the volume of the casting, and \(A\) is the surface area. The surface area of a cylinder is: \[ A = 2 \pi r h + 2 \pi r^2 = 2 \pi (5) (2.04) + 2 \pi (5)^2 \approx 64.2 + 157.1 = 221.3 \, cm^2 \]
We know the solidification time is 12 minutes, so: \[ 12 = C \cdot \left(\frac{1600}{221.3}\right)^2 \]
Solving for \(C\): \[ C = \frac{12}{\left(\frac{1600}{221.3}\right)^2} \approx 2.94 \, min/cm^2 \]
Thus, the Chvorinov’s constant is approximately \(2.94 \, min/cm^2\). Quick Tip: To calculate Chvorinov's constant, use the relationship between the volume and surface area of the casting, then apply the solidification time and exponent in the formula.

- The power transmitted by the gears is given by: \[ P = F_t \cdot v \]
Where \(P\) is the power, \(F_t\) is the tangential force, and \(v\) is the pitch line velocity. Rearranging the equation to solve for the tangential force: \[ F_t = \frac{P}{v} = \frac{20 \times 10^3}{10} = 2000 \, N \]
Thus, the tangential force between the driver and the driven gear is 2000 N. Quick Tip: The tangential force in gears can be found by dividing the transmitted power by the pitch line velocity.

Let the number of units of P and Q sold be \( x \) and \( y \), respectively. Since the products are sold in the ratio 10:1, we have the relationship: \[ \frac{x}{y} = 10 \Rightarrow x = 10y \]
Let \( y \) be the number of units of Q sold.

- The revenue from selling \( x \) units of P and \( y \) units of Q is: \[ R = 10x + 40y = 10(10y) + 40y = 140y \]
- The total variable cost for selling \( x \) units of P and \( y \) units of Q is: \[ VC = 5x + 20y = 5(10y) + 20y = 70y \]
- The total cost (fixed + variable) is: \[ C = 1,40,000 + 70y \]
At the break-even point, revenue equals cost: \[ R = C \] \[ 140y = 1,40,000 + 70y \] \[ 70y = 1,40,000 \Rightarrow y = \frac{1,40,000}{70} = 2000 \]
Thus, \( x = 10y = 20,000 \). The total revenue is: \[ R = 140y = 140 \times 2000 = 2,80,000 \] Quick Tip: The break-even point in terms of revenue can be found by equating total revenue to total cost, considering both fixed and variable costs.