GATE PI 2022

This question tests your knowledge of homophones, specifically "tyre" and "tire." For the first blank, we need a noun for something that burns. Both 'tire' and 'tyre' mean a rubber wheel covering, and 'tyre' is the standard British English spelling often used in such exams. For the second blank, we need a verb to describe the effect of inhaling smoke; 'tire' (verb) means to make weary or exhausted, which fits the sentence. 'Tier' (noun), meaning a level or rank, is incorrect here. , "tyre / tire," correctly completes the sentence.

To perfectly fit a sphere of radius $r$ inside a cubical box, the minimum side length of the cube must be equal to the sphere's diameter. The diameter of the sphere is $2 \times r = 2r$ cm. So, the minimum side length of the cube, $s$, must be $2r$ cm.

The volume of a cube is calculated using the formula $V = s^3$.
Substituting $s = 2r$ into the volume formula gives:
$V_{min} = (2r)^3$
$V_{min} = 2^3 \times r^3$
$V_{min} = 8r^3$
Therefore, the minimum volume of the cubical box required is $8r^3$ cubic centimeters.

This problem involves understanding the concept of work rate, specifically how pipes fill or empty a tank. We need to find the tank's total capacity.

First, let's define the rates of each pipe.
Pipe P fills the tank in 10 minutes, so its filling rate is $\frac{1}{10}$ of the tank per minute.
Pipe Q fills the tank in 6 minutes, so its filling rate is $\frac{1}{6}$ of the tank per minute.
Pipe R empties water at a constant rate of 34 litres per minute.

Let $C$ be the capacity of the tank in litres. We can express the filling rates of P and Q in litres per minute:
Rate of Pipe P (filling) = $\frac{C}{10}$ litres per minute.
Rate of Pipe Q (filling) = $\frac{C}{6}$ litres per minute.

When all three pipes operate simultaneously, the net rate of change in the tank's volume is the sum of the filling rates minus the emptying rate:
$\text{Net Rate} = \left( \frac{C}{10} + \frac{C}{6} \right) - 34 \text{ litres/minute}$
We are given that a full tank empties completely in 1 hour (which is 60 minutes) when all pipes are operating. This means the net emptying rate is $\frac{C}{60}$ litres per minute. Since it's emptying, we represent this as a negative net rate:
$\text{Net Rate} = -\frac{C}{60} \text{ litres/minute}$
Now, we equate the two expressions for the Net Rate:
$\left( \frac{C}{10} + \frac{C}{6} \right) - 34 = -\frac{C}{60}$
To solve for $C$, we find a common denominator for the fractions, which is 60:
$\left( \frac{6C}{60} + \frac{10C}{60} \right) - 34 = -\frac{C}{60}$
$\frac{16C}{60} - 34 = -\frac{C}{60}$
Rearranging the terms to isolate $C$:
$\frac{16C}{60} + \frac{C}{60} = 34$
$\frac{17C}{60} = 34$
Solving for $C$:
$17C = 34 \times 60$
$C = \frac{34 \times 60}{17}$
$C = 2 \times 60$
$C = 120$
The capacity of the storage tank is 120 litres.

We have 6 people (P, Q, R, S, T, U) sitting around a circular table facing the center. We need to find Q's immediate neighbors.

Let's break down the given statements:

1. "P sits next to S and T": This tells us P is exactly between S and T. So, the sequence will look like S-P-T or T-P-S.
2. "Q sits diametrically opposite to P": In a circular arrangement of 6 people, "diametrically opposite" means there are 2 people between them on either side.
3. "The shortest distance between S and R is equal to the shortest distance between T and U": Here, "distance" refers to the number of people sitting between the two specified individuals along the shorter path around the table. If they are adjacent, the distance is $0$.

Let's start placing them:

• We can fix P at position 1. Due to statement 2, Q must be at position 4 (P(1) -- Person -- Person -- Q(4)).
• Now, using statement 1, S and T must be next to P. This leads to two initial scenarios for placing S and T:
  • Scenario A: S is at position 2, and T is at position 6. Our current partial arrangement is P(1), S(2), Q(4), T(6).
  • Scenario B: T is at position 2, and S is at position 6. Our current partial arrangement is P(1), T(2), Q(4), S(6).
• The remaining people are R and U, and the remaining positions are 3 and 5.

Now, let's test these scenarios with statement 3:

Scenario A (P-S-?-Q-?-T)
Positions: P(1), S(2), Q(4), T(6). Remaining positions for R and U are 3 and 5.

• Possibility A1: R at 3, U at 5. The full arrangement is P(1), S(2), R(3), Q(4), U(5), T(6).

  • Distance(S, R): S(2) and R(3) are adjacent, so 0 people between them. Distance = $0$.
  • Distance(T, U): T(6) and U(5) are adjacent, so 0 people between them. Distance = $0$.
  • Statement 3 (Distance = $0$ for both) is satisfied.
  • In this possibility, Q(4)'s neighbors are R(3) and U(5).

• Possibility A2: U at 3, R at 5. The full arrangement is P(1), S(2), U(3), Q(4), R(5), T(6).

  • Distance(S, R): S(2) to R(5). The people between them are U(3) and Q(4). So, 2 people. Distance = $2$.
  • Distance(T, U): T(6) to U(3). The people between them (clockwise) are P(1), S(2). So, 2 people. Distance = $2$.
  • Statement 3 (Distance = $2$ for both) is satisfied.
  • In this possibility, Q(4)'s neighbors are U(3) and R(5).

Scenario B (P-T-?-Q-?-S)
Positions: P(1), T(2), Q(4), S(6). Remaining positions for R and U are 3 and 5.

• Possibility B1: R at 3, U at 5. The full arrangement is P(1), T(2), R(3), Q(4), U(5), S(6).

  • Distance(S, R): S(6) to R(3). The people between them (clockwise) are P(1), T(2). So, 2 people. Distance = $2$.
  • Distance(T, U): T(2) to U(5). The people between them are R(3), Q(4). So, 2 people. Distance = $2$.
  • Statement 3 (Distance = $2$ for both) is satisfied.
  • In this possibility, Q(4)'s neighbors are R(3) and U(5).

• Possibility B2: U at 3, R at 5. The full arrangement is P(1), T(2), U(3), Q(4), R(5), S(6).

  • Distance(S, R): S(6) and R(5) are adjacent, so 0 people between them. Distance = $0$.
  • Distance(T, U): T(2) and U(3) are adjacent, so 0 people between them. Distance = $0$.
  • Statement 3 (Distance = $0$ for both) is satisfied.
  • In this possibility, Q(4)'s neighbors are U(3) and R(5).

In all valid arrangements (A1, A2, B1, B2), the neighbors of Q are consistently R and U.

To determine the minimum number of doors to open to travel from point P to point Q, we need to find the shortest path in terms of door crossings. By carefully examining the provided building layout, we identify the most efficient route:

1. From the room containing P, open the door connecting it to the next room. This is the first door.
2. Proceed through this room and open the door that leads towards point Q. This is the second door.
   This path requires opening $2$ doors to reach point Q. Therefore, the minimum number of doors to be opened is $2$.

This question asks us to identify the most logical conclusion drawn directly from the provided text. The passage states that climate change threatens regular rice production, and Green Super Rice (GSR) is being developed to be tough ($resilient$) in bad weather and still give more grain ($higher$ $yields$).

Let's break down each option:

• Option A says GSR grows only in extreme weather. The passage says GSR is $resilient$ in extreme weather, meaning it can handle it, not that it exclusively grows there. So, this isn't right.
• Option B suggests GSR could be used later to fight the bad impacts of climate change. This perfectly matches the passage's point that GSR is a solution to the threat climate change poses to rice. This is a strong logical inference.
• Option C claims GSR produces less than regular rice. The passage clearly states GSR gives $higher$ $yields$, which contradicts this option. Thus, this is incorrect.
• Option D states regular rice will keep producing well even in extreme weather. The passage explicitly says climate change $poses$ $a$ $threat$ to sustained rice availability, directly opposing this idea. So, this is also incorrect.

is the correct logical inference.

To find the probability that the sum of the numbers on the two disks is $8$, we first determine the total possible outcomes. Each disk has $8$ equally likely outcomes (numbers $1$ to $8$). Since the two disks are spun independently, the total number of possible outcomes is $8 \times 8 = 64$. Next, we identify the favorable outcomes where the sum of the numbers from the two disks is $8$. These combinations are $(1, 7)$, $(2, 6)$, $(3, 5)$, $(4, 4)$, $(5, 3)$, $(6, 2)$, and $(7, 1)$, totaling $7$ favorable outcomes. Therefore, the probability is calculated as:
$\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{7}{64}$

To determine which expression involving $p+q$ satisfies both inequalities, we'll combine them. We are given:
(i) $3p - q < 4$
(ii) $3q - p < 12$
Adding the two inequalities together:
$(3p - q) + (3q - p) < 4 + 12$
Simplifying the left side by combining like terms ($3p - p$ and $-q + 3q$) and the right side:
$2p + 2q < 16$
Factoring out $2$ from the left side:
$2(p + q) < 16$
Finally, dividing both sides by $2$ to isolate $p+q$:
$p + q < 8$
This result indicates that $p+q$ must be strictly less than $8$ for both original inequalities to hold true. Comparing this with the given options, Option A, $p+q < 8$, is the direct match.

This problem tests your understanding of syllogisms. We'll evaluate each conclusion based on the given statements.

Statements:

1. Some engineers are writers.
2. No writer is an actor.
3. All actors are engineers.

Conclusion I: Some writers are engineers.
This is correct. Statement 1, "Some engineers are writers," establishes that there's an overlap. If at least one engineer is a writer, it logically follows that at least one writer is an engineer. This is a direct inference from an 'I' type (Some A are B) statement.

Conclusion II: All engineers are actors.
This is incorrect. Statement 3, "All actors are engineers," means the set of actors is a subset of engineers. It does not imply that all engineers are actors. Furthermore, from Statement 1, some engineers are writers, and from Statement 2, no writer is an actor. This directly contradicts the idea that all engineers are actors.

Conclusion III: No actor is a writer.
This is correct. Statement 2, "No writer is an actor," clearly states there's no overlap between writers and actors. If no writer is an actor, then it necessarily means no actor can be a writer. These two statements are logically equivalent (e.g., No A is B $\Leftrightarrow$ No B is A).

Conclusion IV: Some actors are writers.
This is incorrect. Statement 2, "No writer is an actor," directly contradicts this conclusion. If there is absolutely no shared element between the set of writers and the set of actors, then it is impossible for some actors to also be writers.

Based on this analysis, only Conclusion I and Conclusion III are logically inferred from the statements.

To form a square with a single round hole near the center without overlapping pieces, we must find a set of shapes that fit together perfectly.
Option 1 and Option 2 show irregular shapes that geometrically cannot form a precise square boundary with a distinct central hole.
Option 3 provides a central piece with a round cutout and four identical edge pieces. When these four edge pieces are arranged around the central piece, they precisely complete a square, with the central cutout forming the required single round hole. This arrangement satisfies all conditions.
Option 4 contains pieces that are not suitable for assembling the desired square with a central circular void.

In Injection Blow Molding (IBM) for plastic beverage bottles, we first create a preform (parison) by injection molding. This heated preform is then inflated into the final bottle shape inside a mold. This inflation, or blowing, is done by introducing compressed hot air into the parison. Hot air is preferred because it's easily available, allows for precise control over temperature and pressure, and is safe for plastic inflation. Other options like hot water, hot oil, or alcohol are not suitable due to potential issues like contamination, control difficulties, viscosity, flammability, and safety concerns. Therefore, hot air is used for blowing in injection blow molding.

The order of a differential equation is determined by the order of its highest derivative. In the given differential equation, $\left(\frac{dy}{dx}\right)^2 + 5\frac{dy}{dx} + 4y = 5x^3$ the only derivative present is $\frac{dy}{dx}$. This is a first derivative, meaning its order is 1. Since this is the highest (and only) derivative, the order of the differential equation is 1.

The question asks for the expansion of the vector triple product $(a \times b) \times c$. This is a direct application of a fundamental identity in vector algebra. The Vector Triple Product Identity is given by:
$(a \times b) \times c = (a \cdot c)b - (b \cdot c)a$
This identity shows that the result is a vector lying in the plane formed by vectors $a$ and $b$, and it's expressed as a linear combination of $a$ and $b$. Comparing this standard identity with the provided options, we find that Option C, which is $(a \cdot c)b - (b \cdot c)a$, exactly matches the formula.

For numerical integration methods, we often talk about their "order of exactness." This refers to the highest degree polynomial for which the method gives an exact result. The Trapezoidal Rule provides exact results for polynomials of degree 1 or less. Simpson's 1/3 Rule is more accurate and provides exact results for polynomials of degree 3 or less. The given function, $y = 2x + 5$, is a linear function, which is a polynomial of degree 1. Since the degree of the function (1) is less than or equal to the exactness limit for both the Trapezoidal Rule (1) and Simpson's 1/3 Rule (3), both methods will compute the integral exactly.

To determine which metal has a Face-Centered Cubic (FCC) structure, we need to recall the typical crystal structures of the given options. Alpha iron ($\alpha$-Fe) exhibits a Body-Centered Cubic (BCC) structure at room temperature. Chromium (Cr) also typically forms a Body-Centered Cubic (BCC) structure. Magnesium (Mg) is known to crystallize in a Hexagonal Close-Packed (HCP) structure. Aluminum (Al), however, is well-known for its Face-Centered Cubic (FCC) structure. Therefore, Aluminum is the correct answer.

For an isotropic material, Young's modulus ($E$), shear modulus ($G$), and Poisson's ratio ($\nu$) are related by the formula $E = 2G(1 + \nu)$. We know that for any physically realistic material, Poisson's ratio $\nu$ must satisfy $0 \le \nu < 0.5$. To find the maximum possible value of $E$, we substitute the maximum possible value of $\nu$ into this equation. As $\nu$ approaches its upper limit of $0.5$:
$E_{max} = 2G(1 + 0.5)$
$E_{max} = 2G(1.5)$
$E_{max} = 3G$
Therefore, the maximum possible value of Young's modulus for the material is $3G$.

Let's match each casting method to its typical products.

P: Continuous Casting is a high-volume process used to produce long, solid shapes directly from molten metal without interruption. This makes it ideal for manufacturing slabs and strips, which are continuous uniform cross-sections. Thus, P-3.

Q: Investment Casting, often called the lost-wax process, is renowned for its ability to produce parts with exceptional detail and surface finish. It's perfectly suited for creating thin and intricate shaped components where precision is paramount. Thus, Q-1.

R: Centrifugal Casting utilizes rotational forces to distribute molten metal against the inner walls of a rotating mold. This method is highly effective for forming hollow axisymmetric parts, such as pipes or tubes, with uniform wall thickness and high density. Thus, R-2.

Combining these matches gives us P-3, Q-1, R-2.

In Electro-Discharge Machining (EDM), the energy ($E$) dissipated per spark is given by the product of the discharge voltage ($V_b$) and the charge ($Q$) transferred, so $E = V_b \times Q$. The charge ($Q$) stored in the circuit's capacitance ($C$) and transferred during the spark is proportional to the discharge voltage ($V_b$), specifically $Q = C V_b$. Substituting this into the energy formula, we get $E = V_b \times (C V_b)$, which simplifies to $E = C V_b^2$, or more precisely, $E \approx \frac{1}{2} C V_b^2$. Therefore, assuming the capacitance ($C$) remains constant for a spark, the energy dissipated per spark ($E$) is directly proportional to the square of the discharge voltage ($V_b$).
$E \propto V_b^2$

To match the CNC codes with their functions, we need to understand the standard definitions for each:

• $G91$ ($P$) is specifically used for programming in incremental coordinates, meaning each new coordinate is defined relative to the previous one.
• $M02$ ($Q$) is a miscellaneous function that signals the end of the program, often including an automatic reset.
• $G32$ ($R$) is a preparatory function command for performing thread cutting operations in turning, synchronizing the spindle rotation with the tool feed.
• $M05$ ($S$) is a miscellaneous function used to stop the rotation of the machine spindle.

Based on these definitions, the correct matches are:

• $P-2$ ($G91$ - Programming in incremental coordinates)
• $Q-1$ ($M02$ - End of program)
• $R-4$ ($G32$ - Thread cutting in turning)
• $S-3$ ($M05$ - Spindle stop)

Comparing these matches with the given options, we identify that Option B: P-2, Q-1, R-4, S-3 is the correct choice.

When we're tracking quality in manufacturing, different types of data require specific control charts. The question asks which chart is appropriate for monitoring the "number of defects" found in a single welded joint.

Let's break down the options:

• The c-chart is specifically used to monitor the number of defects per unit when the unit size for inspection remains constant. Since a welded joint is considered one unit and we're counting its defects, the c-chart is ideal.
• The $\bar{X}$ - chart monitors the average (mean) of a process characteristic, typically for continuous data such as length or weight.
• The R - chart tracks the variation (range) within subgroups of a process variable, also for continuous data.
• The p - chart monitors the proportion or percentage of defective units, especially when the sample size isn't constant. This chart focuses on whether a unit is good or bad, not how many defects are within it.

Therefore, to monitor the total number of defects found in each welded joint, the c-chart is the correct choice.

Concurrent Engineering is a product development strategy where different tasks are performed at the same time (in parallel) rather than one after another (sequentially). This parallel and integrated approach, involving various teams like design and manufacturing from the beginning, aims to reduce the total time needed to launch a product. Options A and B are incorrect because concurrent engineering is an integrated approach that performs tasks in parallel, not sequentially or non-integrated. Option D is also incorrect, as concurrent engineering is specifically designed to decrease development time. , stating that concurrent engineering reduces the lead time for product development, accurately describes a primary benefit of this methodology.

Therbligs are the basic elements of motion in work study. To match the given symbols with their meanings: "Search" is represented by P, "Inspect" by Q, "Rest for overcoming fatigue" by R, and "Avoidable delay" by S. Thus, the correct matching is P-4, Q-3, R-1, S-2.

Let's match the types of production layouts with their suitable production types.

First, understand the layouts:

• Process layout (P) groups similar machines or workstations together based on their function. Think of a workshop where all drilling machines are in one area, all welding machines in another, etc. This offers high flexibility to handle varied products.
• Product layout (Q) arranges machines or workstations in a sequential order according to the steps needed to produce a specific product. This is like an assembly line, designed for efficient, continuous flow.
• Fixed position layout (R) means the product itself remains stationary, and workers, materials, and equipment are brought to the product. This is used when the product is too large or heavy to move.

Next, understand the production types:

• Job production (1) involves making unique, custom products, often one at a time, based on specific customer orders.
• Batch production (2) involves producing goods in groups or batches. Each batch may be slightly different, but all units within a batch are identical. It's a compromise between mass production and job production.
• Mass production (3) is about manufacturing large quantities of standardized products using repetitive processes.

Now, let's match them:

• P (Process layout), with its flexibility, is best suited for 2 (Batch production). It can efficiently switch between different batches requiring varying sequences of operations.
• Q (Product layout), with its linear, specialized flow, is ideal for 3 (Mass production). It maximizes efficiency for producing large volumes of identical items.
• R (Fixed position layout) is essential for 1 (Job production), especially when the unique product is massive and cannot be moved, such as building a ship, an airplane, or a custom-designed bridge.

Therefore, the correct mapping is P-2, Q-3, R-1.

To find the determinant of matrix $A$, we first perform the matrix multiplication.
$A = \begin{bmatrix} 3 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times 4 \\ 2 \times 1 & 2 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & 12 \\ 2 & 8 \end{bmatrix}$
Now, using the formula $\det(A) = ad - bc$ for a $2 \times 2$ matrix, we calculate the determinant:
$\det(A) = (3)(8) - (12)(2) = 24 - 24 = 0$
Therefore, the value of $\det(A)$ is $0$.

This problem asks for the probability of drawing three non-defective gears without replacement. The key concept here is conditional probability combined with sampling without replacement, meaning the total number of items and the number of desired items change after each selection.

Understanding the Setup:
We have a total of 10 spur gears.

• Total gears ($N$): 10
• Defective gears ($D$): 1
• Non-defective gears ($N_D$): $10 - 1 = 9$
  We need to draw 3 gears ($n=3$) such that all are non-defective.

Step-by-Step Probability Calculation:
We will calculate the probability for each draw, considering the reduction in the total number of gears and non-defective gears after each successful draw.

1. Probability of the 1st gear being non-defective:
   There are 9 non-defective gears out of 10 total.
   $P(\text{1st is non-defective}) = \frac{9}{10}$

2. Probability of the 2nd gear being non-defective (given the 1st was non-defective):
   After taking one non-defective gear, 8 non-defective gears remain out of a total of 9 gears.
   $P(\text{2nd is non-defective | 1st was non-defective}) = \frac{8}{9}$

3. Probability of the 3rd gear being non-defective (given the first two were non-defective):
   After taking two non-defective gears, 7 non-defective gears remain out of a total of 8 gears.
   $P(\text{3rd is non-defective | 1st \& 2nd were non-defective}) = \frac{7}{8}$

Overall Probability:
To get the probability that all three are non-defective, we multiply these individual probabilities:
$P(\text{All 3 non-defective}) = \frac{9}{10} \times \frac{8}{9} \times \frac{7}{8}$
$P(\text{All 3 non-defective}) = \frac{7}{10}$

Final Result:
Converting the fraction to a decimal and rounding to two decimal places:
$\frac{7}{10} = 0.7$
Rounded to two decimal places, the probability is $0.70$.

To find the average speed of the piston, we first need to determine the total distance it travels and the time taken for that travel. We are given the crank speed $N = 60$ revolutions per minute, the crank radius $r = 30$ mm, and the connecting rod length $l = 120$ mm.

The stroke length of the piston is $2r$.
$\text{Stroke} = 2r = 2 \times 30 = 60 \text{ mm}$
For one full revolution of the crank, the piston completes one forward and one return stroke. So, the total distance traveled by the piston is twice the stroke length.
$\text{Total distance} = 2 \times \text{Stroke} = 2 \times 60 = 120 \text{ mm}$
The time taken for one revolution of the crank is $T = \frac{60}{N}$.
$T = \frac{60}{60} = 1 \text{ s}$
Finally, the average speed of the piston is the total distance traveled divided by the total time.
$\text{Average speed} = \frac{\text{Total distance}}{\text{Time}} = \frac{120}{1} = 120 \text{ mm/s}$
Thus, the average speed of the piston over one revolution of the crank, rounded off to the nearest integer, is $120$ mm/s.

To find the Reynolds number, we first need the average velocity. For laminar, fully developed flow in a circular pipe, the average velocity ($V_{avg}$) is half of the maximum axial velocity ($V_{max}$). Given $V_{max} = 0.5 \text{ m/s}$, we calculate $V_{avg} = \frac{0.5 \text{ m/s}}{2} = 0.25 \text{ m/s}$. Now, we use the Reynolds number formula $Re = \frac{V_{avg} D}{\nu}$.
Substituting the given values: $D = 8 \text{ mm} = 8 \times 10^{-3} \text{ m}$, $\nu = 1 \times 10^{-6} \text{ m}^2/\text{s}$, and our calculated $V_{avg} = 0.25 \text{ m/s}$.
$Re = \frac{(0.25 \text{ m/s}) \times (8 \times 10^{-3} \text{ m})}{1 \times 10^{-6} \text{ m}^2/\text{s}}$
$Re = \frac{2 \times 10^{-3}}{1 \times 10^{-6}}$
$Re = 2 \times 10^{3}$
$Re = 2000$
Rounding to the nearest integer, the Reynolds number is 2000.

The von Mises yield criterion states that a material starts to yield when its von Mises equivalent stress ($\sigma_v$) reaches the material's uniaxial yield stress ($\sigma_y$). The formula for calculating von Mises stress from principal stresses ($\sigma_1, \sigma_2, \sigma_3$) is:

$\sigma_v = \sqrt{\frac{1}{2} \$[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]\$}$

Given the principal stresses as $\sigma_1 = 200$ MPa, $\sigma_2 = 100$ MPa, and $\sigma_3 = 100$ MPa, we substitute these values into the formula:

\sigma_v = \sqrt{\frac{1}{2} \[(200 - 100)^2 + (100 - 100)^2 + (100 - 200)^2]$}$$\sigma_v = \sqrt{\frac{1}{2} $[(100)^2 + (0)^2 + (-100)^2]$}$$\sigma_v = \sqrt{\frac{1}{2} [10000 + 0 + 10000]}$$\sigma_v = \sqrt{\frac{1}{2} [20000]}$$\sigma_v = \sqrt{10000}$$\sigma_v = 100$ MPa

Since yielding begins when $\sigma_v = \sigma_y$, the yield stress ($\sigma_y$) of the material is $100$ MPa. Rounding to the nearest integer, the yield stress is 100 MPa.

To determine the shear angle ($\phi$) in orthogonal machining, we'll use the relationship between the chip thickness ratio ($r_t$) and the rake angle ($\alpha$). We are given the chip-compression ratio ($r = 1.25$) and a zero rake angle ($\alpha = 0^\circ$).

First, we calculate the chip thickness ratio ($r_t$) using the given chip-compression ratio:
$r_t = \frac{1}{r} = \frac{1}{1.25} = 0.8$
Now, we apply the shear angle formula:
$\tan(\phi) = \frac{r_t \cos(\alpha)}{1 - r_t \sin(\alpha)}$
Substitute the calculated $r_t = 0.8$ and the given $\alpha = 0^\circ$:
$\tan(\phi) = \frac{0.8 \times \cos(0^\circ)}{1 - 0.8 \times \sin(0^\circ)}$
Since $\cos(0^\circ) = 1$ and $\sin(0^\circ) = 0$, the equation simplifies to:
$\tan(\phi) = \frac{0.8 \times 1}{1 - 0.8 \times 0} = \frac{0.8}{1} = 0.8$
Finally, we find the shear angle $\phi$:
$\phi = \arctan(0.8) \approx 38.66^\circ$
Rounding to one decimal place, the shear angle is $38.7^\circ$.

When cutting a thread on a lathe, the core idea is to synchronize the workpiece's rotation with the lead screw's rotation. This synchronization ensures the cutting tool moves precisely to create the desired thread. The relationship is defined by the ratio of pitches being equal to the ratio of revolutions:

$\frac{\text{Pitch of Workpiece Thread} (p_w)}{\text{Pitch of Lead Screw} (p_{ls})} = \frac{\text{Revolutions of Lead Screw} (N_{ls})}{\text{Revolutions of Workpiece} (N_w)}$

We are given a workpiece thread pitch ($p_w$) of $2 \text{ mm}$ and a lead screw pitch ($p_{ls}$) of $4 \text{ mm}$. We need to find the lead screw revolutions ($N_{ls}$) for one revolution of the workpiece ($N_w = 1$). Plugging these values into the formula:

$\frac{2 \text{ mm}}{4 \text{ mm}} = \frac{N_{ls}}{1 \text{ revolution}}$

Solving this equation for $N_{ls}$:

$0.5 = N_{ls}$

Therefore, for one revolution of the workpiece, the lead screw must complete $0.5$ revolutions.

This question asks us to calculate the Root Mean Square (RMS) value of surface roughness, often denoted as $R_q$. This is a common way to quantify surface texture based on deviations from a datum line.

The formula for RMS roughness is:
$R_q = \sqrt{\frac{\sum_{i=1}^{N} y_i^2}{N}}$
Here, $y_i$ represents each absolute deviation, and $N$ is the total number of points.

Let's apply this to our given data: $10, 15, 12, 10, 13, 12, 20, 25 \ \mu \text{m}$.
The number of points, $N=8$.

1. Square each deviation ($y_i^2$):
   $10^2=100$, $15^2=225$, $12^2=144$, $10^2=100$, $13^2=169$, $12^2=144$, $20^2=400$, $25^2=625$.

2. **Sum the squared deviations (\sum y_i^2$):**$\sum y_i^2 = 100 + 225 + 144 + 100 + 169 + 144 + 400 + 625 = 1807$.

3. Calculate the mean of the squared deviations:
   $\frac{\sum y_i^2}{N} = \frac{1807}{8} = 225.875$.

4. Take the square root to find $R_q$:
   $R_q = \sqrt{225.875} \approx 15.0291346 \ \mu \text{m}$.

Rounding to one decimal place, the RMS value of surface roughness is $15.0 \ \mu \text{m}$. This falls within the specified answer range of 15 to 16.

Here, the machine works as long as at least one motor is operational. This means the machine fails only if both motors fail.
First, we find the individual failure probabilities:
Failure probability of motor 1 is $1 - R_1 = 1 - 0.90 = 0.10$.
Failure probability of motor 2 is $1 - R_2 = 1 - 0.70 = 0.30$.
The probability that both motors fail simultaneously is $P(\text{both fail}) = 0.10 \times 0.30 = 0.03$.
The overall reliability of the machine is the complement of this failure probability:
$R_{\text{machine}} = 1 - P(\text{both fail}) = 1 - 0.03 = 0.97$
Thus, the overall reliability of the machine, rounded off to two decimal places, is $0.97$.

The problem asks for the probability of no customer arriving in a 15-minute period, given an average arrival rate of 8 customers per hour and exponential interarrival times. Exponential interarrival times imply that the number of arrivals in a given interval follows a Poisson distribution.

First, let's find the average number of arrivals ($\mu$) in the 15-minute interval.
Given arrival rate $\lambda = 8$ customers/hour.
Time interval $t = 15$ minutes.
We convert the rate to customers per minute: $\lambda = \frac{8 \text{ customers}}{60 \text{ minutes}}$.
Then, $\mu = \lambda \times t = \left( \frac{8}{60} \right) \times 15 = \frac{120}{60} = 2$ customers.
So, on average, 2 customers arrive every 15 minutes.

Now, we use the Poisson probability formula for $P(X=k) = \frac{e^{-\mu} \mu^k}{k!}$ to find the probability of zero arrivals ($k=0$) in 15 minutes, with $\mu=2$.
$P(X=0) = \frac{e^{-2} \times 2^0}{0!}$
Since $2^0 = 1$ and $0! = 1$:
$P(X=0) = \frac{e^{-2} \times 1}{1} = e^{-2}$
Calculating the numerical value:
$e^{-2} \approx 0.135335$
Rounding to two decimal places, $P(X=0) \approx 0.14$.

To find the useful life of the machine using the straight-line depreciation method, we use the formula:
$\text{Useful Life (\in years)} = \frac{\text{Cost of Machine} - \text{Salvage Value}}{\text{Annual Depreciation Cost}}$
First, calculate the total amount that will be depreciated over the machine's life, which is the depreciable amount:
$\text{Depreciable Amount} = ₹65000 - ₹5000 = ₹60000$
Now, divide this depreciable amount by the annual depreciation cost to find the useful life:
$\text{Useful Life} = \frac{₹60000}{₹10000 \text{ per year}} = 6 \text{ years}$
Thus, the useful life of the machine is 6 years.

To find the standard deviation of the project duration, we first understand that only activities on the critical path contribute to its variance. The total project variance is the sum of the variances of these critical path activities.

The problem states the critical path consists of activities B, E, and G. From the given data, their individual variances are:

• Variance of Activity B: $1$ minute$^2$
• Variance of Activity E: $4$ minutes$^2$
• Variance of Activity G: $4$ minutes$^2$

Now, we sum these variances to get the total project variance:
$\text{Variance}_{\text{Project}} = \text{Variance}_B + \text{Variance}_E + \text{Variance}_G$
$\text{Variance}_{\text{Project}} = 1 + 4 + 4$
$\text{Variance}_{\text{Project}} = 9 \text{ minutes}^2$

Finally, the standard deviation of the project duration is the square root of the total project variance:
$\text{Standard Deviation}_{\text{Project}} = \sqrt{\text{Variance}_{\text{Project}}}$
$\text{Standard Deviation}_{\text{Project}} = \sqrt{9}$
$\text{Standard Deviation}_{\text{Project}} = 3 \text{ minutes}$
Rounding to two decimal places, the standard deviation is $3.00$ minutes.

Let's understand how squaring a matrix affects its eigenvalues and eigenvectors.

We start with the definition of an eigenvalue $\lambda$ and its corresponding eigenvector $v$ for a matrix $A$:
$Av = \lambda v$
Now, let's multiply both sides by $A$ from the left:
$A(Av) = A(\lambda v)$
Using the associative property on the left and factoring out the scalar $\lambda$ on the right, we get:
$A^2v = \lambda (Av)$
Substitute $Av = \lambda v$ back into the equation:
$A^2v = \lambda (\lambda v)$
Simplifying the right side gives us:
$A^2v = \lambda^2 v$
This equation shows that $v$ is still an eigenvector for $A^2$, but its corresponding eigenvalue is $\lambda^2$. Therefore, when a matrix is squared, its eigenvalues get squared, but its eigenvectors are retained.

This is a second-order linear homogeneous ordinary differential equation (ODE) with constant coefficients: $4\frac{d^2y}{dx^2}-4 \frac{dy}{dx} + y = 0$. To find the general solution, we form the characteristic equation by replacing $\frac{d^2y}{dx^2}$ with $r^2$ and $\frac{dy}{dx}$ with $r$, which gives $4r^2 - 4r + 1 = 0$. Factoring this quadratic equation yields $(2r - 1)^2 = 0$, indicating a repeated real root $r = \frac{1}{2}$. For a repeated real root $r$, the general solution of such an ODE is $y = (c_1 + c_2x)e^{rx}$. Substituting $r = \frac{1}{2}$ into this form, we get $y = (c_1 + c_2x)e^{\frac{1}{2}x}$ or equivalently $y = (c_1 + c_2x)e^{x/2}$.

This problem asks us to find the new sample size needed to make a confidence interval narrower, while keeping the same confidence level.

The core idea is how the margin of error ($E$) of a confidence interval relates to the sample size ($n$). For a normal distribution, the margin of error is given by the formula:
$E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$
Here, $Z_{\alpha/2}$ is a constant for a given confidence level, and $\sigma$ is the population standard deviation, which is also constant. This means the margin of error is inversely proportional to the square root of the sample size: $E \propto \frac{1}{\sqrt{n}}$.
Consequently, the sample size ($n$) is inversely proportional to the square of the margin of error ($E^2$): $n \propto \frac{1}{E^2}$.

Let's calculate the initial margin of error ($E_1$) and the desired margin of error ($E_2$):

1. Initial Scenario:

   • Confidence Interval: $[500, 700]$
   • Mean: $600$
   • Initial Margin of Error ($E_1$): Half the width of the interval.
     $E_1 = \frac{700 - 500}{2} = \frac{200}{2} = 100$
   • Initial Sample Size ($n_1$): $1000$

2. Desired Scenario:

   • Desired Confidence Interval: $[550, 650]$
   • Desired Margin of Error ($E_2$): Half the width of the new interval.
     $E_2 = \frac{650 - 550}{2} = \frac{100}{2} = 50$

Now, using the relationship $n \propto \frac{1}{E^2}$, we can set up a ratio for the sample sizes and margins of error:
$\frac{n_2}{n_1} = \left(\frac{E_1}{E_2}\right)^2$
Substitute the known values:
$\frac{n_2}{1000} = \left(\frac{100}{50}\right)^2$
$\frac{n_2}{1000} = (2)^2$
$\frac{n_2}{1000} = 4$
$n_2 = 4 \times 1000 = 4000$
To reduce the margin of error from $100$ to $50$ (i.e., by half), the sample size needs to increase by a factor of $2^2 = 4$. The new sample size required is $4000$.

The transformation structure of eutectoid steel depends heavily on the cooling rate from 100% austenite at 750 °C. For water quenching (P), which is the fastest cooling method, diffusion is significantly suppressed, leading to the formation of martensite below the martensite start temperature ($M_s$). Thus, P matches with 3 (Martensite). Oil quenching (Q) is slower than water quenching but faster than air cooling, resulting in very fine pearlite due to limited diffusion. So, Q matches with 4 (Very fine pearlite). Air cooling (R) is a moderate cooling rate, allowing enough time for fine pearlite to form, making R match with 2 (Fine pearlite). Furnace cooling (S) is the slowest method, providing ample time for atomic diffusion, which leads to coarse pearlite. Therefore, S matches with 1 (Coarse pearlite). The final matching is P-3, Q-4, R-2, S-1.

To find the force in member BC, we apply the method of joints at joint B. Since AC = BC, the truss forms an isosceles right triangle, meaning the angle between members AB and AC is $45^\circ$. We use the equilibrium conditions: $\Sigma F_x = 0$ (sum of horizontal forces) and $\Sigma F_y = 0$ (sum of vertical forces).

At joint B:

1. Vertical forces ($\Sigma F_y = 0$): The only member with a vertical component is AC. For equilibrium, $F_{AC} \sin(45^\circ) = 0$. This directly implies $F_{AC} = 0$.

2. Horizontal forces ($\Sigma F_x = 0$): The external force is 10 kN to the right. Member BC has a horizontal force $F_{BC}$ (assuming tensile). Member AC, which we found has zero force, would have a horizontal component $F_{AC} \cos(45^\circ)$.
   So, $F_{BC} - F_{AC} \cos(45^\circ) = 10 \, \text{kN}$.
   Substituting $F_{AC} = 0$: $F_{BC} - 0 = 10 \, \text{kN}$.
   Therefore, $F_{BC} = 10 \, \text{kN}$.

Since we assumed $F_{BC}$ to be tensile (pulling away from the joint) and got a positive value, the force in member BC is 10 kN (tensile).

This question asks us to match common powder metallurgy processing steps with their descriptions. Let's break down each step:

• P - Atomization: This is the method used to create metal powders from molten metal. So, it matches description 2: "A process for producing metal powder."
• Q - Sintering: This involves heating the compacted powder below its melting point to bond the particles together. Therefore, it matches description 3: "Metal powders are heated below their melting points to allow bonding."
• R - Compaction: In this step, loose metal powders are pressed into a specific shape using dies and pressure. This corresponds to description 1: "Blended powders are pressed into shapes using dies and pressure."
• S - Infiltration: Here, a lower melting point metal is introduced into the pores of a pre-sintered part to fill them, improving density and properties. This aligns with description 4: "A slug of low melting point metal is placed in contact with the sintered part and heated."

Based on this analysis, the correct matching is P-2, Q-3, R-1, S-4.

To determine the type of fit, we first calculate the maximum and minimum possible sizes for both the hole and the shaft.

1. Calculate Size Limits:

• Hole Dimensions:

  • Nominal Size = $25.000$ mm
  • Tolerance = $^{+0.002}_{-0.001}$ mm
  • Maximum Hole Size ($H_{max}$): $25.000 \, \text{mm} + 0.002 \, \text{mm} = 25.002 \, \text{mm}$
  • Minimum Hole Size ($H_{min}$): $25.000 \, \text{mm} - 0.001 \, \text{mm} = 24.999 \, \text{mm}$

• Shaft Dimensions:

  • Nominal Size = $25.000$ mm
  • Tolerance = $^{+0.001}_{-0.003}$ mm
  • Maximum Shaft Size ($S_{max}$): $25.000 \, \text{mm} + 0.001 \, \text{mm} = 25.001 \, \text{mm}$
  • Minimum Shaft Size ($S_{min}$): $25.000 \, \text{mm} - 0.003 \, \text{mm} = 24.997 \, \text{mm}$

2. Analyze Fit Conditions:

Let's compare these limits to define the fit:

| Component | Minimum Size | Maximum Size |
| :-------- | :----------------- | :----------------- |
| Hole | $24.999$ mm | $25.002$ mm |
| Shaft | $24.997$ mm | $25.001$ mm |

• Clearance Fit: Requires $H_{min} > S_{max}$. Here, $24.999 \, \text{mm}$ is not greater than $25.001 \, \text{mm}$. So, it's not purely a clearance fit.
• Interference Fit: Requires $S_{min} > H_{max}$. Here, $24.997 \, \text{mm}$ is not greater than $25.002 \, \text{mm}$. So, it's not purely an interference fit.
• Transition Fit: Occurs when the tolerance ranges overlap, meaning there can be either clearance or interference depending on the manufactured sizes. This condition is met if $H_{max} > S_{min}$ AND $S_{max} > H_{min}$.
  • Is $H_{max} > S_{min}$? $25.002 \, \text{mm} > 24.997 \, \text{mm}$ (True)
  • Is $S_{max} > H_{min}$? $25.001 \, \text{mm} > 24.999 \, \text{mm}$ (True)

Since both conditions for a transition fit are true, the tolerance ranges overlap, indicating that a transition fit is present.

3. Conclusion on Fit Type:

As the maximum possible hole size ($H_{max}$) is larger than the minimum possible shaft size ($S_{min}$), and the maximum possible shaft size ($S_{max}$) is larger than the minimum possible hole size ($H_{min}$), the tolerance ranges overlap. This overlap allows for scenarios where the hole is larger than the shaft (clearance) or where the shaft is larger than the hole (interference), which defines a Transition Fit.

To determine which products meet the Six Sigma standard, we first need to understand what it means. The Six Sigma standard requires a process to produce no more than 3.4 Defects Per Million Opportunities (DPMO). We'll calculate the DPMO for each product using the formula:

$\text{DPMO} = \frac{\text{Number of Defects}}{\text{Total Units \in Batch}} \times 10^6$

Given the batch size for each product is $2 \times 10^7$ units:

• For Product P: $\text{DPMO} = \frac{60}{2 \times 10^7} \times 10^6 = 3$. Since $3 \le 3.4$, Product P conforms to Six Sigma.
• For Product Q: $\text{DPMO} = \frac{71}{2 \times 10^7} \times 10^6 = 3.55$. Since $3.55 > 3.4$, Product Q does not conform.
• For Product R: $\text{DPMO} = \frac{80}{2 \times 10^7} \times 10^6 = 4$. Since $4 > 3.4$, Product R does not conform.
• For Product S: $\text{DPMO} = \frac{55}{2 \times 10^7} \times 10^6 = 2.75$. Since $2.75 \le 3.4$, Product S conforms to Six Sigma.

Based on these calculations, only products P and S conform to the Six Sigma standard.

This question tests your understanding of key product development processes. Let's match each process to its defining characteristic:

• P: Product synthesis matches with 2: Process related to design conceptualization. Product synthesis is about bringing together initial ideas and concepts to form the preliminary design.
• Q: Product simplification matches with 4: Process of reducing the number of parts without losing the functionalities. Simplification aims to make a product less complex while retaining its core functions, often by reducing components.
• R: Product analysis matches with 1: Process of conversion of conceptual design into engineering science based model. Product analysis involves a detailed examination to transform an abstract concept into a technically sound, engineering model.
• S: Product standardization matches with 3: Process of maintaining uniformity and consistency. Standardization ensures that products or parts are uniform and consistent in their design and manufacturing.

Based on these specific matches, the correct pairing is P-2, Q-4, R-1, and S-3.

Let's find the value of $\lim_{x\to1} \frac{x^{3-}3x+2}{x^{3-}x^{2-}x+1}$.

First, we substitute $x=1$ into the expression to check the form.
Numerator: $1^3 - 3(1) + 2 = 1 - 3 + 2 = 0$
Denominator: $1^3 - 1^2 - 1 + 1 = 1 - 1 - 1 + 1 = 0$
Since we have the indeterminate form $\frac{0}{0}$, we can use factorization.

Because $x=1$ makes both numerator and denominator zero, $(x-1)$ must be a factor of both.
Factor the Numerator: $x^3 - 3x + 2$
We can see that $x^3 - 3x + 2 = (x-1)(x^{2+}x-2)$.
Further factoring the quadratic part gives $x^{2+}x-2 = (x-1)(x+2)$.
So, the numerator becomes $(x-1)(x-1)(x+2) = (x-1)^2(x+2)$.

Factor the Denominator: $x^3 - x^2 - x + 1$
Group terms: $x^2(x-1) - 1(x-1) = (x^{2-}1)(x-1)$.
Using the difference of squares formula, $x^{2-}1 = (x-1)(x+1)$.
So, the denominator becomes $(x-1)(x+1)(x-1) = (x-1)^2(x+1)$.