GATE PI 2014

The question tests our understanding of verb tenses, specifically how to describe a chronic condition's ongoing symptoms. Alzheimer's disease is a chronic condition, and short-term memory loss is a continuous, characteristic symptom.

1. **Experienced (Simple Past Tense):** Implies the memory loss happened in the past and is now over, which is incorrect for an ongoing condition.
2. **Has experienced (Present Perfect Tense):** Suggests an action completed in the past with current relevance or continuing to the present. While it connects to the present, the simple present is more suitable for a defining, continuous characteristic.
3. **Is experiencing (Present Continuous Tense):** Indicates an action happening at the exact moment. The sentence describes a general fact about the disease, not just what's occurring instantly.
4. **Experiences (Simple Present Tense):** Used for facts, general truths, or characteristics. Since short-term memory loss is a defining and ongoing symptom for someone with Alzheimer's, "experiences" is the most appropriate verb. It conveys that this is a regular and expected occurrence. Additionally, "a person" is singular, and "experiences" ensures correct subject-verb agreement.

Therefore, "experiences" correctly conveys that short-term memory loss is a constant and characteristic feature for someone with Alzheimer's disease.

To complete the sentence "____________ is the key to their happiness; they are satisfied with what they have," we need a word that means being happy and satisfied with one's current situation. "Contentment" perfectly describes this state, aligning directly with "satisfied with what they have." In contrast, "ambition" and "hunger" imply wanting more, contradicting satisfaction, and "perseverance" relates to persistence, not this specific feeling of happiness with what one possesses.

The statement "As a woman, I have no country" is metaphorical, implying a deeper identity beyond national borders rather than a literal lack of citizenship. It suggests a primary connection or loyalty that transcends specific nations, highlighting a universal bond or shared experience among women globally.
* **Option A: Women have no country.** This is too literal and misinterprets the metaphorical meaning; it doesn't mean women are stateless.
* **Option B: Women are not citizens of any country.** Similar to Option A, this is a literal misinterpretation regarding legal status, not identity or allegiance.
* **Option C: Women's solidarity knows no national boundaries.** This option accurately captures the essence. "Solidarity" (unity based on common interests) transcending "national boundaries" aligns with the idea that a woman's connection to other women globally can be more profound than her connection to a specific country.
* **Option D: Women of all countries have equal legal rights.** While important, this focuses on legal status and rights, not the core meaning of shared identity and allegiance beyond nationalistic ties expressed in the original quote.
is the most fitting interpretation, emphasizing that shared experiences among women transcend national divisions.

The core concept here is understanding the relationship between the probability of an event and its recurrence interval. We can approximate this relationship using the formula $P = \frac{1}{T}$, where $P$ is the probability of an event occurring in a given year, and $T$ is the average time between successive occurrences (recurrence interval).

Given that the probability of an earthquake greater than Magnitude 6 occurring in the Garhwal Himalayas in any given year is $0.04$.
So, $P = 0.04$.
Using the formula, the recurrence interval $T = \frac{1}{P}$.
$T = \frac{1}{0.04}$
$T = 25$ years.
Therefore, the average time between successive occurrences of such earthquakes is 25 years.

To find the doubling time for the population, we can use the compound growth formula. The initial population (5 million) is not needed for this calculation, only the growth rate matters.

First, let's use the "Rule of 72" for a quick estimate.
Estimated Years to Double = $\frac{72}{\text{Growth Rate (\%)}}$
Estimated Years to Double = $\frac{72}{20} = 3.6$ years.

For a precise calculation, we use the compound growth formula: $P(t) = P_0 \times (1 + r)^t$, where $P_0$ is the initial population, $r$ is the annual growth rate (as a decimal), and $t$ is the number of years.
We want to find $t$ when $P(t) = 2 \times P_0$.
$2 \times P_0 = P_0 \times (1 + r)^t$
$2 = (1 + r)^t$
Given $r = 20\% = 0.20$:
$2 = (1 + 0.20)^t$
$2 = (1.2)^t$
To solve for $t$, take the natural logarithm ($\ln$) of both sides:
$\ln(2) = \ln((1.2)^t)$
$\ln(2) = t \times \ln(1.2)$
$t = \frac{\ln(2)}{\ln(1.2)}$
Using a calculator:
$t \approx \frac{0.6931}{0.1823}$
$t \approx 3.80$ years.

Both the estimate (3.6 years) and the precise calculation (3.80 years) fall within the range of 3-4 years.

We start by listing the initial facts about the ages of the four children: Som (S), Riaz (R), Shiv (Sh), and Ansu (A).

1. Som is younger than Riaz: $R > S$
2. Shiv is elder than Ansu: $Sh > A$
3. Ansu is the youngest in the group: This means $A < S$, $A < R$, and $A < Sh$.
   Our goal is to find the eldest child.

**Analyzing Statement 1 Alone:**
Statement 1 says: Shiv is younger than Riaz ($R > Sh$).
Let's combine this with our initial facts:
* We have $R > Sh$ (from Statement 1).
* We know $Sh > A$ (from initial facts).
* We also know $R > S$ (from initial facts).
Since $R > Sh$ and $Sh > A$, we can clearly establish the order $R > Sh > A$. This tells us Riaz is older than Shiv and Ansu.
Coupling this with $R > S$, we see that Riaz is also older than Som.
Therefore, Riaz is older than Shiv, Ansu, and Som. This means **Riaz is the eldest child**.
So, Statement 1 by itself is sufficient to determine the eldest child.

**Analyzing Statement 2 Alone:**
Statement 2 says: Shiv is elder than Som ($Sh > S$).
Let's combine this with our initial facts:
* We have $Sh > S$ (from Statement 2).
* We know $R > S$ (from initial facts).
* We know $Sh > A$ and $A$ is the youngest (from initial facts).
From these, we know both Riaz and Shiv are older than Som. Shiv is also older than Ansu. However, we do not know the age relationship between Riaz and Shiv.
Consider two possibilities:

1. If $R > Sh$, the order could be $R > Sh > S > A$. In this case, Riaz is eldest.
2. If $Sh > R$, the order could be $Sh > R > S > A$. In this case, Shiv is eldest.
   Since we cannot definitively determine who is the eldest between Riaz and Shiv using only Statement 2 and initial conditions, Statement 2 alone is not sufficient.

Based on this analysis, Statement 1 by itself determines the eldest child.

This paragraph highlights how our approach to data needs to change with the advent of big data. Historically, in the "information-deprived analog era" where data was "sparse," "every data point was critical," leading to an "obsession with exactness" to prevent bias. However, the text argues that applying this "conventional mindset of measurement" (exactness) to the "digital, connected world of the twenty-first century" and big data is to "miss a crucial point," implying that such exactitude is no longer as critical. Therefore, the main point is that exactitude is not critical in dealing with big data.

We need to find the ratio of revenue per kilogram for Item 1 to Item 4. First, let's convert the given total quantities and revenues to standard units:
Total quantity of exports $= 5 \text{ lakh tonnes } = 5 \times 10^5 \text{ tonnes } = 5 \times 10^5 \times 1000 \text{ kg } = 5 \times 10^8 \text{ kg}$.
Total revenues $= 250 \text{ crore rupees } = 250 \times 10^7 \text{ rupees}$.

Now, let's calculate the quantity and revenue for Item 1:
Quantity of Item 1 exported $= (11/100) \times (5 \times 10^8 \text{ kg}) = 55 \times 10^6 \text{ kg}$.
Revenue from Item 1 exported $= (12/100) \times (250 \times 10^7 \text{ rupees}) = 3 \times 10^8 \text{ rupees}$.
Revenue per kg for Item 1 $= \frac{3 \times 10^8}{55 \times 10^6} = \frac{300}{55} \text{ rupees/kg}$.

Next, for Item 4:
Quantity of Item 4 exported $= (22/100) \times (5 \times 10^8 \text{ kg}) = 11 \times 10^7 \text{ kg}$.
Revenue from Item 4 exported $= (6/100) \times (250 \times 10^7 \text{ rupees}) = 15 \times 10^7 \text{ rupees}$.
Revenue per kg for Item 4 $= \frac{15 \times 10^7}{11 \times 10^7} = \frac{15}{11} \text{ rupees/kg}$.

Finally, the required ratio is:
$\frac{\text{Revenue per kg for Item 1}}{\text{Revenue per kg for Item 4}} = \frac{\frac{300}{55}}{\frac{15}{11}} = \frac{300}{55} \times \frac{11}{15} = \frac{4}{1}$
Thus, the ratio is $4:1$.

This problem requires us to find the distance between two points, X and Q, given their relative positions to other points. We'll solve this using coordinate geometry by assigning $(x, y)$ coordinates to each point and then applying the distance formula.

Let's begin by placing Z at the origin, $(0, 0)$, to simplify calculations.

1. **Coordinates of Z:** Z is at $(0, 0)$.
2. **Coordinates of W:** W is $1$ km west of Z. Moving west decreases the x-coordinate. So, W is at $(-1, 0)$.
3. **Coordinates of Y:** Y is $1$ km southeast of Z. Southeast movement from $(0, 0)$ implies equal positive change in x and negative change in y. For a $1$ km distance, if the coordinates are $(a, -a)$, then $\sqrt{a^2 + (-a)^2} = 1$, which simplifies to $\sqrt{2a^2} = 1$, so $a = 1/\sqrt{2}$. Therefore, Y is at $(1/\sqrt{2}, -1/\sqrt{2})$.
4. **Coordinates of X:** X is $1$ km northeast of Y. From Y's position $(1/\sqrt{2}, -1/\sqrt{2})$, moving $1$ km northeast means adding $(1/\sqrt{2}, 1/\sqrt{2})$ to Y's coordinates.
   X's x-coordinate: $1/\sqrt{2} + 1/\sqrt{2} = 2/\sqrt{2} = \sqrt{2}$.
   X's y-coordinate: $-1/\sqrt{2} + 1/\sqrt{2} = 0$.
   So, X is at $(\sqrt{2}, 0)$.
5. **Coordinates of P:** P is $1$ km south of W. From W at $(-1, 0)$, moving south decreases the y-coordinate. So, P is at $(-1, 0 - 1) = (-1, -1)$.
6. **Coordinates of Q:** Q is $1$ km east of P. From P at $(-1, -1)$, moving east increases the x-coordinate. So, Q is at $(-1 + 1, -1) = (0, -1)$.

Now we have the coordinates:
X: $(\sqrt{2}, 0)$
Q: $(0, -1)$

To find the distance between X and Q, we use the distance formula:
$\text{Distance} = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$
Let $(x_1, y_1) = (\sqrt{2}, 0)$ and $(x_2, y_2) = (0, -1)$.
$\text{Distance XQ} = \sqrt{((0 - \sqrt{2})^2 + (-1 - 0)^2)}$
$\text{Distance XQ} = \sqrt{((-\sqrt{2})^2 + (-1)^2)}$
$\text{Distance XQ} = \sqrt{(2 + 1)}$
$\text{Distance XQ} = \sqrt{3}$
The distance between X and Q is $\sqrt{3}$ km.

This problem can be solved using Bayes' Theorem. Bayes' Theorem helps us calculate the probability of an event based on prior knowledge of conditions that might be related to the event. The formula is given by:

$P(E_i | A) = \frac{P(E_i) \times P(A | E_i)}{\sum_{i=1}^n P(E_i) \times P(A | E_i)}$

Here, we want to find the probability that an individual is actually HIV positive given that the test result is positive.

Let's define the events:
* Let $A$ be the event that the person is actually HIV+.
* Let $B$ be the event that the person is actually HIV-.
* Let $E$ be the event that the diagnostic kit shows a positive result.

From the problem statement:
* The probability that a person is HIV+ is $P(A) = 0.1$.
* The probability that a person is HIV- is $P(B) = 1 - P(A) = 1 - 0.1 = 0.9$.
* The kit correctly identifies HIV+ individuals 95% of the time, so $P(E | A) = 0.95$.
* The kit correctly identifies HIV- individuals 89% of the time. This means it incorrectly identifies HIV- individuals as positive 11% of the time. So, $P(E | B) = 1 - 0.89 = 0.11$.

We need to find the probability that the individual is actually positive given a positive test result, i.e., $P(A | E)$.

Using Bayes' Theorem:
$P(A | E) = \frac{P(A) \times P(E | A)}{[P(A) \times P(E | A)] + [P(B) \times P(E | B)]}$

Substituting the values:
$P(A | E) = \frac{0.1 \times 0.95}{(0.1 \times 0.95) + (0.9 \times 0.11)}$
$P(A | E) = \frac{0.095}{0.095 + 0.099}$
$P(A | E) = \frac{0.095}{0.194}$
$P(A | E) = \frac{95}{194}$
$P(A | E) \approx 0.4897$

Thus, the probability that the individual is actually positive, given a positive test result, is approximately $0.4897$.

To determine the nature of solutions for a system of linear equations $AX = B$, we analyze the determinant of the coefficient matrix $A$. If $\det(A) \neq 0$, the system has a unique solution. If $\det(A) = 0$, it could have no solution or infinitely many solutions.

First, let's form the coefficient matrix $A$ from the given equations:
$A = \begin{pmatrix} 1 & 2 & 4 \\ 1 & 3 & 1 \\ 3 & 2 & 3 \end{pmatrix}$
Now, we calculate the determinant of $A$:
$\det(A) = 1 \cdot \begin{vmatrix} 3 & 1 \\ 2 & 3 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 1 \\ 3 & 3 \end{vmatrix} + 4 \cdot \begin{vmatrix} 1 & 3 \\ 3 & 2 \end{vmatrix}$
$\det(A) = 1((3 \times 3) - (1 \times 2)) - 2((1 \times 3) - (1 \times 3)) + 4((1 \times 2) - (3 \times 3))$
$\det(A) = 1(9 - 2) - 2(3 - 3) + 4(2 - 9)$
$\det(A) = 1(7) - 2(0) + 4(-7)$
$\det(A) = 7 - 0 - 28 = -21$
Since $\det(A) = -21 \neq 0$, the system of linear equations has a unique solution.

To find the direction where the directional derivative of a scalar function $\phi$ is maximum, we need to calculate its gradient vector, $\nabla\phi$, at the given point. The gradient vector points in the direction of the maximum rate of increase of the function.

Given the function $\phi = 2xz - y^2$, its gradient $\nabla\phi$ is found by taking partial derivatives:
$\nabla\phi = \frac{\partial\phi}{\partial x}\mathbf{i} + \frac{\partial\phi}{\partial y}\mathbf{j} + \frac{\partial\phi}{\partial z}\mathbf{k}$
Calculating the partial derivatives:
$\frac{\partial\phi}{\partial x} = 2z$
$\frac{\partial\phi}{\partial y} = -2y$
$\frac{\partial\phi}{\partial z} = 2x$
So, the gradient vector is $\nabla\phi = (2z)\mathbf{i} - (2y)\mathbf{j} + (2x)\mathbf{k}$.

Next, we evaluate this gradient at the point (1, 3, 2) by substituting $x=1$, $y=3$, and $z=2$:
$x$-component: $2z = 2(2) = 4$
$y$-component: $-2y = -2(3) = -6$
$z$-component: $2x = 2(1) = 2$
Thus, the gradient vector at (1, 3, 2) is $\nabla\phi\bigg|_{(1,3,2)} = 4\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}$.

This vector, $4\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}$, represents the direction in which the directional derivative of $\phi$ is maximum at the point (1, 3, 2). Comparing this with the given options, it matches Option B.

The Biot number (\text{Bi}$) helps us understand if the internal temperature resistance of an object is significant compared to the external heat transfer resistance. It's defined as$\text{Bi} = \frac{h L_c}{K_{\text{solid}}}$. Here,$h$is the heat transfer coefficient of the fluid,$K_{\text{solid}}$is the thermal conductivity of the sphere, and$L_c$is the characteristic length. For a sphere, the characteristic length$L_c = \frac{\text{Volume}}{\text{Surface Area}} = \frac{\frac{4}{3}\pi R^3}{4\pi R^2} = \frac{R}{3}$.

Given: Diameter $D = 0.1 \text{ m}$, so radius $R = 0.05 \text{ m}$. Thermal conductivity of the sphere $K_{\text{sphere}} = 10 \text{ W/m-K}$. Heat transfer coefficient $h = 10 \text{ W/m}^2\text{-K}$.

First, calculate the characteristic length:
$L_c = \frac{R}{3} = \frac{0.05}{3} \text{ m}$.

Now, calculate the Biot number:
$\text{Bi} = \frac{h L_c}{K_{\text{sphere}}} = \frac{10 \times \frac{0.05}{3}}{10} = \frac{0.05}{3} = 0.01666... \approx 0.0167$.

The value of the Biot number is approximately $0.0167$.

Maintainability refers to how easily and quickly a system can be repaired. Let's analyze the options:
* **Downtime** is the total time a system is unavailable, which is a *result* of failures and repairs, not a measure of repair efficiency itself.
* **Mean Time Between Failure (MTBF)** measures reliability (how often a system fails), not how fast it can be fixed.
* **Mean Time To Repair (MTTR)** is the average time taken to fix a failed system and return it to operation. A lower MTTR directly indicates better maintainability.
* **System availability** is the overall uptime, often calculated as $\text{Availability} = \frac{\text{MTBF}}{\text{MTBF} + \text{MTTR}}$. While influenced by maintainability, it's a measure of operational readiness, not maintainability itself.

Thus, **Mean Time To Repair (MTTR)** is the direct quantitative measure of maintainability.

To analyze the cause and effect of product failure, we look for a technique that systematically links potential ways a product can fail (failure modes) to their underlying reasons (causes) and their impact (effects).

* **Quality Function Deployment (QFD)** translates customer needs into design specifications; it's preventative, not primarily for post-failure analysis.
* **Fault Tree Analysis (FTA)** deductively works backward from a top-level failure to identify root causes using Boolean logic, but its focus isn't as directly on the specific effects of each failure mode.
* **Value Analysis (VA)** optimizes cost and function; it doesn't systematically analyze failure causes and effects.
* **Failure Mode and Effect Analysis (FMEA)** is designed to identify potential failure modes, their causes, and their effects. It systematically documents how a product might fail, why it might fail, and what the consequences of that failure would be, making it ideal for cause-and-effect analysis.

Therefore, FMEA is the most suitable technique as it explicitly identifies failure modes, determines their effects, and pinpoints their causes, providing a comprehensive map for understanding and mitigating product failures.

The Product Life Cycle (PLC) has four stages: Introduction, Growth, Maturity, and Decline. Initially, product variety is low in the Introduction stage. It increases during the Growth and Maturity stages, peaking in Maturity as companies offer a wide range of variations to compete. However, as the product passes through ALL stages, the question focuses on the overall trend. In the Decline stage, demand drops, leading companies to streamline operations and discontinue unpopular or unprofitable variations. This results in a significant decrease in product variety. Therefore, considering the entire progression, product variety ultimately decreases, especially in the Decline stage due to product line rationalization.

To determine depreciation using the declining balance method, we use the formula for book value at the end of $n$ years: $BV_n = P(1-f)^n$, where $P$ is the purchasing value, $f$ is the fixed depreciation rate, and $BV_n$ is the scrap value.
Given $P = 100,000$, $n = 10$ years, and $BV_{10} = 20,000$.
$20,000 = 100,000(1-f)^{10}$
Solving for $f$, we get $f = 0.149$.
The book value at the end of the first year ($BV_1$) is $P(1-f) = 100,000(1-0.149) = 85,100$.
The depreciation during the first year is the purchasing value minus the book value at the end of the first year: $100,000 - 85,100 = 14,900$.

To determine the required number of observations ($n$) for estimating a proportion with a specified precision and confidence, we use the formula derived from the margin of error. The margin of error ($A$) is given by $A = Z\sqrt{\frac{{P\left( {1 - P} \right)}}{n}}$, where $Z$ is the standard normal variate (z-value) for the desired confidence level, $P$ is the estimated proportion of the activity, and $n$ is the number of observations. Rearranging this formula to solve for $n$, we get $n=Z^2\left[\frac{P(1-P)}{A^2}\right]$.

Given:
* The proportion of idle time (which represents $1-P$ in our formula if $P$ is the "occurrence" of idle time, or $P$ if $P$ is the "occurrence" of working time) was 0.16. Let's assume $P$ here refers to the *proportion of actual activity* rather than idle time, so $P = 1 - 0.16 = 0.84$. (Alternatively, if $P$ was idle time, then $P(1-P)$ would be $0.16(1-0.16)$). Let's stick to the given calculation where $P$ is the "percentage occurrence of an activity" that is *not* idle, so $P = 1 - \text{idle time} = 1 - 0.16 = 0.84$.
* The desired limit of accuracy ($A$) is 0.03.
* The confidence level is 95%, for which the z-value ($Z$) is 1.96.

Substituting these values into the formula:
$n = \frac{{{1.96^2}\times 0.84\left( {1 - 0.84} \right)}}{{{0.03^{2\;}}}} = 574$
Thus, the company needs 574 observations.

The core concept here is that the total time spent producing items (Net Production Time) is equal to the time it takes to produce one item (Cycle Time) multiplied by the total number of units produced.

Given that the plant operates for $2$ shifts of $8$ hours each, the Net Production Time in seconds is calculated as $2 \times 8 \times 3600 = 57600$ seconds. The total Number of Units Produced is $30000$.
Using the formula:
$Cycle\; time = \frac{Net \;Production\; time}{Number\;of\;Units\;Produced}$
We substitute the values:
$Cycle\; time = \frac{57600}{30000} = 1.92$
Therefore, the Cycle time is $1.92$ seconds, which can be rounded to $1.9$ to $2$ seconds.

To find the standard error of the mean, we first need to understand the relationship between the number of observations, the coefficient of variation, and the allowable percentage error. The number of observations, $N$, is given by the formula $N = {\left( {\frac{{{C_V}}}{\varepsilon }} \right)^2}$, where $\varepsilon$ is the allowable percentage error of the mean and $C_V$ is the coefficient of variation. The coefficient of variation is calculated as $C_v= \frac{\sigma}{\bar{x}} \times 100$, where $\sigma$ is the standard deviation and $\bar{x}$ is the mean.

Given $\bar{x} = 80$, $\sigma = 12$, and $N = 100$:

First, calculate the coefficient of variation ($C_V$):
${C_v} = \frac{12}{80} \times 100 = 15\%$

Next, use the formula for $N$ to find $\varepsilon$:
$100 = {\left( {\frac{{15}}{\varepsilon }} \right)^2}$
$\varepsilon = 1.5\%$

Finally, the standard error of the mean is calculated by multiplying the mean by the allowable percentage error (expressed as a decimal):
$\text{Standard error of mean} = 80 \times \frac{1.5}{100} = 1.2$

To determine the appropriate control chart, we first classify the data type. The diameter of a shaft is a continuous measurement, making it **variable data**, not attribute data (which involves counts or proportions).

Let's evaluate the given control chart options:
* **p chart:** Monitors the *proportion* of non-conforming items (attribute data). Not suitable for measuring diameter.
* **c chart:** Monitors the *number of defects* per unit (attribute data). Not suitable for measuring diameter.
* **$\bar{X}$ and R chart:** These charts are specifically designed for **variable data**. The $\bar{X}$ (X-bar) chart tracks the process average (mean diameter), while the R (Range) chart tracks the process variability (difference between max and min measurements within samples). Together, they monitor both the central tendency and the spread, which is essential for ensuring the shaft diameter stays within the tolerance of $0.02$.
* **U chart:** Monitors the *number of defects* per unit when the sample size varies (attribute data). Not suitable for measuring diameter.

Therefore, the $\bar{X}$ and R chart is the correct choice as it effectively monitors both the average diameter and its variation, ensuring the process is in statistical control for the given tolerance.

A Just-In-Time (JIT) manufacturing system focuses on minimizing waste and improving efficiency by receiving materials and producing goods only when needed. Let's analyze the given options in the context of JIT characteristics:

* **A. Reduction of lot sizes:** JIT systems strive to produce in smaller batches, ideally single units, to reduce inventory holding, shorten lead times, and increase flexibility. This is a core JIT principle.
* **C. Small but frequent deliveries:** To support minimal inventory, JIT relies on frequent, small deliveries from suppliers that precisely match production requirements. This aligns with JIT.
* **D. Higher productivity:** By eliminating waste (e.g., excess inventory, waiting time, defects) and streamlining processes, JIT systems inherently lead to increased overall productivity. This is a direct outcome of JIT implementation.
* **B. Efficient use of buffer inventory:** The fundamental principle of JIT is to minimize or eliminate all forms of inventory, including buffer inventory (extra stock held to cover variations or disruptions). JIT aims for a "zero inventory" state, relying on strong supplier relationships, high quality control, flexible workforce, and preventive maintenance to achieve smooth production flow without buffers. Therefore, "efficient use of buffer inventory" contradicts the JIT philosophy, which seeks to reduce, not just efficiently manage, buffer inventory.

For materials that follow Hooke's Law (meaning stress is proportional to strain within their elastic limit) and have consistent properties in all directions (isotropic), there's a specific relationship between their elastic constants: Young's Modulus (E), Shear Modulus (G), and Poisson's Ratio ($\mu$). Young's Modulus ($E$) measures stiffness in tension/compression, Shear Modulus ($G$) measures resistance to twisting or shearing, and Poisson's Ratio ($\mu$) describes how much a material thins out when stretched. The fundamental relationship connecting these three is given by:

$E = 2G(1 + \mu)$

Let's check the given options:
A. $E=\frac{G}{(2~+~\mu)}$ (Incorrect)
B. $E=\frac{2G}{(1~+~\mu)}$ (Incorrect)
C. $E = G (1 + \mu)$ (Incorrect, missing factor of 2)
D. $E = 2G (1 + \mu)$ (Correct)

Therefore, the correct relationship is $E = 2G(1 + \mu)$.

Stress relief annealing aims to reduce internal stresses in a metal alloy without significantly changing its microstructure. Option A, heating above recrystallization and air cooling, leads to recrystallization and is characteristic of normalizing or full annealing, not just stress relief. Option C, heating to recrystallization and rapid cooling, describes hardening or quenching, which introduces new stresses. Option D, repeated heating and cooling cycles, is used for grain refinement or phase transformations, not standard stress relief. Option B accurately describes stress relief annealing: heating the material below its recrystallization temperature allows internal stresses to relax through atomic movement without causing significant microstructural changes like recrystallization, holding at this temperature ensures sufficient time for stress reduction, and slow furnace cooling prevents the reintroduction of new stresses.

Let's break down the two main types of Computer-Aided Process Planning (CAPP) you need to know for GATE:

1. **Retrieval (or Variant) CAPP:** Imagine a library of existing manufacturing plans. This approach finds a plan for a similar part (using techniques like **Part coding using Group Technology (i)**) and then modifies it. It relies heavily on a **database of standard process plans for part families (iii)**.

2. **Generative CAPP:** This is like a smart engineer creating a brand new plan from scratch every time. It doesn't look for old plans. Instead, it builds the plan based on the part's exact design. Key to this are **Part feature recognition and extraction (ii)** (identifying holes, slots, etc.) and the **Geometric modelling of a part (iv)** (the detailed 3D design).

The question asks what Generative CAPP is *NOT* based on. From our understanding:

* **Part coding using Group Technology (i)** is a core concept of Retrieval CAPP, not Generative.
* **Part feature recognition and extraction (ii)** is absolutely essential for Generative CAPP to understand the part's design.
* **Database of standard process plans for part families (iii)** is the defining feature of Retrieval CAPP, exactly what Generative CAPP avoids.
* **Geometric modelling of a part (iv)** is the fundamental input for Generative CAPP to start building the plan.

Therefore, Generative CAPP is NOT based on **(i) Part coding using Group Technology** and **(iii) Database of standard process plans for part families**.

To produce metal powders, we need to break down bulk metal into fine particles. Atomization involves disintegrating molten metal using high-velocity fluid or centrifugal force, allowing the droplets to solidify into powder. Machining and grinding reduce solid metals into powder through mechanical processes like milling. Electrolysis is an electrochemical method where metal ions deposit onto a cathode and are then collected as powder. Compaction, on the other hand, is a process where already produced metal powder is pressed under high pressure into a solid shape (green compact). Therefore, compaction consolidates existing powder rather than producing it.

The Basic Length Unit (BLU) is the smallest possible increment of movement the machine can achieve. It's determined by the mechanical design. We are given that each angular step moves the table by one BLU. The angular step is determined by the stepper motor's resolution, not the pulse frequency. Therefore, the BLU is constant and does not depend on the pulse generator's frequency. Doubling the frequency of the pulse generator will only double the speed of the motor and thus the speed of the table movement, but the size of each step (BLU) remains unchanged. So, BLU will remain the same.

To reduce surface roughness in longitudinal turning, especially during a finishing pass, the most impactful parameter is the feed rate. The theoretical surface roughness ($R_t$) is often approximated by the formula $R_t \approx \frac{f^2}{8r_e}$, where $f$ is the feed rate and $r_e$ is the tool nose radius. This formula clearly shows that surface roughness is directly proportional to the square of the feed rate. Therefore, reducing the feed of the tool (e.g., from $0.3 \text{ mm/revolution}$ to a smaller value) will significantly reduce the surface roughness by minimizing the height of the cusps left on the workpiece. While spindle speed ($300 \text{ rpm}$) affects cutting conditions, its direct influence on surface roughness is generally less pronounced than that of the feed rate.

Chills are strategically placed inserts, typically metallic, with higher thermal conductivity than the mould material. Their primary function is to accelerate heat extraction from specific regions of the molten metal during casting.

* **Option A: Achieve directional solidification** is correct. Chills rapidly cool localized areas, causing solidification to begin there and progress towards the riser. This controlled, "directional" solidification ensures the riser feeds molten metal to compensate for shrinkage, preventing defects like porosity in the final casting.
* **Option B: Reduce the roughness of the top surface of the cast product** is incorrect. Surface roughness is mainly influenced by mould properties, pouring temperature, and gating design, not directly by chills.
* **Option C: Increase the solidification time** is incorrect. Chills, due to their high thermal conductivity, *decrease* the local solidification time by drawing heat away faster.
* **Option D: Reserve excess molten metal** is incorrect. Risers are used to reserve excess molten metal to compensate for shrinkage, whereas chills promote and direct solidification.

Therefore, the key role of chills is to promote directional solidification, leading to a sound and defect-free casting.

A mandrel is a tool used in metalworking to shape or support material, especially for hollow items like tubes. Let's analyze the options for where a moving mandrel is typically used:

* **Wire Drawing:** This process reduces wire diameter. While specialized cases might use mandrels (e.g., hollow wire), a *moving mandrel* is not a standard or defining feature of general wire drawing.
* **Tube Drawing:** This process reduces the diameter and/or wall thickness of a tube. Here, a mandrel is often used inside the tube to control the internal diameter and ensure uniform wall thickness. Types include fixed, plug, and floating mandrels. A **floating mandrel** is a key example of a *moving mandrel*, as it isn't fixed but moves freely with the tube, optimizing its position due to the forces involved. This significantly aids in achieving precise inner dimensions and smooth finishes.
* **Metal Cutting:** This involves material removal. Mandrels might hold workpieces, but they aren't directly involved in the cutting action itself, and a *moving mandrel* is not a typical tool here.
* **Forging:** This shapes metal using compressive forces. While mandrels can create holes in specific forging operations, a *moving mandrel* is not a characteristic technique of forging in general.

Therefore, the application where a moving mandrel is a significant and characteristic component is **Tube drawing**, where it is crucial for maintaining dimensional accuracy and surface quality of the tube's interior during the operation.

Brazing and soldering are classified as heterogeneous joining methods because they utilize a filler metal whose chemical composition is distinctly different from that of the base metals being joined. In both processes, a molten filler metal (solder for soldering, brazing filler metal for brazing) is used to create a bond through capillary action without melting the base metals. Soldering employs filler metals with a melting point typically below $450^\circ\text{C}$ ($842^\circ\text{F}$), while brazing uses filler metals with a melting point above $450^\circ\text{C}$ ($842^\circ\text{F}$) but below the base metals' melting point. Heterogeneous joining specifically refers to cases where the filler material has a different composition than the base metals, unlike homogeneous joining (where filler is similar to base metal or base metals are fused directly) or autogenous joining (where base metals are fused without filler or with metallurgically identical filler).

Let's break down these common mechanical mechanisms and where you'll find them in machine tools, which is a frequently tested concept in GATE.

**P. Quick return $\rightarrow$ 2. Shaping:** The quick return mechanism is a defining characteristic of shaping machines. Its purpose is to make the non-cutting return stroke of the ram much faster than the cutting stroke, which saves time and improves machining efficiency.

**Q. Apron $\rightarrow$ 1. Lathe:** You'll find the apron as part of the carriage on a lathe. This component houses the gears and controls that enable the automatic longitudinal and cross feeds, crucial for operations like turning and threading.

**R. Intermittent indexing $\rightarrow$ 4. Milling:** Intermittent indexing involves rotating the workpiece by a specific, fixed amount after each cutting pass. This is widely used in milling machines, often with a dividing head, to create evenly spaced features like gear teeth, splines, or holes.

**S. Differential mechanism $\rightarrow$ 3. Gear hobbing:** The differential mechanism is vital in gear hobbing, especially when cutting helical gears. It ensures the precise synchronization between the workpiece's rotation and the hob's rotation, allowing for the accurate generation of the helix angle.

Based on these associations:
* P matches with 2 (Shaping)
* Q matches with 1 (Lathe)
* R matches with 4 (Milling)
* S matches with 3 (Gear hobbing)

This combination leads to the correct mapping.

Reaming is a machining operation that uses a reamer to improve the quality of an existing hole, typically after drilling. Its primary purpose is to enlarge the hole slightly to achieve a precise, final dimension, improve the surface finish, and enhance the accuracy of the hole's diameter and roundness. Unlike drilling, which creates the initial hole, reaming removes a very small amount of material to refine the hole's precision and surface finish. Therefore, reaming is fundamentally a finishing process applied to existing holes to meet tight tolerances and quality standards.

The part shown has features like square and circular holes, which often require precise and complex machining, especially if the material is hard or needs a good surface finish. Electrical Discharge Machining (EDM) is a non-traditional machining process ideal for such tasks on conductive materials, regardless of their hardness, because it uses electrical sparks instead of direct tool contact.

There are two main types of EDM relevant here:

1. **Die-sinking EDM:** This process uses a shaped electrode (the "die") to create a negative impression (like the circular holes) in the workpiece. The electrode doesn't touch the workpiece; sparks between them erode material.
2. **CNC Wire-cut EDM:** This uses a thin, continuously moving wire as an electrode to precisely cut complex shapes (like the square holes or intricate profiles) following a programmed path.

Both die-sinking EDM for features like circular holes and CNC wire-cut EDM for features like square holes are capable of producing the complex and precise geometries shown in the figure, particularly if the material is difficult to machine by conventional methods. Therefore, using both **Die-sinking and CNC Wire-cut EDM processes** is the most suitable combination.

In open die forging of a circular disc, the exponential decay of normal stress from the center to the periphery indicates how friction behaves. If the disc were frictionless, the normal stress would be uniform. If there was only sticking friction, the stress distribution would be different from exponential decay. The presence of exponential decay suggests that the material is always sliding relative to the platens, meaning the shear stress at the interface is always less than the shear strength required for sticking. This implies that there is no sticking friction anywhere on the flat face of the disc. Therefore, the correct answer is that there is no sticking friction anywhere on the flat face of the disc.

To find $\nabla^2\varphi$ for the given scalar function $\varphi = 2x^3y^2z^4$, we need to compute the Laplacian. The Laplacian operator in Cartesian coordinates is defined as the sum of the second partial derivatives with respect to each spatial variable:
$\nabla^2\varphi = \frac{\partial^2\varphi}{\partial x^2} + \frac{\partial^2\varphi}{\partial y^2} + \frac{\partial^2\varphi}{\partial z^2}$
Let's calculate each term step-by-step.

**1. Calculate $\frac{\partial^2\varphi}{\partial x^2}$:**
First, we find the partial derivative with respect to $x$:
$\frac{\partial\varphi}{\partial x} = \frac{\partial}{\partial x} (2x^3y^2z^4) = 2y^2z^4 \cdot (3x^2) = 6x^2y^2z^4$
Next, we differentiate this result again with respect to $x$:
$\frac{\partial^2\varphi}{\partial x^2} = \frac{\partial}{\partial x} (6x^2y^2z^4) = 6y^2z^4 \cdot (2x) = 12xy^2z^4$

**2. Calculate $\frac{\partial^2\varphi}{\partial y^2}$:**
First, we find the partial derivative with respect to $y$:
$\frac{\partial\varphi}{\partial y} = \frac{\partial}{\partial y} (2x^3y^2z^4) = 2x^3z^4 \cdot (2y) = 4x^3yz^4$
Next, we differentiate this result again with respect to $y$:
$\frac{\partial^2\varphi}{\partial y^2} = \frac{\partial}{\partial y} (4x^3yz^4) = 4x^3z^4 \cdot (1) = 4x^3z^4$

**3. Calculate $\frac{\partial^2\varphi}{\partial z^2}$:**
First, we find the partial derivative with respect to $z$:
$\frac{\partial\varphi}{\partial z} = \frac{\partial}{\partial z} (2x^3y^2z^4) = 2x^3y^2 \cdot (4z^3) = 8x^3y^2z^3$
Next, we differentiate this result again with respect to $z$:
$\frac{\partial^2\varphi}{\partial z^2} = \frac{\partial}{\partial z} (8x^3y^2z^3) = 8x^3y^2 \cdot (3z^2) = 24x^3y^2z^2$

**4. Sum the partial derivatives:**
Finally, we add the three second partial derivatives to get the Laplacian:
$\nabla^2\varphi = 12xy^2z^4 + 4x^3z^4 + 24x^3y^2z^2$
Comparing this result with the given options, we find it matches Option C.

The Simpson's 1/3rd rule is used to approximate definite integrals by fitting quadratic curves through sets of three points. The general formula for Simpson's 1/3rd rule is:
\int_{x_0}^{x_n} ydx = \frac{h}{3}\left\{ {\left( {{y_0} + {y_n} \right) + 4\left( {{y_1} + {y_3} + {y_5} + \ldots } \right) + 2\left( {{y_2} + {y_4} + {y_6} + \ldots } \right)} \right\}}
where $h = \frac{{b - a}}{n}$. For a polynomial function of degree less than 3, this rule gives the exact integral value.

Given the data:
$x$: 1, 3, 5
$y$: 2, 6, 4
Here, $a = 1$, $b = 5$, and $n = 2$.
First, calculate $h$: $h = \frac{{b - a}}{n} = \frac{{5 - 1}}{2} = 2$.
From the given data, we have $y_0 = 2$, $y_1 = 6$, and $y_2 = 4$.
Applying Simpson's 1/3rd rule for three points ($n=2$), the formula simplifies to:
$\int ydx = \frac{h}{3}\left[ {{y_0} + 4{y_1} + {y_2}} \right]$
Substitute the values:
$\int ydx = \frac{{2}}{3}\left[ {2+ 4 (6)+4} \right] = 20$

Let's solve the equation $\sin(x) = x^2$ using the Newton-Raphson method. This method helps us find roots of an equation $f(x)=0$ iteratively. The formula for each iteration is:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

First, we need to define our function $f(x)$. From $\sin(x) = x^2$, we get $f(x) = x^2 - \sin(x)$.
Next, we find its derivative, $f'(x) = 2x - \cos(x)$.

We are given the initial guess $x_0 = 1$. Remember, all trigonometric functions operate on radians, so ensure your calculator is in radian mode or convert degrees appropriately (as noted in the "Mistake Points").

**First Iteration (n = 0):**

Substitute $x_0 = 1$ into $f(x)$ and $f'(x)$:
$f(x_0) = f(1) = (1)^2 - \sin(1) = 1 - \sin(1) \approx 0.159$
$f'(x_0) = f'(1) = 2(1) - \cos(1) = 2 - \cos(1) \approx 1.459$

Now, apply the Newton-Raphson formula to find $x_1$:
$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{0.159}{1.459} \approx 1 - 0.10897875 \approx 0.891$

**Second Iteration (n = 1):**

Now we use $x_1 = 0.891$ for the next iteration.
Substitute $x_1 = 0.891$ into $f(x)$ and $f'(x)$:
$f(x_1) = f(0.891) = (0.891)^2 - \sin(0.891) \approx 0.793881 - 0.77818 \approx 0.015701 \approx 0.0161$ (Rounding difference in original explanation)
$f'(x_1) = f'(0.891) = 2(0.891) - \cos(0.891) = 1.782 - 0.6288 \approx 1.1532 \approx 1.153$

Finally, apply the Newton-Raphson formula to find $x_2$:
$x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 0.891 - \frac{0.0161}{1.153} \approx 0.891 - 0.01396357 \approx 0.877036 \approx 0.87$

The value of $x$ after 2 iterations is approximately $0.87$.

To find the change in entropy when a substance's temperature changes, we use the formula for total entropy change: $\Delta S = mC\ln \left( \frac{T_2}{T_1} \right)$. Here, mass of water $m = 2 \text{ kg}$, specific heat $C = 4.18 \text{ kJ/kg-K}$. The initial temperature $T_1 = 20^\circ\text{C} = 273 + 20 = 293 \text{ K}$ and the final temperature $T_2 = 40^\circ\text{C} = 273 + 40 = 313 \text{ K}$. Substituting these values, the change in entropy is $\Delta S = 2 \times 4.18 \times \ln \left( \frac{313}{293} \right) = 0.552 \text{ kJ/K}$.

The work done in a closed system for a polytropic process is given by $W = \int Pdv$.

Given initial pressure ($P_1$) = $500 \text{ kPa}$, initial volume ($V_1$) = $0.75 \text{ m}^3$. The final volume ($V_2$) is double the initial volume, so $V_2 = 2 \times 0.75 = 1.5 \text{ m}^3$. The process follows $P \propto V^{-1.2}$, which means $PV^{1.2} = C$ (a constant).

First, we find the final pressure ($P_2$) using the polytropic relation:
$P_1V_1^{1.2} = P_2V_2^{1.2}$
$500 \times 0.75^{1.2} = P_2 \times 1.5^{1.2}$
$P_2 = 217.6376 \text{ kPa}$

Now, we calculate the work done for a polytropic process using the formula:
$W = \frac{P_1V_1 - P_2V_2}{n - 1}$
$W = \frac{500 \times 0.75 - 217.64 \times 1.5}{1.2 - 1}$
$W = 242.7 \text{ kJ}$

In steady-state heat flow, the heat transfer rate through both bars is equal. We apply Fourier's Law of Heat Conduction, $Q = - kA\frac{{dT}}{{dx}}$, where $Q$ is the heat transfer rate, $k$ is thermal conductivity, $A$ is the cross-sectional area, and $\frac{dT}{dx}$ is the temperature gradient.

For steady-state heat flow, $Q_P = Q_Q$. Let $t_f$ be the junction temperature. Since the bars are geometrically identical, their cross-sectional areas ($A$) and lengths ($\delta$) are the same.
Thus, $k_1\frac{{t_1 -t_f}}{{\delta_1}} = k_2\frac{{t_f -t_2}}{{\delta_2}}$.
As $A_1 = A_2$ and $\delta_1 = \delta_2$, this simplifies to $k_1(t_1 - t_f) = k_2(t_f - t_2)$.

Given: $t_1 = 40^\circ C$, $t_2 = 10^\circ C$, $k_1 = 5 \text{ W/m-K}$, $k_2 = 10 \text{ W/m-K}$.
Substituting these values:
$5 \times (40 - t_f) = 10 \times (t_f - 10)$
$40 - t_f = 2(t_f - 10)$
$40 - t_f = 2t_f - 20$
$60 = 3t_f$
$t_f = 20^\circ C$.

To determine the cooling time, we'll use the Lumped Parameter Analysis, which applies when internal conductive resistance is much smaller than surface convective resistance. This means the sphere's temperature is uniform throughout at any given time. The temperature distribution is described by:

$\frac{{{T}_{i}}-{{T}_{\infty }}}{T-{{T}_{\infty }}~}=~{{e}^{\left( \frac{hA}{\rho V{{C}_{p}}} \right)t}}$

Rearranging this for time, we get:

$\frac{hA}{\rho V{{C}_{p}}}t=\ln\left( \frac{{{T}_{i}}-{{T}_{\infty }}}{T-{{T}_{\infty }}~} \right)$

Here, $T_i$ is the initial temperature, $T$ is the temperature at time $t$, $T_{\infty }$ is the ambient fluid temperature, $h$ is the heat transfer coefficient, $A$ is the surface area, $\rho$ is the density, $V$ is the volume, and $C_p$ is the specific heat. For a sphere, the ratio of volume to surface area is $\frac{V}{A} = \frac{r}{3}$.

Given values are: $C_p = 314 \text{ J/kgK}$, $h = 10 \text{ W/m}^2\text{K}$, $m = 1 \text{ kg}$, $A = 0.0314 \text{ m}^2$, $T_i = 50^\circ\text{C}$, $T_{\infty } = 10^\circ\text{C}$, and $T = 20^\circ\text{C}$.

First, calculate the radius $r$ and density $\rho$:
From the surface area of a sphere, $A = 4\pi r^2 = 0.0314 \text{ m}^2$, we get $r = 0.05 \text{ m}$.
The density is $\rho = \frac{mass}{volume} = \frac{m}{\frac{4}{3}\pi r^3} = \frac{1}{\frac{4}{3}\pi(0.05)^3} = 1909.86 \text{ kg/m}^3$.

Now, substitute these values into the rearranged equation:

$\ln\left( {\frac{{50 - 10}}{{20 - 10}}} \right) = \left( {\frac{{10 \times 3 }}{{0.05 \times 1909.86 \times 314}}} \right) \times t$

Solving for $t$:

$t= \frac{{1.386 \times \;0.05 \times 1909.86 \times 314}}{{10 \times 3}}$

$t = 1385.29 \text{ seconds}$

Therefore, the time taken for the sphere to cool down to $20^\circ\text{C}$ is $1385.29 \text{ seconds}$.

To find the principal stresses and their direction, we use Mohr's Circle concepts: the center, radius, and the angle for the principal plane.

Given: $\sigma_x = 5 \text{ kN/mm}^2$, $\sigma_y = 3 \text{ kN/mm}^2$, and $\tau = 1 \text{ kN/mm}^2$.

First, calculate the Center of Mohr's Circle:
$\text{Center} = \frac{{\sigma_x + \sigma_y}}{2} = \frac{{5 + 3}}{2} = 4 \text{ kN/mm}^2$
Next, calculate the Radius of Mohr's Circle, which represents the maximum shear stress:
$\text{Radius} (R) = \sqrt{{{\left( {\frac{{\sigma_x - \sigma_y}}{2}} \right)}^2} + {{\left( {\tau_{xy}} \right)}^2}} = \sqrt {{{\left( {\frac{{5 - 3}}{2}} \right)}^2} + {{\left( {1} \right)}^2}} = \sqrt{{(1)^2 + (1)^2}} = \sqrt{2} \approx 1.414 \text{ kN/mm}^2$
The principal stresses $\sigma_1$ and $\sigma_2$ are then:
$\sigma_1 = \text{Center} + \text{Radius} = 4 + 1.414 = 5.414 \text{ kN/mm}^2$
$\sigma_2 = \text{Center} - \text{Radius} = 4 - 1.414 = 2.586 \text{ kN/mm}^2$
Finally, to find the angle $\phi$ (or $\theta_p$) of the principal plane, use the formula:
$\tan 2\theta_p = \frac{{2\tau_{xy}}}{{(\sigma_x - \sigma_y)}} = \frac{{2 \times 1}}{{5 - 3}} = \frac{2}{2} = 1$
$2\theta_p = \arctan(1) = 45^\circ$
$\theta_p = 22.5^\circ$
Since we consider the rotation in the clockwise direction for $\tau_{xy}$ being positive relative to $(\sigma_x - \sigma_y)$ when $\sigma_x > \sigma_y$, the angle is negative: $\phi = -22.5^\circ$.
Thus, the principal stresses are $5.41 \text{ kN/mm}^2$, $2.58 \text{ kN/mm}^2$, and the direction is $-22.5^\circ$.