GATE PI 2015

To complete the sentence "Apparent lifelessness ___________ dormant life," we need a word that describes how a seemingly inactive state can contain or hide life that is temporarily inactive. "Harbours" (A) perfectly fits this, meaning to shelter or hold something often hidden, implying that the lifeless appearance conceals active life. In contrast, "lead to" (B) suggests causation, which doesn't accurately describe dormant life existing within apparent lifelessness. While "supports" (C) indicates providing necessary conditions, "harbours" better conveys the idea of containing it. "Affects" (D) implies influence, which is too general and doesn't capture the essence of concealment. Therefore, "harbours" is the most precise choice, as it highlights that the inactive life is present within the seemingly lifeless state.

The question requires us to select the idiom that best describes someone feeling uncomfortable or out of place. Let's analyze the options: "Dog out of herd" and "Sheep from the heap" are not standard English idioms. While "Bird from the flock" describes separation, it doesn't convey the strong sense of discomfort or awkwardness implied by being completely outside one's natural environment. "Fish out of water" is a common idiom meaning a person is in an unfamiliar situation, causing them to feel uncomfortable or awkward, just as a fish struggles outside of water. This perfectly fits the scenario of a town boy in a sleepy village, suggesting he felt awkward and out of place. Therefore, the completed sentence is: "That boy from the town was a fish out of water in the sleepy village."

This question tests your understanding of two commonly confused word pairs: "elude" vs. "allude" and "elusive" vs. "allusive." Let's break down each word's meaning and then analyze the options.

• Elude (verb): To cleverly escape or avoid something. Think of a tricky villain trying to get away.

• Elusive (adjective): Describes something that is difficult to find, catch, or achieve. If something "eludes" you, it is "elusive."

• Allude (verb): To indirectly hint at or refer to something. You're not saying it directly, but you're dropping clues.

• Allusive (adjective): Describes something that contains or makes indirect references. If someone "alludes" to something, their statement is "allusive."

Now, let's examine each option:

• A. "When the teacher eludes to different authors, he is being elusive." This is incorrect. If the teacher is hinting, the correct verb is "alludes to." Also, "elusive" means hard to find, not making hints. The teacher would be "allusive" if making indirect references.

• B. "When the thief keeps eluding the police, he is being elusive." This is correct. "Eluding the police" means the thief is escaping or avoiding capture. Since the thief is difficult for the police to catch, he is accurately described as "elusive." The adjective "elusive" perfectly fits the action of "eluding."

• C. "Matters that are difficult to understand, identify or remember are allusive." This is incorrect. Matters that are hard to understand or remember are typically "elusive" or "obscure," not "allusive." "Allusive" relates to indirect references.

• D. "Mirages can be allusive, but a better way to express them is illusory." This is incorrect. While a mirage might in a very specific, stretched context be considered "allusive" (as in indirectly suggesting something), its core characteristic is being an illusion, hence "illusory." A mirage is something that is difficult to grasp physically, making "elusive" a more fitting description than "allusive" for its general nature.

correctly uses "eluding" (implied action) and "elusive" (descriptive adjective) in context.

This problem tests our understanding of logical deductions, specifically the transitive property of inequalities.

Let's represent the ages: Tanya's age as $T$, Eric's age as $E$, and Cliff's age as $C$.

1. Statement 1: "Tanya is older than Eric" translates to $T > E$.
2. Statement 2: "Cliff is older than Tanya" translates to $C > T$.
3. Statement 3: "Eric is older than Cliff" translates to $E > C$.

We are given that Statement 1 ($T > E$) and Statement 2 ($C > T$) are true.

Combining these two true statements using the transitive property (if $A > B$ and $B > C$, then $A > C$):
Since $C > T$ and $T > E$, it logically follows that $C > E$. This means Cliff is older than Eric.

Now, let's compare this derived conclusion ($C > E$) with the third statement ($E > C$). The third statement claims that Eric is older than Cliff, which is the direct opposite of our logical deduction that Cliff is older than Eric. Therefore, the third statement ($E > C$) must be false if the first two statements are true.

This problem asks for the total number of matches when 5 teams play each other exactly once. Since the order of teams in a match doesn't matter (Team A vs. Team B is the same as Team B vs. Team A), this is a combination problem.

Method 1: Using the Combinations Formula

We need to choose 2 teams for each match from a total of 5 teams. The formula for combinations is:
$C(n, k) = \frac{n!}{k!(n-k)!}$
Here, $n = 5$ (total teams) and $k = 2$ (teams per match).
$C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)}$
$C(5, 2) = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10$
Therefore, 10 matches will be held.

Method 2: Step-by-Step Listing

Alternatively, we can list the matches systematically:

• Team 1 plays 4 other teams.
• Team 2 plays 3 new teams (since it already played Team 1).
• Team 3 plays 2 new teams.
• Team 4 plays 1 new team.
• Team 5 has already played all other teams.

Total matches = $4 + 3 + 2 + 1 = 10$.

Both methods confirm that 10 matches are required.

Okay, let's break down this sentence correction problem, which is very common in GATE and other competitive exams. It tests your understanding of parts of speech, specifically adjectives and adverbs.

The original sentence is: "Increased productivity necessary reflects greater efforts made by the employees." We need to fix the underlined part, "Increased productivity necessary."

1. Analyze the original underlined part: "Increased productivity necessary" acts as the subject of the verb "reflects." "Increased productivity" is a noun phrase, which is fine. However, "necessary" is an adjective. An adjective describes a noun. Here, it's placed awkwardly before the verb "reflects" and doesn't correctly modify "increased productivity" in this context. We need something that tells how the productivity reflects efforts.

2. Evaluate Option A: "Increase in productivity necessary": This changes "Increased productivity" to "Increase in productivity," which is grammatically sound as a noun phrase. But it still keeps the adjective "necessary" in an incorrect position. The sentence would read: "Increase in productivity necessary reflects..." This is still grammatically awkward because "necessary" isn't modifying the right word or isn't in the right form.

3. Evaluate Option B: "Increase productivity is necessary": This option creates a clause "Increase productivity is necessary." While this clause is grammatically correct on its own, if we try to insert it, the sentence becomes: "Increase productivity is necessary reflects greater efforts..." This doesn't make sense syntactically. The phrase "Increase productivity is necessary" cannot function as the subject for the verb "reflects." It fundamentally changes the structure and meaning of the original sentence.

4. Evaluate Option C: "Increase in productivity necessarily": Here, "Increase in productivity" serves as the subject. Crucially, "necessarily" is an adverb. Adverbs modify verbs, adjectives, or other adverbs. In this case, "necessarily" correctly modifies the verb "reflects," telling us how the productivity reflects the efforts - it reflects them in a necessary way. The sentence becomes: "Increase in productivity necessarily reflects greater efforts made by the employees." This structure is grammatically sound and clearly conveys that the reflection of efforts is a necessary consequence of increased productivity.

5. Evaluate Option D: "No improvement required": This is incorrect because, as established, the original sentence has a grammatical error with the placement and form of "necessary."

is the most appropriate replacement because "necessarily" is an adverb that correctly modifies the verb "reflects," making the sentence grammatically sound and preserving the intended meaning.

Let's break down these statements and conclusions like we would for a GATE exam. We need to determine if the conclusions are always true based on the statements.

Understanding the Statements:

• Statement I: No manager is a leader. This means the group of 'Managers' and the group of 'Leaders' have absolutely no overlap. Think of them as two separate, distinct circles in a Venn diagram. We can represent this as $M \cap L = \emptyset$.
• Statement II: All leaders are executives. This tells us that the entire group of 'Leaders' is completely contained within the larger group of 'Executives'. Every single person who is a leader is also an executive. In a Venn diagram, the 'Leaders' circle is entirely inside the 'Executives' circle. This is represented as $L \subseteq E$.

Analyzing the Conclusions:

• Conclusion I: No manager is an executive. This claims that 'Managers' and 'Executives' have no overlap at all.
• Conclusion II: No executive is a manager. This is just another way of saying 'Managers' and 'Executives' have no overlap. If one is true, the other is true.

Logical Deduction (and why the conclusions might be wrong):
We know Managers are separate from Leaders ($M \cap L = \emptyset$), and Leaders are a part of Executives ($L \subseteq E$). However, this doesn't tell us if Managers overlap with the rest of the Executives (those who are not Leaders).

Consider this counter-example scenario:

• Managers (M): {Alice, Bob}
• Leaders (L): {Charlie, David}
• Executives (E): {Charlie, David, Alice, Eve}

Let's check if our statements hold true in this scenario:

• Statement I: No manager is a leader. (Alice, Bob are not Charlie, David). True.
• Statement II: All leaders are executives. (Charlie, David are in Charlie, David, Alice, Eve). True.

Now, let's check the conclusions with this scenario:

• Conclusion I: No manager is an executive. Alice is a manager AND an executive in our example. This makes the conclusion FALSE.
• Conclusion II: No executive is a manager. Since Alice is both an executive and a manager, this conclusion is also FALSE.

Since we found a scenario where the statements are true but both conclusions are false, neither conclusion logically follows. Therefore, neither conclusion I nor II follows.

This problem leverages the concept of how areas of triangles with a common vertex or sharing a base relate to the ratio of their bases or heights. Since $\angle Q$ is a right angle, $\triangle PQR$ is a right-angled triangle. Its area can be calculated as $\frac{1}{2} \times PQ \times QR$.

Let the area of $\triangle PQR$ be $A$. We are given the ratios $PS:QS = 3:1$ and $RT:QT = 5:2$.
From $PS:QS = 3:1$, we have $PQ = PS + QS = 3QS + QS = 4QS$, so $QS = \frac{1}{4} PQ$.
Thus, $\frac{SQ}{PQ} = \frac{1}{4}$.
From $RT:QT = 5:2$, we have $QR = QT + TR = QT + 5QT/2 = 7QT/2$, so $QT = \frac{2}{7} QR$.
Thus, $\frac{QT}{QR} = \frac{2}{7}$.

The area of $\triangle SQT$ is given as $20 \text{ cm}^2$. Since $\angle Q$ is a right angle, $\triangle SQT$ is also a right-angled triangle.
$\text{Area of } \triangle SQT = \frac{1}{2} \times SQ \times QT$
Substitute the expressions for $SQ$ and $QT$ in terms of $PQ$ and $QR$:
$\text{Area of } \triangle SQT = \frac{1}{2} \times \left( \frac{1}{4} PQ \right) \times \left( \frac{2}{7} QR \right)$
$= \frac{1}{4} \times \frac{2}{7} \times \left( \frac{1}{2} \times PQ \times QR \right)$
Notice that $\left( \frac{1}{2} \times PQ \times QR \right)$ is the area of $\triangle PQR$.
$= \frac{1}{14} \times \text{Area of } \triangle PQR$
We are given $\text{Area of } \triangle SQT = 20 \text{ cm}^2$.
$20 \text{ cm}^2 = \frac{1}{14} \times A$
Therefore, $A = 14 \times 20 = 280 \text{ cm}^2$.

We need to construct right triangle PQR where the right angle is at P, and line PR is parallel to the x-axis. The coordinates $(x, y)$ of P, Q, and R must be integers within the ranges $-4 \le x \le 5$ and $6 \le y \le 16$.

First, let's find the total number of possible integer values for x and y.
For x-coordinates, the integers are $-4, -3, \dots, 4, 5$. The count is $5 - (-4) + 1 = 10$.
For y-coordinates, the integers are $6, 7, \dots, 15, 16$. The count is $16 - 6 + 1 = 11$.

Since the right angle is at P and PR is parallel to the x-axis, it means PR is a horizontal line segment. For PQ to be perpendicular to PR, PQ must be a vertical line segment.
Let $P = (x_p, y_p)$.
Since PR is parallel to the x-axis, R must have the same y-coordinate as P. So, $R = (x_r, y_p)$, where $x_r \ne x_p$.
Since PQ is perpendicular to PR, PQ must be vertical, meaning Q must have the same x-coordinate as P. So, $Q = (x_p, y_q)$, where $y_q \ne y_p$.

Now, let's calculate the number of ways to choose the coordinates for P, R, and Q:

1. Choosing Point P $(x_p, y_p)$:
   There are 10 choices for $x_p$ and 11 choices for $y_p$.
   Number of ways to choose P = $10 \times 11 = 110$.

2. Choosing Point R $(x_r, y_p)$:
   The y-coordinate $y_p$ is fixed by the choice of P.
   For $x_r$, there are 10 possible x-coordinates, but $x_r$ cannot be equal to $x_p$.
   Number of choices for $x_r = 10 - 1 = 9$.

3. Choosing Point Q $(x_p, y_q)$:
   The x-coordinate $x_p$ is fixed by the choice of P.
   For $y_q$, there are 11 possible y-coordinates, but $y_q$ cannot be equal to $y_p$.
   Number of choices for $y_q = 11 - 1 = 10$.

To find the total number of different triangles, we multiply the number of ways to choose P by the number of valid choices for $x_r$ and $y_q$.
Total Triangles = (Number of ways to choose P) $\times$ (Number of choices for $x_r$) $\times$ (Number of choices for $y_q$)
Total Triangles = $(10 \times 11) \times 9 \times 10$
Total Triangles = $110 \times 90$
Total Triangles = $9,900$.

To determine independence or dependence of events, we first list all possible outcomes for three coin tosses.
The total sample space $S$ has $2^3 = 8$ equally likely outcomes: $S = \{\text{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\}$. Each outcome has a probability of $1/8$.

Now, let's define the given events and their probabilities:

• Event X: Head occurs in each of the first two tosses.
  • Outcomes for X: $\{\text{HHH, HHT}\}$
  • $P(X) = \frac{2}{8} = \frac{1}{4}$
• Event Y: A tail occurs on the third toss.
  • Outcomes for Y: $\{\text{HHT, HTT, THT, TTT}\}$
  • $P(Y) = \frac{4}{8} = \frac{1}{2}$
• Event Z: Two tails occur in three tosses.
  • Outcomes for Z: $\{\text{HTT, THT, TTH}\}$
  • $P(Z) = \frac{3}{8}$

Two events A and B are independent if $P(A \cap B) = P(A) \times P(B)$. Otherwise, they are dependent.

Let's check the independence for each pair mentioned in the options:

1. Checking X and Y:

   • Intersection $X \cap Y$: The outcomes common to both X and Y.
     • $X \cap Y = \{\text{HHT}\}$
     • $P(X \cap Y) = \frac{1}{8}$
   • Product of individual probabilities:
     • $P(X) \times P(Y) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$
   • Since $P(X \cap Y) = P(X) \times P(Y)$, events X and Y are independent. is FALSE.

2. Checking Y and Z:

   • Intersection $Y \cap Z$: The outcomes common to both Y and Z.
     • $Y \cap Z = \{\text{HTT, THT}\}$
     • $P(Y \cap Z) = \frac{2}{8} = \frac{1}{4}$
   • Product of individual probabilities:
     • $P(Y) \times P(Z) = \frac{1}{2} \times \frac{3}{8} = \frac{3}{16}$
   • Since $P(Y \cap Z) = \frac{1}{4}$ and $P(Y) \times P(Z) = \frac{3}{16}$, we see that $P(Y \cap Z) \neq P(Y) \times P(Z)$. Therefore, events Y and Z are dependent. is TRUE and option C is FALSE.

3. Checking X and Z:

   • Intersection $X \cap Z$: The outcomes common to both X and Z.
     • $X \cap Z = \{\}$ (empty set, as there are no common outcomes)
     • $P(X \cap Z) = \frac{0}{8} = 0$
   • Product of individual probabilities:
     • $P(X) \times P(Z) = \frac{1}{4} \times \frac{3}{8} = \frac{3}{32}$
   • Since $P(X \cap Z) = 0$ and $P(X) \times P(Z) = \frac{3}{32}$, we see that $P(X \cap Z) \neq P(X) \times P(Z)$. Therefore, events X and Z are dependent. is FALSE.

Simpson's rule is a numerical integration technique used to approximate definite integrals. Unlike simpler methods, Simpson's rule approximates the function within each pair of subintervals using a parabola, which is a quadratic curve defined by a second-degree polynomial. By fitting a parabola through three points (the two endpoints and the midpoint of the interval), it integrates the area under this quadratic approximation. This approach generally yields more accurate results than methods like the Trapezoidal rule (which uses straight lines) or the Rectangle rule (which uses constants).

Work is done when a force causes an object to move in the direction of the force. When a constant force $\vec{f}$ displaces an object by a distance $\vec{d}$, and the angle between the force and displacement is $\theta$, the work done ($W$) is given by the dot product of the force and displacement vectors. This is expressed as $W = \vec{f}.\vec{d} = fd\cos\theta$. Therefore, the work done by a force is the scalar product of the force and the displacement. The SI unit for work is Joule (J).

To find the probability of getting a correct sequence, we need to divide the number of favorable outcomes (correct sequences) by the total number of possible outcomes (all distinct sequences).

Given:

• Total number of distinct components = 5
• Number of correct sequential assembly ways = 2

The total number of possible sequences for 5 distinct components is given by $5!$:
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
The number of favorable sequences (correct assembly ways) is 2.

Using the probability formula, $P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$:
$P(\text{Correct Sequence}) = \frac{2}{5!}$

The function is defined as $f(x) = x + x + x + \dots$ ($x$ times). This means $x$ is added to itself $x$ times, which can be written as the summation: $f(x) = \sum_{i=1}^{x} x$ This simplifies to $f(x) = x × x = x^2$. However, the crucial part is the phrase "($x$ \times )". In mathematics, performing an operation a certain "number of \times " implies that this number must be a positive integer. We can add something 3 \times or 5 \times , but not $\sqrt{2}$ \times or $-4$ \times \in this context. Therefore, for the expression "$x + x + x + \dots$ ($x$ \times )" to be meaningful, $x$ must be a positive integer ($x \in \{1, 2, 3, \dots\}$).

• Option A (All real values of $x$): Incorrect. While $x^2$ is defined for all real numbers, the operation "added $x$ times" requires $x$ to be a positive integer.
• Option B (Only at positive integer values of $x$): Correct. This aligns with the requirement that the number of times an operation is performed must be a positive integer.
• Option C (Only at negative integer values of $x$): Incorrect. Adding something '$-3$ times' is not a standard interpretation for such a definition.
• Option D (Only at rational values of $x$): Incorrect. Rational numbers include fractions and negative numbers, which are not allowed for the "($x$ times)" constraint.

A perfectly plastic material initially requires stress to deform, after which it deforms continuously under constant stress. Examining the provided stress-strain curves, material S shows an initial elastic region where stress increases with strain, followed by a region where it deforms at a constant stress value, characteristic of a perfectly plastic material. The initial elastic region, though not explicitly mentioned in the original definition, is typically present before the perfectly plastic flow begins.

To solve this problem, we need to understand the definitions of true stress and engineering stress.
True stress ($\sigma_T$) is defined as the applied force divided by the deformed (current) cross-sectional area:
$\sigma_T = \frac{\text{Applied Force}}{\text{Deformed Area}}$
Engineering stress ($\sigma_E$) is defined as the applied force divided by the initial cross-sectional area:
$\sigma_E = \frac{\text{Applied Force}}{\text{Initial Area}}$
Given:

• True stress ($\sigma_T$) = 700 MPa
• Initial diameter ($D_0$) = 13 mm
• Diameter at fracture (deformed diameter, $D_f$) = 10 mm

From the true stress definition, we can find the applied force:
$\text{Applied Force} = \sigma_T \times \text{Deformed Area}$
$\text{Applied Force} = 700 \times \text{Deformed Area}$
Now, substitute this into the engineering stress formula:
$\sigma_E = \frac{700 \times \text{Deformed Area}}{\text{Initial Area}}$
The area of a circular cross-section is given by $\frac{\pi}{4} \times (\text{diameter})^2$.
$\sigma_E = \frac{700 \times \frac{\pi}{4} \times (10 \text{ mm})^2}{\frac{\pi}{4} \times (13 \text{ mm})^2}$
$\sigma_E = \frac{700 \times 10^2}{13^2}$
$\sigma_E = \frac{700 \times 100}{169}$
$\sigma_E = \frac{70000}{169} \approx 414.20 \text{ MPa}$

To find the maximum shear stress ($\tau_{max}$) when principal stresses are given, we use the formula:

$\tau_{max}=\rm{Max}\left[\frac{\sigma_1\;-\;\sigma_2}{2}\;,\;\frac{\sigma_2\;-\;\sigma_3}{2}\;,\;\frac{\sigma_3\;-\;\sigma_1}{2}\right]$

Given principal stress values are $\sigma_1 = 120$ MPa, $\sigma_2 = -50$ MPa, and $\sigma_3 = 10$ MPa. Let's calculate the absolute differences divided by two:

$\frac{\sigma_1\;-\;\sigma_2}{2}=\frac{120\;-\;(-50)}{2}=85\;MPa$
$\frac{\sigma_2\;-\;\sigma_3}{2}=\frac{-50\;-\;10}{2}=-30\;MPa$
$\frac{\sigma_3\;-\;\sigma_1}{2}=\frac{10\;-\;120}{2}=-55\;MPa$

The maximum shear stress is the largest of these absolute values, which is $85$ MPa.

This question tests your understanding of the various components in a foundry molding system and their specific roles in creating a defect-free casting. Let's break down each component:

• P. Sprue: The sprue is the vertical channel that guides the molten metal downwards from the pouring basin into the rest of the gating system. So, its function is '2. feeds molten metal from the pouring basin'.
• Q. Riser: As the molten metal solidifies, it shrinks. The riser acts as a reservoir of hot molten metal, feeding the solidifying casting to compensate for this liquid shrinkage and prevent voids. Thus, its function is '4. supplies molten metal to compensate for liquid shrinkage'.
• R. Gate: The gate is the final, narrow passageway through which molten metal enters the actual mold cavity. It's designed to control the flow rate and direction, ensuring a smooth fill. Therefore, its function is '1. regulates the flow of molten metal into the mold cavity'.
• S. Pouring basin: This is the initial entry point where molten metal is poured from the ladle. It serves as a temporary holding area, helping to reduce turbulence and allow some impurities to settle before the metal enters the sprue. Hence, its function is '3. acts as a reservoir for molten metal'.

Matching these, we get: P-2, Q-4, R-1, S-3.

In strip rolling, the relative velocity ($V_{rel}$) of the strip with respect to the roller is defined as $V_{rel} = V_S - V_R$, where $V_S$ is the strip velocity and $V_R$ is the roller surface velocity. At the entry plane, the strip moves slower than the roller ($V_S < V_R$), so $V_{rel} = V_S - V_R$ is negative. At the exit plane, due to reduction in thickness, the strip moves faster than the roller ($V_S > V_R$), making $V_{rel} = V_S - V_R$ positive.

To find the strain hardening coefficient ($n$), we use the relationship between maximum reduction per pass ($R$) and the strain hardening coefficient: $R = 1 - e^{-(n+1)}$
Given the maximum reduction $R = 77\% = 0.77$, we substitute this into the equation:
$0.77 = 1 - e^{-(n+1)}$
Rearranging the terms, we get:
$e^{-(n+1)} = 1 - 0.77 = 0.23$
Taking the natural logarithm ($\ln$) on both sides:
$-(n+1) = \ln(0.23) \approx -1.4696$
Solving for $n$:
$-n = -1.4696 + 1$
$n = 0.4696$
Thus, the strain hardening coefficient of the material is approximately $0.4696$.

Let's break down Built-Up Edge (BUE) formation, which is when workpiece material sticks to the cutting tool. BUE occurs due to high pressure and temperature at the tool-chip interface, essentially welding the workpiece material to the tool. While BUE can sometimes protect the tool, it usually causes poor surface finish and dimensional errors. Conditions that reduce temperature, pressure, or friction generally decrease BUE.

1. Low Cutting Speeds: At lower cutting speeds, the chip stays in contact with the tool for a longer time. While temperature might be lower initially, the pressure is often higher. This combination of higher pressure and sufficient contact time allows for more adhesion and diffusion of workpiece material onto the tool. Therefore, BUE formation generally increases at low cutting speeds.
2. Large Positive Rake Angle: A larger positive rake angle reduces the shear angle and cutting forces. This leads to lower temperatures and less friction at the tool-chip interface, thereby decreasing BUE formation.
3. Sharper Tool: A sharper tool reduces deformation and stress concentration, resulting in lower cutting forces and less friction. This, in turn, decreases BUE formation.
4. Cutting Fluid: Cutting fluids cool the interface and reduce friction. By inhibiting the adhesion process, cutting fluids significantly decrease BUE formation.

The question asks for the condition under which BUE formation decreases, EXCEPT for one. Options B, C, and D all describe conditions that decrease BUE. Option A (low cutting speeds) is the exception because it increases BUE formation.

During the turning of mild steel, heat is generated primarily due to plastic deformation and friction. The tool and chip interface experiences the maximum temperature because it combines severe plastic deformation as the chip forms and intense sliding friction between the chip and the tool's rake face. While the primary deformation zone also generates heat, the additional friction at the interface makes it the hottest spot.

Let's analyze each statement regarding the grinding process to find the INCORRECT one.

Statement A: Finer abrasive grains produce better surface finish.
This is CORRECT. Finer grains remove smaller amounts of material with each pass (smaller chip formation and shallower depth of cut per grain), leading to a smoother, improved surface finish.

Statement B: Abrasive grains tend to fracture frequently during the grinding process.
This is CORRECT. Grinding involves significant forces on the abrasive grains, causing them to fracture. This "self-sharpening" action exposes new, sharp cutting edges, which is crucial for maintaining the wheel's cutting ability over time.

Statement C: Specific energy in grinding is higher than that in turning.
This is CORRECT. Specific energy is the energy required to remove a unit volume of material. Grinding typically involves removing very small chips with high friction and high deformation rates, which demands more energy per unit volume compared to turning operations where larger chips are formed.

Statement D: Cutting speed in grinding process is much lower than that in face milling.
This is INCORRECT. Let's compare typical cutting speed ranges:

• Grinding: $1500 - 5000+\text{ m/min}$ (or $30 - 100+\text{ m/s}$)
• Face Milling: $50 - 250\text{ m/min}$ (or $0.8 - 4+\text{ m/s}$)

As you can see, grinding wheels operate at significantly higher peripheral speeds (cutting speeds) than face milling operations. Therefore, the statement claiming grinding speed is much lower than face milling is false.

When multiple components form an assembly, their individual dimension variations (tolerances) contribute to the overall assembly's variation. For $n$ components, each with a dimension following a normal distribution and a tolerance $T_i$, the overall assembly dimension $L_a$ has a tolerance $T_a$. Since individual variations are independent and normally distributed, we use the Root Sum of Squares (RSS) method to calculate the combined tolerance. This method provides a realistic estimate, unlike simple addition (worst-case).

The formula for the RSS method is:
$T_a = \sqrt{\sum_{i=1}^n T_i^2}$
Here, $T_a$ is the overall assembly tolerance, and $T_i$ is the tolerance of the $i$-th component. The nominal overall dimension $L_a$ represents the target size and is not directly incorporated into the calculation of the tolerance $T_a$ itself.

Let's look at why other options are incorrect:
Option A ($T_a = L_a \sqrt {\sum_{i=1}^n \ T_i ^2}$): This option incorrectly multiplies by $L_a$. Tolerances are in units of length (e.g., mm). Multiplying by $L_a$ (also in mm) would result in units of mm$^2$, which is dimensionally inconsistent for a tolerance.
Option C ($T_a = \sqrt {L_a\sum_{i=1}^n \ T_i ^2}$): This option places $L_a$ under the square root. This would lead to incorrect units (e.g., mm$^{3/2}$) and does not align with standard statistical tolerancing methods.
Option D ($T_a = L_a + \sqrt {\sum_{i=1}^n \ T_i ^2}$): This option adds $L_a$ to the RSS value. While $L_a + T_a$ might define an upper limit, this formula does not correctly calculate the tolerance $T_a$. Simple addition of tolerances (\sum T_i$) represents the worst-case scenario, not the statistical combination for normally distributed dimensions.

Electrical Discharge Machining (EDM) removes material through electrical sparks. The Material Removal Rate (MRR) is how fast this happens. Let's break down which factors don't influence it.

1. Hardness of workpiece material: This does not significantly influence MRR. EDM melts and vaporizes material, a process less dependent on mechanical hardness than traditional machining.
2. Melting temperature of workpiece material: This does significantly influence MRR. Lower melting points mean less energy is needed to melt and vaporize, leading to higher MRR.
3. Hardness of tool material: This does not directly influence workpiece MRR. The tool's electrical conductivity and resistance to erosion are more critical than its mechanical hardness.
4. Discharge current and frequency: These do significantly influence MRR. Higher current means more energy per spark and larger craters; higher frequency means more sparks per second, both increasing MRR.

Therefore, the factors that do not influence the material removal rate in EDM are the hardness of the workpiece material (i) and the hardness of the tool material (iii).

Computer-Aided Process Planning (CAPP) uses computer software to automate manufacturing process planning. The question asks about determining a process sequence for any part design without predefined standard plans. Let's look at the different CAPP approaches:

• Retrieval Type Process Planning: This method searches a database for a similar, already-planned part and retrieves its existing plan, which might then be modified. It relies on a library of existing plans.
• Variant Type Process Planning: Similar to retrieval, this type also uses existing plans but systematically modifies them for a new part that is sufficiently similar. It often uses coding and classification to group parts.
• Generative Type Process Planning: This is the key. It does not rely on predefined or existing standard plans. Instead, it automatically creates a process plan directly from the part's design data (like CAD models), manufacturing rules, and available machinery using algorithms and expert knowledge. This directly matches the scenario of determining a process sequence without predefined standard plans.
• Group Technology (GT) Based Process Planning: GT is a classification method for grouping similar parts, often used to support variant or retrieval CAPP systems by finding similar plans. It's a classification technique, not a plan generation method itself.

Therefore, determining the process sequence for any part design without predefined standard plans is known as Generative type process planning.

The rake angle is crucial for efficient material removal and chip formation. For a twist drill, the helix angle is the angle formed by the spiral flute and the drill's axis, and it directly determines the effective rake angle at the cutting edge. A steeper helix angle generally leads to a higher effective rake angle, improving cutting efficiency and chip evacuation. Other angles like lip relief angle (prevents rubbing), chisel edge angle (aids penetration), and point angle (affects penetration rate) do not directly determine the rake angle.

A line balancing problem aims to assign tasks to workstations on an assembly line to equalize the workload, minimizing idle time and bottlenecks for smooth production flow. This problem is inherently solved in the context of a product layout because a product layout (assembly line or flow shop) arranges workstations sequentially for specific product manufacturing steps. Tasks in a product layout are repetitive and standardized, allowing for efficient distribution across workstations to match the desired production rate (takt time). In contrast, a process layout groups similar functions, offering flexibility but lacking a direct sequential flow for line balancing. A fixed position layout involves bringing resources to a stationary product, making line balancing inapplicable, while a production schedule dictates timing rather than the physical arrangement optimized by line balancing.

The assignment problem seeks to pair resources to tasks (e.g., workers to jobs) to optimize an objective, often minimizing total cost. A balanced assignment problem occurs when the number of resources equals the number of tasks. A binary solution means each assignment decision is either 1 (assigned) or 0 (not assigned), represented in a matrix where each row and column has exactly one '1'.

The key reason for the binary nature of the solution to a balanced assignment problem is the uni-modularity property, specifically total uni-modularity, of its constraint matrix. A matrix is totally unimodular if the determinant of every square submatrix is 0, 1, or -1. When a linear programming problem has a totally unimodular constraint matrix and its right-hand side vector consists of integers (as is the case for the assignment problem), the optimal solution obtained from the linear program (even if we initially relax the integer requirement) will automatically be integer-valued. For the assignment problem, this structure specifically forces these integer solutions to be 0s and 1s, leading to a binary solution. This is why algorithms like the Hungarian algorithm guarantee an exact, binary assignment.

Other options are incorrect because:

• Linear formulation alone does not guarantee binary solutions; many linear programs have fractional solutions.
• A non-empty feasible region only means a solution exists, not that it will be binary.
• Approximation algorithms are for problems where exact solutions are too complex; the assignment problem's structure allows for efficient exact, binary solutions.

Material Requirements Planning (MRP) manages inventory and plans production by determining what materials are needed, how much, and when.

• Bill of Materials (BOM): This is a fundamental input, listing all raw materials, components, and quantities required for a product. MRP uses BOM to calculate material needs.
• Inventory Level: Crucial for MRP, this indicates current stock. MRP uses it to calculate net requirements, determining what more needs to be ordered or produced.
• Production Schedule: MRP integrates with this schedule (often from a Master Production Schedule), which outlines products, quantities, and deadlines, driving the procurement and production planning of components.
• Material Price: While material price impacts purchasing and cost, it is not a direct input or calculation component within the core logic of MRP. MRP focuses on material needs ('what', 'how much', 'when'), while pricing is typically handled by purchasing/finance, which uses MRP's output.

Therefore, material price is not included in Material Requirements Planning.

The Ishikawa diagram, also known as a cause-and-effect or fishbone diagram, is a visualization tool designed to identify and categorize the potential causes leading to a specific problem or "effect." It systematically maps out these causes, showing their relationship to the problem.

• Option A: different types of quality defects - This diagram doesn't list defect types; it focuses on the causes of one specific defect.
• Option B: quantitative relation between the extent of defect and a process parameter - This describes statistical analysis tools, not the qualitative cause-and-effect identification of an Ishikawa diagram.
• Option C: relation between defects and their causes - This accurately captures the diagram's purpose: to visually link a problem (defect) to its contributing causes.
• Option D: prioritized quality defects - Prioritization is typically done by tools like Pareto charts, not by the Ishikawa diagram itself.

Therefore, the Ishikawa diagram represents the relation between defects and their causes.

Motion economy principles, developed by Frank and Lillian Gilbreth, focus on optimizing movements for efficiency and reduced fatigue in tasks, primarily by minimizing unnecessary motions and using the smallest possible muscles. A "pivot" in this context refers to a joint enabling rotation or angular movement. The elbow, wrist, and torso (spine) are crucial pivots for manipulation and work tasks, allowing movements like reaching, precise hand actions, bending, and twisting. While the knee is a joint, its primary function is locomotion and weight support, making it less relevant as a "pivot for classified movement" in the specific context of motion economy, which emphasizes optimizing movements for assembly or detailed work. Therefore, the knee is NOT typically considered a primary pivot for manipulative actions under these principles.

In Value Engineering, the 'Value' of a product or service is defined as the ratio of its 'Function' to its 'Cost'. The function represents the benefits derived, while the cost is what is paid for it.

$\text{Value} = \frac{\text{Function}}{\text{Cost}}$

Here, the 'Function' of air travel is the sum of its monetized comfort and the monetary value of time saved:
Monetary Function = Monetized comfort + Monetary time saved
Monetary Function = Rs. 3000 + Rs. 3000 = Rs. 6000

The 'Cost' of air travel is the ticket price, which is Rs. 4000.

Now, substituting these values into the formula:
$\text{Value} = \frac{\text{Rs. 6000}}{\text{Rs. 4000}} = 1.5$

Enterprise Resource Planning (ERP) systems integrate and manage a business's core processes in real-time. Let's look at each option to see if it falls within a typical ERP's scope. General ledger entries are fundamental to accounting and financial management, which is a core ERP function. A materials management system, handling inventory, procurement, and logistics, is a standard operational module found in ERPs. An order management system, tracking sales from placement to fulfillment, is crucial for sales operations and integrated into ERPs. Employee promotion policy, however, is a strategic HR decision and policy-making function, not an operational or transactional process typically managed directly within an ERP system, even though ERPs often include HR modules for transactional tasks like payroll or employee records.

According to Taylor's differential piece-rate wage incentive plan, if a worker's actual output is less than the standard production, the total wage is calculated using the standard piece rate.

Here, the actual output ($18$ units) is less than the standard production ($20$ units).
So, the total wage paid to the worker is:
$\text{Total wage} = \text{Actual no. of units produced} \times \text{Price per unit}$
$\text{Total wage} = 18 \times 500 = 9000$
Therefore, the wage paid to the worker as per Taylor's differential piece-rate wage incentive plan is Rs. $9000$.

To find the curve represented by the solution to the differential equation $6yy' - 25x = 0$, we first rewrite $y'$ as $\frac{dy}{dx}$:
$6y \frac{dy}{dx} - 25x = 0$
Next, we separate the variables by moving the $x$ term to the right side and arranging the differentials:
$6y \frac{dy}{dx} = 25x$
$6y \, dy = 25x \, dx$
Now, we integrate both sides:
$\int 6y \, dy = \int 25x \, dx$
Performing the integration, we get:
$6 \cdot \frac{y^2}{2} = 25 \cdot \frac{x^2}{2} + C_1$
Simplifying the equation yields:
$3y^2 = \frac{25}{2}x^2 + C_1$
Rearranging the terms to the standard form of conic sections, we have:
$3y^2 - \frac{25}{2}x^2 = C_1$
This equation is in the form $Ay^2 + Bx^2 = C$. Here, the coefficient of $y^2$ is $A=3$ (positive), and the coefficient of $x^2$ is $B=-\frac{25}{2}$ (negative). Since the coefficients of $x^2$ and $y^2$ have opposite signs, the equation represents a hyperbola. Depending on the value of $C_1$, this is either a hyperbola opening along the y-axis (if $C_1 > 0$), a hyperbola opening along the x-axis (if $C_1 < 0$), or a pair of intersecting lines (if $C_1 = 0$, which is a degenerate hyperbola). Therefore, the solution represents a family of hyperbolas.

This problem involves a special type of differential equation called a Cauchy-Euler equation. These equations have the form $ax^2y'' + bxy' + cy = 0$. For such equations, we assume a solution of the form $y = x^r$.

Given the differential equation: $x^2y'' + xy' - y = 0$.

1. Assume a solution form: Let $y = x^r$.
2. Find derivatives:
   $y' = \frac{dy}{dx} = rx^{r-1}$
   $y'' = \frac{d^2y}{dx^2} = r(r-1)x^{r-2}$
3. Substitute into the differential equation:
   $x^2(r(r-1)x^{r-2}) + x(rx^{r-1}) - x^r = 0$
4. Simplify and form the characteristic equation:
   $r(r-1)x^r + rx^r - x^r = 0$
   Factor out $x^r$ (assuming $x \neq 0$):
   $x^r [r(r-1) + r - 1] = 0$
   Since $x^r \neq 0$, the expression in the brackets must be zero:
   $r(r-1) + r - 1 = 0$
5. Solve the characteristic equation for 'r':
   $r^2 - r + r - 1 = 0$
   $r^2 - 1 = 0$
   $(r - 1)(r + 1) = 0$
   This gives us two distinct real roots: $r_1 = 1$ and $r_2 = -1$.
6. Form the general solution: For distinct real roots $r_1$ and $r_2$, the general solution to a Cauchy-Euler equation is $y = c_1 x^{r_1} + c_2 x^{r_2}$.
   Substituting our roots:
   $y = c_1 x^1 + c_2 x^{-1}$
   This can be written as:
   $y = c_1x + \frac{c_2}{x}$

Comparing this derived solution with the given options, it matches option C.

Understanding how matrices transform vectors is crucial for competitive exams like GATE. A linear transformation matrix, when multiplied by a vector

, shows us how the vector's components change, which in turn reveals its geometric interpretation.

Let's break down each matrix:

For P:
The matrix is

. When we apply this to a vector

:

.
Notice that the $y$-component becomes $0$, while the $x$-component remains $x$. This means any point $(x, y)$ is mapped to $(x, 0)$, which is a projection onto the x-axis. So, P-3.

For Q:
The matrix is

. Applying it to

:

.
Here, the $x$-component becomes $0$, and the $y$-component remains $y$. This maps $(x, y)$ to $(0, y)$, representing a projection onto the y-axis. So, Q-4.

For R:
The matrix is

. Let's see its effect on

:

.
The $x$-component stays the same ($x$), but the $y$-component is multiplied by $3$ ($3y$). This is a stretch in the y-axis by a factor of $3$. So, R-1.

For S:
The matrix is

. Applying it to

:

.
Both the $x$-component and the $y$-component are scaled by the same factor of $4$. This is a uniform stretch in x and y-axes. So, S-2.

Combining these results, we get P-3, Q-4, R-1, S-2.

We need to find the value of the limit: $\text{lim}_{(x,y) \to (0,0)} \frac{x^2 - xy}{\sqrt{x} - \sqrt{y}}$ As $(x,y)$ approaches $(0,0)$, the expression becomes the indeterminate form $\frac{0}{0}$. The square roots imply $x ≥ 0$ and $y ≥ 0$, and the denominator requires $\sqrt{x} - \sqrt{y} \neq 0$, so $x \neq y$.

To simplify, first factor the numerator: $x^2 - xy = x(x-y)$. The expression becomes:
$\frac{x(x-y)}{\sqrt{x} - \sqrt{y}}$
Next, rationalize the denominator by multiplying the numerator and denominator by its conjugate, $\sqrt{x} + \sqrt{y}$:
$\frac{x(x-y)}{\sqrt{x} - \sqrt{y}} \times \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}$
The denominator simplifies to $(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y}) = (\sqrt{x})^2 - (\sqrt{y})^2 = x - y$.
So the expression becomes:
$\frac{x(x-y)(\sqrt{x} + \sqrt{y})}{x - y}$
Since we are taking the limit as $(x,y) \to (0,0)$ but $(x,y) \neq (0,0)$, we can assume $x \neq y$ and cancel the $(x-y)$ term:
$x(\sqrt{x} + \sqrt{y})$
Now, substitute $x=0$ and $y=0$ into the simplified expression to evaluate the limit:
$\text{lim}_{(x,y) \to (0,0)} x(\sqrt{x} + \sqrt{y}) = 0(\sqrt{0} + \sqrt{0}) = 0(0 + 0) = 0$

The key concept here is the elongation of a bar due to its own weight. For a uniformly prismatic bar, this elongation ($\Delta L$) is given by the formula: $\Delta L = \frac{\gamma L^2}{2E}$ where $\gamma$ is the specific weight, $L$ is the length, and $E$ is Young's modulus.

Given values are: specific weight $\gamma = 100 \text{ kN/m}^3$, length $L = 100 \text{ m}$, and Young's modulus $E = 200 \text{ GPa}$.
Substituting these values into the formula:
$\Delta L = \frac{100 \times 10^3 \times 100^2}{2 \times 200 \times 10^9} = 0.0025 \text{ m}$
Converting this to millimeters, we get $\Delta L = 2.5 \text{ mm}$. Therefore, the elongation of the bar due to its own weight is $2.5 \text{ mm}$.

To find the bending moment at point R, we first resolve the inclined and eccentric loads into their equivalent components. The 6 kN inclined load at R resolves into a horizontal component of $6 \cos(30^\circ) = 5.19 \text{ kN}$ and a vertical component of $6 \sin(30^\circ) = 3 \text{ kN}$. The 10 kN eccentric load at Q is equivalent to a vertical load of 10 kN at Q and a moment of $10 \text{ kNm}$ (since it's applied 1m away from Q, the moment is $10 \text{ kN} \times 1 \text{ m} = 10 \text{ kNm}$).

Now, let's find the reactions at the supports. Let $P_x$ and $P_y$ be the horizontal and vertical reactions at P, and $S$ be the vertical reaction at the roller support.
From the equilibrium of forces in the x-direction:
$P_x - 5.19 = 0 \Rightarrow P_x = 5.19 \text{ kN}$

From the equilibrium of forces in the y-direction:
$P_y - 10 - 3 + S = 0 \Rightarrow P_y + S = 13 \text{ kN}$

Taking moments about support P (to find $S$ directly):
$-10 \text{ kNm}$ (from the eccentric load) $- (10 \text{ kN} \times 2 \text{ m})$ (from the 10 kN vertical load) $- (3 \text{ kN} \times 4 \text{ m})$ (from the 3 kN vertical load at R) $+ (S \times 6 \text{ m}) = 0$
$-10 - 20 - 12 + 6S = 0$
$6S = 42 \Rightarrow S = 7 \text{ kN}$

Finally, the bending moment at point R (due to forces to the right of R) is calculated as:
$M_R = S \times (\text{distance from S to R}) = 7 \text{ kN} \times 2 \text{ m}$
$M_R = 14 \text{ kNm}$

The Quick Return Ratio (QRR) for an offset slider-crank mechanism is defined as the ratio of the angle turned during the forward or cutting stroke ($\alpha$) to the angle turned during the return stroke ($\beta$), given by $QRR = \dfrac{\alpha}{\beta}$. For this mechanism, $\alpha = 180^\circ + \theta$ and $\beta = 180^\circ - \theta$. Therefore, $QRR = \dfrac{180^\circ + \theta}{180^\circ - \theta}$.
Given $\theta = 20^\circ$, we calculate the QRR as:
$QRR = \dfrac{180^\circ + 20^\circ}{180^\circ - 20^\circ} = \dfrac{200^\circ}{160^\circ} = 1.25$.
Since the ratio of forward to return stroke time is $1.25$, which is greater than 1, this mechanism can be used in an application involving quick return.

This problem involves heat transfer in a wall with internal heat generation. We'll use Fourier's Law and the energy balance equation to find the rate of change of energy storage.

Concept:
Fourier's Law describes conductive heat transfer: $\dot{Q} = - kA\frac{{dT}}{{dx}}$ where $\dot{Q}$ is the heat flow rate, $k$ is thermal conductivity, $A$ is the area, and $dT/dx$ is the temperature gradient. The negative sign indicates heat flows from higher to lower temperature.