GATE PI 2013

Q13GATE 2013MCQ1MIndustrial Engineering

The fixed cost and the variable cost of production of a product are Rs. 20000 and Rs. 50 per unit, respectively. The demand for the item is 500 units. To break even, the unit price of the items in Rs. should be

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📖 Explanation

To break even, the total revenue must equal the total costs. The break-even unit price, $P$ , can be found using the formula: Total Revenue = Total Costs, which expands to $P \times Q = \text{Fixed Costs} + (\text{Variable Cost per Unit} \times Q)$ .

Rearranging this formula to solve for $P$ gives us:
$P = \frac{\text{Fixed Costs}}{Q} + \text{Variable Cost per Unit}$
Given Fixed Costs = Rs. 20000, Variable Cost per Unit = Rs. 50, and Quantity ( $Q$ ) = 500 units, we substitute these values into the formula:
$P = \frac{20000}{500} + 50$
$P = 40 + 50$
$P = 90$
Thus, the unit price to break even is Rs. 90.

Q14GATE 2013MCQ1MIndustrial Engineering

Therbligs refer to the

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Q15GATE 2013MCQ1MOperations Research and Operations Management

Customers arrive at a ticket counter at a rate of 50 per $hr$ and tickets are issued in the order of their arrival. The average time taken for issuing a ticket is 1 $min$ . Assuming that customer arrivals form a Poisson process and service times are exponentially distributed, the average waiting time in queue in $min$ is

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📖 Explanation

This is an M/M/1 queuing problem. We need to find the average waiting time in the queue ( $W_q$ ).

First, ensure consistent units. The arrival rate $\lambda = 50 \text{ customers/hour} = \frac{50}{60} \text{ customers/minute} = \frac{5}{6} \text{ customers/minute}$ . The service rate $\mu = \frac{1 \text{ customer}}{1 \text{ minute}} = 1 \text{ customer/minute}$ .

Next, calculate the traffic intensity $\rho = \frac{\lambda}{\mu} = \frac{5/6}{1} = \frac{5}{6}$ . Since $\rho < 1$ , the system is stable.

Finally, use the M/M/1 formula for average waiting time in queue:
$W_q = \frac{\rho}{\mu(1-\rho)}$
Substitute the values:
$W_q = \frac{5/6}{1 \times (1 - 5/6)} = \frac{5/6}{1/6} = 5 \text{ minutes}$
Thus, the average waiting time in the queue is 5 minutes.

Q16GATE 2013MCQ1MManufacturing Processes I

Circular blanks of 10 $mm$ diameter are punched from an aluminum sheet of 2 $mm$ thickness. The shear strength of aluminum is 80 $MPa$ . The minimum punching force required in $kN$ is

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📖 Explanation

To find the minimum punching force, we need to determine the shear force required to cut the material. This is calculated using the formula $F = \tau \times A$ , where $F$ is the force, $\tau$ is the shear strength, and $A$ is the shear area.

First, let's calculate the shear area ( $A$ ). Imagine the punch cutting through the sheet; the area being sheared is the cylindrical surface of the blank.
Diameter ( $d$ ): $10 \, mm$
Thickness ( $t$ ): $2 \, mm$
Circumference ( $C$ ): $\pi \times d = \pi \times 10 \, mm = 10\pi \, mm$
Shear Area ( $A$ ): $C \times t = (10\pi \, mm) \times (2 \, mm) = 20\pi \, mm^2$

Now, we can calculate the punching force ( $F$ ).
Shear Strength ( $\tau$ ): $80 \, MPa = 80 \, N/mm^2$
Force ( $F$ ): $\tau \times A = (80 \, N/mm^2) \times (20\pi \, mm^2) = 1600\pi \, N$

Finally, we convert the force from Newtons to kilonewtons ( $kN$ ).
Force ( $F$ ) in $kN$ : $\frac{1600\pi \, N}{1000 \, N/kN} = 1.6\pi \, kN$
Approximate Value: $1.6 \times \pi \approx 1.6 \times 3.14159 \approx 5.0265 \, kN$

Therefore, the minimum punching force required is approximately $5.03 \, kN$ .

Q17GATE 2013MCQ1MGeneral Engineering

A metric thread of pitch 2 $mm$ and thread angle 60° is inspected for its pitch diameter using 3-wire method. The diameter of the best size wire in $mm$ is

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📖 Explanation

The 3-wire method measures the pitch diameter of a screw thread using cylindrical wires. The 'best size wire' is one that touches the thread flanks exactly at the pitch diameter. Its diameter ( $d$ ) is calculated using the formula: $d = \frac{P}{2 \cos(\alpha)}$ , where $P$ is the pitch and $\alpha$ is half the included thread angle.

Given $P = 2 \text{ mm}$ and the thread angle ( $2\alpha$ ) $= 60^\circ$ , we have $\alpha = 30^\circ$ .
Substituting these values:
$d = \frac{2 \text{ mm}}{2 \cos(30^\circ)}$
Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ :
$d = \frac{2 \text{ mm}}{2 \times \frac{\sqrt{3}}{2}} = \frac{2 \text{ mm}}{\sqrt{3}}$
Calculating the value: $d \approx 1.1547 \text{ mm}$ . Therefore, the best size wire diameter is approximately $1.154 \text{ mm}$ .

Q18GATE 2013MCQ1MGeneral Engineering

Match the CORRECT pairs. Processes Characteristics / Applications P. Friction Welding 1. Non-consumable electrode Q. Gas Metal Arc Welding 2. Joining of thick plates R. Tungsten Inert Gas Welding 3. Consumable electrode wire S. Electroslag Welding 4. Joining of cylindrical dissimilar materials

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Q19GATE 2013MCQ1MManufacturing Processes II

In a rolling process, the state of stress of the material undergoing deformation is

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Q20GATE 2013MCQ1MGeneral Engineering

Consider one-dimensional steady state heat conduction along $x$ -axis ( $0 \le x \le L$ ), through a plane wall; with the boundary surfaces ( $x=0$ and $x=L$ ) maintained at temperatures of 0°C and 100°C. Heat is generated uniformly throughout the wall. Choose the CORRECT statement.

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📖 Explanation

We are dealing with one-dimensional, steady-state heat conduction in a plane wall with uniform heat generation ( $q_g$ ). The governing equation is given by:
$\frac{d^2T}{dx^2} + \frac{q_g}{k} = 0$
Integrating this equation twice yields the general temperature distribution:
$T(x) = -\frac{q_g}{2k}x^2 + C_1x + C_2$
The boundary conditions are $T(0) = 0^\circ C$ and $T(L) = 100^\circ C$ .
Applying $T(0) = 0$ : $-\frac{q_g}{2k}(0)^2 + C_1(0) + C_2 = 0 \implies C_2 = 0$ .
Applying $T(L) = 100$ : $-\frac{q_g}{2k}L^2 + C_1L + 0 = 100$ .
Solving for $C_1$ : $C_1 = \frac{100}{L} + \frac{q_gL}{2k}$ .
Substituting $C_1$ and $C_2$ back into the general solution gives the specific temperature profile:
$T(x) = -\frac{q_g}{2k}x^2 + \left(\frac{100}{L} + \frac{q_gL}{2k}\right)x$
Now, let's compare this with the linear temperature profile ( $T_{linear}(x) = \frac{100x}{L}$ ) that would exist without heat generation ( $q_g = 0$ ):
$T(x) - T_{linear}(x) = \left(-\frac{q_g}{2k}x^2 + \left(\frac{100}{L} + \frac{q_gL}{2k}\right)x\right) - \frac{100x}{L}$
$T(x) - T_{linear}(x) = -\frac{q_g}{2k}x^2 + \frac{q_gL}{2k}x = \frac{q_g}{2k}(Lx - x^2)$
Since heat is generated, $q_g > 0$ . Thermal conductivity $k > 0$ . For $0 < x < L$ , the term $(Lx - x^2)$ is always positive. Therefore, $T(x) - T_{linear}(x)$ is always positive for $0 < x < L$ . This means the actual temperature $T(x)$ is always greater than the linear temperature profile within the wall. Since the linear profile varies from $0^\circ C$ to $100^\circ C$ , and $T(x)$ is always elevated above it, the temperature must exceed $100^\circ C$ somewhere inside the wall. Thus, the maximum temperature inside the wall must be greater than $100^\circ C$ .

Q21GATE 2013MCQ1MGeneral Engineering

A cylinder contains 5 $m^3$ of an ideal gas at a pressure of 1 $bar$ . This gas is compressed in a reversible isothermal process till its pressure increases to 5 $bar$ . The work in $kJ$ required for this process is

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📖 Explanation

To find the work done during a reversible isothermal compression of an ideal gas, we use the formula:
$W = P_1 V_1 \ln \frac{P_1}{P_2}$
Given values are $P_1 = 1 \text{ bar}$ , $V_1 = 5 \text{ m}^3$ , and $P_2 = 5 \text{ bar}$ .
Substituting these values:
$W = (1 \text{ bar}) (5 \text{ m}^3) \ln \frac{1 \text{ bar}}{5 \text{ bar}}$
$W = 5 \text{ bar} \cdot \text{m}^3 \ln(0.2)$
Since $\ln(0.2) \approx -1.6094$ :
$W \approx 5 \times (-1.6094) \text{ bar} \cdot \text{m}^3$
$W \approx -8.047 \text{ bar} \cdot \text{m}^3$
To convert this to kilojoules (kJ), use the conversion factor $1 \text{ bar} \cdot \text{m}^3 = 100 \text{ kJ}$ :
$W \approx -8.047 \times 100 \text{ kJ}$
$W \approx -804.7 \text{ kJ}$
The negative sign indicates work is done *on* the system during compression. The work required is the magnitude, which is $804.7 \text{ kJ}$ .

Q22GATE 2013MCQ1MGeneral Engineering

A planar closed kinematic chain is formed with rigid links PQ = 2.0 $m$ , QR = 3.0 $m$ , RS = 2.5 $m$ and SP = 2.7 $m$ with all revolute joints. The link to be fixed to obtain a double rocker (rocker- rocker) mechanism is

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📖 Explanation

To determine the link to be fixed for a double rocker mechanism, we first identify the lengths of the rigid links: $PQ = 2.0 \text{ m}$ , $QR = 3.0 \text{ m}$ , $RS = 2.5 \text{ m}$ , and $SP = 2.7 \text{ m}$ .
Next, we apply Grashof's Law by identifying the shortest link ( $s$ ), longest link ( $l$ ), and the other two links ( $p$ and $q$ ). Here, $s = PQ = 2.0 \text{ m}$ (shortest), $l = QR = 3.0 \text{ m}$ (longest), $p = RS = 2.5 \text{ m}$ , and $q = SP = 2.7 \text{ m}$ .
We then calculate the sums: $s + l = 2.0 + 3.0 = 5.0 \text{ m}$ and $p + q = 2.5 + 2.7 = 5.2 \text{ m}$ .
For a mechanism to be a double rocker (rocker-rocker), where all movable links oscillate, the condition $s + l > p + q$ must ideally be met. In this case, $s + l = 5.0 \text{ m}$ and $p + q = 5.2 \text{ m}$ , which means $s + l < p + q$ . This indicates a Grashof mechanism. However, for problems like this asking for a double rocker, fixing the link opposite to the shortest link, or an intermediate link, can result in the desired motion. Based on standard analysis and the provided answer, fixing link $RS$ (one of the intermediate links) will result in a double rocker mechanism.

Q23GATE 2013MCQ1MEngineering Mathematics

Let $X$ be a normal random variable with mean 1 and variance 4. The probability $P\{X<0\}$ is

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📖 Explanation

We're given a normal random variable $X$ with mean $\mu = 1$ and variance $\sigma^2 = 4$ . We need to find $P\{X < 0\}$ .

First, calculate the standard deviation: $\sigma = \sqrt{\text{variance}} = \sqrt{4} = 2$ .
Next, we standardize $X$ to a standard normal variable $Z$ using the formula $Z = \frac{X - \mu}{\sigma}$ .
For $X = 0$ , the Z-score is $Z = \frac{0 - 1}{2} = -0.5$ .
So, $P\{X < 0\}$ is equivalent to $P\{Z < -0.5\}$ . The standard normal distribution is symmetric around its mean ( $Z=0$ ), and the total area under the curve is 1. Since $-0.5$ is to the left of the mean (0), the probability $P\{Z < -0.5\}$ must be less than $P\{Z < 0\}$ , which is $0.5$ . Also, probabilities cannot be negative, so $P\{Z < -0.5\} > 0$ . Therefore, $0 < P\{Z < -0.5\} < 0.5$ . Thus, the probability $P\{X < 0\}$ is greater than zero and less than 0.5.

Q24GATE 2013MCQ1MEngineering Mathematics

Choose the CORRECT set of functions, which are linearly dependent.

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📖 Explanation

To check for linear dependence, we look for constants $c_1, c_2, ..., c_n$ , where at least one is non-zero, such that $c_1 f_1(x) + c_2 f_2(x) + ... + c_n f_n(x) = 0$ for all $x$ . This means one function can be expressed as a combination of the others.

Let's examine each option:
* **Option A:** $\sin x$ , $\sin^2 x$ , $\cos^2 x$ . The identity $\sin^2 x + \cos^2 x = 1$ involves a constant and not $\sin x$ . These functions are generally independent.
* **Option B:** $\cos x$ , $\sin x$ , $\tan x$ . These are independent trigonometric functions.
* **Option C:** $\cos 2x$ , $\sin^2 x$ , $\cos^2 x$ . We use the double angle identity: $\cos 2x = \cos^2 x - \sin^2 x$ . Rearranging this, we get $1 \cdot \cos 2x - 1 \cdot \cos^2 x + 1 \cdot \sin^2 x = 0$ . Since the coefficients $(1, -1, 1)$ are not all zero, this set is linearly dependent.
* **Option D:** $\cos 2x$ , $\sin x$ , $\cos x$ . These functions are generally linearly independent.

Therefore, the set $\cos 2x$ , $\sin^2 x$ , $\cos^2 x$ is linearly dependent.

Q25GATE 2013MCQ1MEngineering Mathematics

The eigenvalues of a symmetric matrix are all

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📖 Explanation

A symmetric matrix $A$ is defined as a square matrix where $A = A^T$ . A fundamental property of symmetric matrices in linear algebra is that all their eigenvalues are real. This means that for any symmetric matrix, its eigenvalues will not have any imaginary components. Therefore, among the given options, the correct one is that the eigenvalues of a symmetric matrix are all real.

Q26GATE 2013MCQ1MEngineering Mathematics

The partial differential equation $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2}$ is a

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📖 Explanation

Let's break down this Partial Differential Equation (PDE) step-by-step to determine its order and linearity.

First, to find the **order** of the PDE, we look for the highest derivative present.
* The term $\frac{\partial u}{\partial t}$ is a first-order derivative.
* The term $u \frac{\partial u}{\partial x}$ involves a first-order derivative ( $\frac{\partial u}{\partial x}$ ).
* The term $\frac{\partial^2 u}{\partial x^2}$ is a second-order derivative.
Since the highest derivative is $\frac{\partial^2 u}{\partial x^2}$ , the order of the PDE is **2**.

Next, to check for **linearity**, a PDE is linear if the dependent variable ( $u$ ) and all its derivatives appear only to the first power, and there are no products of $u$ or its derivatives. If any of these conditions are not met, the PDE is non-linear.
* The terms $\frac{\partial u}{\partial t}$ and $\frac{\partial^2 u}{\partial x^2}$ are linear components.
* However, the term $u \frac{\partial u}{\partial x}$ contains a product of the dependent variable $u$ and its partial derivative $\frac{\partial u}{\partial x}$ . This product makes the equation **non-linear**.

Therefore, the given PDE, $\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2}$ , is a **non-linear equation of order 2**.

Q27GATE 2013MCQ1MEngineering Mathematics

Match the CORRECT pairs. Numerical Integration Scheme Order of Fitting Polynomial P. Simpson's 3/8 Rule 1. First Q. Trapezoidal Rule 2. Second R. Simpson's 1/3 Rule 3. Third

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Q28GATE 2013MCQ1MGeneral Engineering

A rod of length $L$ having uniform cross-sectional area $A$ is subjected to a tensile force $P$ as shown in the figure below. If the Young's modulus of the material varies linearly from $E_1$ to $E_2$ along the length of the rod, the normal stress developed at the section-SS is

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📖 Explanation

Let's break down how to find the normal stress in this rod.

**Key Concept: Normal Stress**
Normal stress ( $\sigma$ ) is simply the force applied perpendicular to a surface, divided by the area over which it's applied. Think of it as how much force is spread over each unit of area.
$\sigma = \frac{\text{Force}}{\text{Area}} = \frac{P}{A}$

**Applying it to the Problem:**
The rod has a uniform cross-sectional area $A$ and is subjected to a tensile force $P$ . We need to find the normal stress at section-SS.

Even though the Young's modulus (which tells us about the material's stiffness) varies along the rod from $E_1$ to $E_2$ , this variation affects *how much the rod deforms*, not the *stress distribution* itself when a uniform axial force is applied to a uniform cross-section. At any point along the rod, including section-SS, the entire force $P$ is being carried by the full cross-sectional area $A$ .

**Conclusion:**
Therefore, the normal stress developed at section-SS is directly calculated as the applied force divided by the cross-sectional area:
$\sigma = \frac{P}{A}$

Q29GATE 2013MCQ1MGeneral Engineering

For steady, fully developed flow inside a straight pipe of diameter $D$ , neglecting gravity effects, the pressure drop $\Delta p$ over a length $L$ and the wall shear stress $\tau_w$ are related by

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📖 Explanation

To find the relationship between pressure drop ( $\Delta p$ ) and wall shear stress ( $\tau_w$ ) for steady, fully developed flow in a pipe, we perform a force balance on a cylindrical fluid section of length $L$ and diameter $D$ . The upstream pressure force acting on the fluid cylinder's cross-sectional area is $F_p = \Delta p \times \frac{\pi D^2}{4}$ . The wall shear force, which resists motion, acts on the pipe wall surface area and is given by $F_s = \tau_w \times (\pi D L)$ . For steady, fully developed flow, these forces must balance:
$\Delta p \times \frac{\pi D^2}{4} = \tau_w \times \pi D L$
Rearranging to solve for $\tau_w$ :
$\tau_w = \frac{\Delta p \times \frac{\pi D^2}{4}}{\pi D L}$
Simplifying this expression yields:
$\tau_w = \frac{\Delta p D}{4L}$

Q30GATE 2013MCQ1MGeneral Engineering

For a ductile material, toughness is a measure of

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Q31GATE 2013MCQ1MManufacturing Processes I

A cube shaped casting solidifies in 5 $min$ . The solidification time in $min$ for a cube of the same material, which is 8 times heavier than the original casting, will be

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📖 Explanation

This problem asks us to find the solidification time for a heavier cube casting, given the solidification time of an original cube casting.

The key concept here is Chvorinov's Rule, which states that solidification time ( $T_s$ ) is proportional to the square of the ratio of the casting's Volume ( $V$ ) to its Surface Area ( $A$ ). Mathematically, this is expressed as:
$T_s \propto \left(\frac{V}{A}\right)^2$
For a cube with side length $L$ , the volume is $V = L^3$ and the surface area is $A = 6L^2$ . Substituting these into the ratio:
$\frac{V}{A} = \frac{L^3}{6L^2} = \frac{L}{6}$
So, the solidification time for a cube is proportional to the square of its side length:
$T_s \propto \left(\frac{L}{6}\right)^2 \propto L^2$
Alternatively, we can express $L$ in terms of $V$ ( $L = V^{1/3}$ ). Substituting this into the proportionality:
$T_s \propto (V^{1/3})^2 = V^{2/3}$
This means that solidification time is proportional to the volume raised to the power of $2/3$ .

Now, let's apply this to the problem:
We are given that the new casting is 8 times heavier than the original. Since the material is the same, weight is directly proportional to volume.
So, if the original casting has volume $V_1$ and the new casting has volume $V_2$ :
$W_2 = 8W_1 \Rightarrow V_2 = 8V_1$
Let $T_{s1}$ be the solidification time for the original casting and $T_{s2}$ for the new casting. Using the proportionality $T_s \propto V^{2/3}$ :
$\frac{T_{s2}}{T_{s1}} = \left(\frac{V_2}{V_1}\right)^{2/3}$
We are given $T_{s1} = 5$ min and we found $V_2/V_1 = 8$ . Substituting these values:
$\frac{T_{s2}}{5 \text{ min}} = \left(8\right)^{2/3}$
First, calculate $8^{2/3}$ :
$8^{2/3} = (8^{1/3})^2 = (2)^2 = 4$
Now, substitute this back into the equation:
$\frac{T_{s2}}{5 \text{ min}} = 4$
Solving for $T_{s2}$ :
$T_{s2} = 4 \times 5 \text{ min} = 20 \text{ min}$
Therefore, the solidification time for the new, heavier cube casting is 20 minutes.

Q32GATE 2013MCQ1MManufacturing Processes II

A steel bar 200 $mm$ in diameter is turned at a feed of 0.25 $mm/rev$ with a depth of cut of 4 $mm$ . The rotational speed of the workpiece is 160 $rpm$ . The material removal rate in $mm^3/s$ is

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📖 Explanation

To determine the Material Removal Rate (MRR) in turning, we first need to ensure all units are consistent. The rotational speed ( $N$ ) is given in revolutions per minute ( $rpm$ ), so we convert it to revolutions per second ( $rev/s$ ):
$N_{rps} = \frac{N_{rpm}}{60 \, s/min} = \frac{160 \, rpm}{60 \, s/min} = \frac{16}{6} \, rev/s = \frac{8}{3} \, rev/s$
The formula for MRR in turning is given by $MRR = \pi \times D \times d_c \times f \times N_{rps}$ , where $D$ is the diameter, $d_c$ is the depth of cut, and $f$ is the feed. Substituting the given values:
$MRR = \pi \times (200 \, mm) \times (4 \, mm) \times (0.25 \, mm/rev) \times \left(\frac{8}{3} \, rev/s\right)$
$MRR = \pi \times 200 \times (4 \times 0.25) \times \frac{8}{3} \, mm^3/s$
$MRR = \pi \times 200 \times 1 \times \frac{8}{3} \, mm^3/s = \pi \times \frac{1600}{3} \, mm^3/s$
Calculating the numerical value, $MRR \approx 3.14159 \times \frac{1600}{3} \, mm^3/s \approx 1675.5 \, mm^3/s$ .

Q33GATE 2013MCQ1MManufacturing Processes II

In the 3-2-1 principle of fixture design, 3 refers to the number of

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📖 Explanation

The 3-2-1 principle in fixture design aims to constrain a workpiece's six degrees of freedom (three translational and three rotational) using strategically placed locators. The '3' in the 3-2-1 principle refers to the **3 locators** placed on the primary datum face. These three locators establish the foundational positioning and restrict movement along the three axes perpendicular to this face (e.g., $X$ , $Y$ , $Z$ translation). The '2' refers to two locators on the secondary datum face, restricting two rotational movements, and the '1' refers to one locator on the tertiary datum face, restricting the final rotational movement.

Q34GATE 2013MCQ1MOperations Research and Operations Management

In simple exponential smoothing forecasting, to give higher weightage to recent demand information, the smoothing constant must be close to

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📖 Explanation

In Simple Exponential Smoothing (SES), the forecast for the next period, $F_{t+1}$ , is calculated using the formula: $F_{t+1} = \alpha D_t + (1 - \alpha) F_t$ , where $D_t$ is the actual demand of the current period, $F_t$ is the forecast for the current period, and $\alpha$ is the smoothing constant, with $0 \leq \alpha \leq 1$ . The value of $\alpha$ directly controls the weight given to the most recent actual demand ( $D_t$ ) versus the previous forecast ( $F_t$ ). To give higher weightage to recent demand information ( $D_t$ ), $\alpha$ must be large. If $\alpha$ is close to 1, the term $(1-\alpha)$ becomes small, reducing the influence of $F_t$ and increasing the influence of $D_t$ . Therefore, to prioritize recent demand, $\alpha$ must be close to 1.

Q35GATE 2013MCQ1MEngineering Mathematics

A company manufactures 1000 toys every day. On an average, 10% of the toys are defective and 40% of the defective toys can be reworked into defect-free ones. The average number of defect-free toys manufactured daily is

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📖 Explanation

To find the average number of defect-free toys manufactured daily, we first calculate the number of defective toys. The company manufactures $1000$ toys, and $10\%$ are defective, so the number of defective toys is $\frac{10}{100} \times 1000 = 100$ . Out of these, $40\%$ can be reworked into defect-free toys, meaning the number of reworkable toys is $\frac{40}{100} \times 100 = 40$ . The number of defect-free toys initially produced is $1000 - 100 = 900$ . Adding the reworked toys, the total number of defect-free toys manufactured daily is $900 + 40 = 940$ .

Q36GATE 2013MCQ1MQuality and Reliability

The type of control chart used to monitor the amount of dispersion in a sample is

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📖 Explanation

Control charts are vital tools in Statistical Process Control (SPC) for monitoring different aspects of a process's performance. The question asks which chart specifically monitors the *dispersion* or *spread* within a sample. Let's look at the options:

* **c-chart:** This chart tracks the *number of defects per unit*.
* **p-chart:** This chart monitors the *proportion of nonconforming items*.
* ** $\bar{x}$ -chart:** This chart is used to monitor the *average (mean)* of the process data.
* **R-chart (Range Chart):** This chart monitors the *variability within subgroups* by tracking the *range* (which is the maximum value minus the minimum value) of each sample.

The R-chart directly measures the spread or dispersion of data points within samples, making it the appropriate choice for monitoring the amount of dispersion.

Q37GATE 2013MCQ2MQuality and Reliability

A company plans to purchase a machine whose uptime needs to be atleast 95%. They have shortlisted two models of the machine with the following operational characteristics: Machine MTBF (hr) MTTR (hr) Model M 60 4 Model N 48 2 The company should buy

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📖 Explanation

To determine which machine meets the company's requirement of at least 95% uptime, we first need to understand how uptime is calculated. Uptime represents the percentage of time a machine is expected to be operational. It's calculated using two key metrics: Mean Time Between Failures (MTBF), which is the average time a machine works before failing, and Mean Time To Repair (MTTR), which is the average time it takes to fix the machine after a failure.

The formula for uptime percentage is:
$Uptime = \left(\frac{MTBF}{MTBF + MTTR} \times 100\%\right)$

Now, let's apply this formula to each machine model:

**For Model M:**
* MTBF = 60 hr
* MTTR = 4 hr
* $Uptime (M) = \left(\frac{60}{60 + 4} \times 100\%\right) = \left(\frac{60}{64} \times 100\%\right) = 0.9375 \times 100\% = 93.75\%$

**For Model N:**
* MTBF = 48 hr
* MTTR = 2 hr
* $Uptime (N) = \left(\frac{48}{48 + 2} \times 100\%\right) = \left(\frac{48}{50} \times 100\%\right) = 0.96 \times 100\% = 96\%$

Finally, we compare these calculated uptimes with the required minimum of 95%:
* Model M's uptime is 93.75%, which is less than 95%.
* Model N's uptime is 96.00%, which is greater than or equal to 95%.

Since only Model N meets the specified uptime requirement, the company should buy only Model N.

Q38GATE 2013MCQ2MQuality and Reliability

A manufacturer produces bars designed to be of 10 mm diameter with a tolerance of $\pm0.1$ mm. Historical data indicates that manufactured bars have an average diameter of 9.98 mm with a standard deviation of 0.15 mm. The process capability index is

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📖 Explanation

The **Process Capability Index ( $C_{pk}$ )** tells us how well a manufacturing process is performing against its specified design limits, considering if the process is centered.

First, let's identify the given values:
* Nominal Diameter (target): $10$ mm
* Tolerance: $\pm 0.1$ mm
* Average Diameter ( $\mu$ ): $9.98$ mm
* Standard Deviation ( $\sigma$ ): $0.15$ mm

Now, we calculate the **Upper Specification Limit (USL)** and **Lower Specification Limit (LSL)**:
* USL = Nominal + Tolerance = $10 + 0.1 = 10.1$ mm
* LSL = Nominal - Tolerance = $10 - 0.1 = 9.9$ mm

The formula for $C_{pk}$ is:
$C_{pk} = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right)$

Substitute the values into the formula:
* Term 1: $\frac{10.1 - 9.98}{3 \times 0.15} = \frac{0.12}{0.45} \approx 0.267$
* Term 2: $\frac{9.98 - 9.9}{3 \times 0.15} = \frac{0.08}{0.45} \approx 0.178$

Now, find the minimum of these two values:
$C_{pk} = \min(0.267, 0.178) = 0.178$

Rounding $0.178$ to two decimal places gives $0.18$ .

Q39GATE 2013MCQ2MOperations Research and Operations Management

Let (P) denote the linear programming formulation of a transportation problem with $m$ sources and $n$ destinations. Then, the dual linear program of (P) has

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📖 Explanation

To determine the structure of the dual linear program for a transportation problem, we first analyze the primal problem. A transportation problem with $m$ sources and $n$ destinations has $m \times n$ variables, denoted by $x_{ij}$ , representing the quantity shipped from source $i$ to destination $j$ . It has $m+n$ constraints: $m$ supply constraints (one for each source) and $n$ demand constraints (one for each destination). According to the duality principle in linear programming, the number of variables in the dual problem equals the number of constraints in the primal problem, which is $m+n$ . Similarly, the number of constraints in the dual problem equals the number of variables in the primal problem, which is $m \times n$ . Therefore, the dual linear program of (P) has $m+n$ variables and $m \times n$ constraints.

Q40GATE 2013MCQ2MIndustrial Engineering

Following data refers to an automat and a center lathe, which are being compared to machine a batch of parts in a manufacturing shop. Automat Center Lathe Machine Set-up Time in min 120 30 Machine Set-up Cost in Rs./min 800 150 Machining Time per piece in min 2 25 Machining Cost in Rs./min 500 100 Automat will be economical if the batch size exceeds

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📖 Explanation

To determine when the Automat is more economical, we need to find the batch size ( $N$ ) where its total cost is less than or equal to the Center Lathe's total cost.

First, calculate the total cost for each machine:

For the Automat:
Set-up Cost $= 120 \text{ min} \times 800 \text{ Rs./min} = 96000 \text{ Rs.}$
Machining Cost per piece $= 2 \text{ min/piece} \times 500 \text{ Rs./min} = 1000 \text{ Rs./piece}$
Total Cost for Automat ( $C_A$ ) is given by:
$C_A(N) = 96000 + 1000N$

For the Center Lathe:
Set-up Cost $= 30 \text{ min} \times 150 \text{ Rs./min} = 4500 \text{ Rs.}$
Machining Cost per piece $= 25 \text{ min/piece} \times 100 \text{ Rs./min} = 2500 \text{ Rs./piece}$
Total Cost for Center Lathe ( $C_L$ ) is given by:
$C_L(N) = 4500 + 2500N$

Next, find the breakeven point where the costs are equal:
$C_A(N) = C_L(N)$
$96000 + 1000N = 4500 + 2500N$
$96000 - 4500 = 2500N - 1000N$
$91500 = 1500N$
$N = \frac{91500}{1500}$
$N = 61$

The breakeven batch size is 61. For the Automat to be economical, its cost must be less than the Center Lathe's cost. Comparing the cost equations, $C_A(N)$ has a higher fixed cost but a lower variable cost per piece than $C_L(N)$ . Thus, the Automat becomes more economical when the batch size exceeds 61.

Q41GATE 2013MCQ2MManufacturing Processes II

Cylindrical pins of $25^{+0.020}_{+0.010}$ mm diameter are electroplated in a shop. Thickness of the plating is $30 ^{\pm 2.0}$ micron. Neglecting gage tolerances, the size of the GO gage in mm to inspect the plated components is

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📖 Explanation

To size a GO gage for a plated component, we need to ensure it checks the *maximum material condition* of the part. This means the GO gage must correspond to the largest possible diameter of the pin *after* plating. For a cylindrical pin, the plating adds thickness to the surface. Since the pin has a circular cross-section, plating increases the diameter by twice the plating thickness (one thickness on each side).

**Maximum Pin Diameter:**
The pin diameter is $25^{+0.020}_{+0.010}$ mm.
Maximum Pin Diameter = Nominal Diameter + Upper Tolerance
Maximum Pin Diameter = $25 + 0.020 = 25.020$ mm.
**Maximum Plating Thickness:**
The plating thickness is $30^{\pm 2.0}$ micron.
Maximum Plating Thickness = Nominal Thickness + Upper Tolerance
Maximum Plating Thickness = $30 + 2.0 = 32$ micron.
Converting to mm: $32 \times 0.001 = 0.032$ mm.
**GO Gage Size Calculation:**
For a cylindrical component, the plating adds to the diameter on both sides. Therefore, the total increase in diameter due to plating is $2 \times$ Maximum Plating Thickness.
GO Gage Size = Maximum Pin Diameter + $(2 \times$ Maximum Plating Thickness $)$
GO Gage Size = $25.020 \text{ mm} + (2 \times 0.032 \text{ mm})$
GO Gage Size = $25.020 \text{ mm} + 0.064 \text{ mm}$
GO Gage Size = $25.084 \text{ mm}$ .

Q42GATE 2013MCQ2MManufacturing Processes II

During the electrochemical machining (ECM) of iron (atomic weight = 56, valency = 2) at current of 1000 A with 90% current efficiency, the material removal rate was observed to be 0.26 gm/s. If Titanium (atomic weight = 48, valency = 3) is machined by the ECM process at the current of 2000 A with 90% current efficiency, the expected material removal rate in gm/s will be

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📖 Explanation

This problem involves Electrochemical Machining (ECM) and asks us to find the Material Removal Rate (MRR) for Titanium based on given data for Iron. The fundamental principle governing material removal in ECM is Faraday's laws of electrolysis, which can be expressed by the formula for MRR:
$\text{MRR} = \frac{I \cdot A \cdot \eta}{n \cdot F}$
Here, $I$ is the current, $A$ is the atomic weight, $\eta$ is the current efficiency, $n$ is the valency, and $F$ is Faraday's constant ( $96500 \text{ C/mol}$ ).

We are given the following:
For Iron: $A_{Fe} = 56$ , $n_{Fe} = 2$ , $I_{Fe} = 1000 \, \text{A}$ , $\text{MRR}_{Fe} = 0.26 \, \text{g/s}$ .
For Titanium: $A_{Ti} = 48$ , $n_{Ti} = 3$ , $I_{Ti} = 2000 \, \text{A}$ , and $\eta = 0.9$ (for both materials).

Using the formula for Titanium:
$\text{MRR}_{Ti} = \frac{I_{Ti} \cdot A_{Ti} \cdot \eta}{n_{Ti} \cdot F}$
Substituting the values:
$\text{MRR}_{Ti} = \frac{2000 \times 48 \times 0.9}{3 \times 96500}$
Calculating the value:
$\text{MRR}_{Ti} = \frac{86400}{289500} \approx 0.298 \, \text{g/s} \approx 0.3 \, \text{g/s}$
Thus, the expected material removal rate for Titanium is approximately $0.3 \, \text{g/s}$ .

Q43GATE 2013MCQ2MGeneral Engineering

Specific enthalpy and velocity of steam at inlet and exit of a steam turbine, running under steady state, are as given below: Specific enthalpy (kJ/kg) Velocity (m/s) Inlet steam condition 3250 180 Exit steam condition 2360 5 The rate of heat loss from the turbine per kg of steam flow rate is 5 kW. Neglecting changes in potential energy of steam, the power developed in kW by the steam turbine per kg of steam flow rate, is

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📖 Explanation

To find the power developed by the steam turbine, we apply the Steady Flow Energy Equation (SFEE), neglecting changes in potential energy. The SFEE for a turbine is given by $w = (h_{\in} - h_{out}) + \frac{V_{\in}^2 - V_{out}^2}{2} + q$ , where $w$ is power developed per unit mass flow rate, $h$ is specific enthalpy, $V$ is velocity, and $q$ is heat transfer per unit mass (heat loss is negative).

Given:
$h_{\in} = 3250 \text{ kJ/kg}$
$V_{\in} = 180 \text{ m/s}$
$h_{out} = 2360 \text{ kJ/kg}$
$V_{out} = 50 \text{ m/s}$
Rate of heat loss $= 5 \text{ kW}$ , which means $q = -5 \text{ kJ/kg}$ for 1 kg/s steam flow.

First, calculate the change in enthalpy:
$\Delta h = h_{\in} - h_{out} = 3250 \text{ kJ/kg} - 2360 \text{ kJ/kg} = 890 \text{ kJ/kg}$

Next, calculate the change in kinetic energy, remembering to convert units to kJ/kg by dividing by $2 \times 1000$ :
$\Delta KE = \frac{V_{\in}^2 - V_{out}^2}{2 \times 1000} = \frac{(180 \text{ m/s})^2 - (50 \text{ m/s})^2}{2000} = \frac{32400 - 2500}{2000} = \frac{29900}{2000} = 14.95 \text{ kJ/kg}$

Finally, substitute these values into the SFEE to find the power developed:
$w = \Delta h + \Delta KE + q = 890 \text{ kJ/kg} + 14.95 \text{ kJ/kg} + (-5 \text{ kJ/kg}) = 890 + 14.95 - 5 = 899.95 \text{ kJ/kg}$

The power developed is $899.95 \text{ kW}$ (since 1 kJ/kg for 1 kg/s flow rate equals 1 kW). This value is closest to option A.

Q44GATE 2013MCQ2MGeneral Engineering

A simply supported beam of length $L$ is subjected to a varying distributed load $\sin (3 \pi x/L)$ Nm $^{-1}$ , where the distance $x$ is measured from the left support. The magnitude of the vertical reaction force in N at the left support is

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📖 Explanation

To find the vertical reaction force at the left support ( $R_A$ ) of the simply supported beam, we use the principle of static equilibrium, specifically the moment equilibrium equation. We will take moments about the right support (B) to eliminate the unknown reaction $R_B$ .

The distributed load is given by $w(x) = \sin(3\pi x/L)$ Nm $^{-1}$ . The moment equilibrium equation about point B is:
$\sum M_B = R_A \times L - \int_0^L x \cdot w(x) \, dx = 0$
Substituting $w(x)$ , we get:
$R_A \times L = \int_0^L x \sin\left(\frac{3\pi x}{L}\right) \, dx$
Now, let's evaluate the integral using integration by parts, $\int u \, dv = uv - \int v \, du$ .
Let $u = x$ and $dv = \sin\left(\frac{3\pi x}{L}\right) dx$ .
Then, $du = dx$ and $v = -\frac{L}{3\pi} \cos\left(\frac{3\pi x}{L}\right)$ .
Applying integration by parts:
$\int_0^L x \sin\left(\frac{3\pi x}{L}\right) \, dx = \left[ x \left(-\frac{L}{3\pi} \cos\left(\frac{3\pi x}{L}\right)\right) \right]_0^L - \int_0^L \left(-\frac{L}{3\pi} \cos\left(\frac{3\pi x}{L}\right)\right) \, dx$
$= \left[ -\frac{Lx}{3\pi} \cos\left(\frac{3\pi x}{L}\right) \right]_0^L + \frac{L}{3\pi} \int_0^L \cos\left(\frac{3\pi x}{L}\right) \, dx$
Evaluating the first term:
$\left[-\frac{Lx}{3\pi} \cos\left(\frac{3\pi x}{L}\right)\right]_0^L = \left(-\frac{L^2}{3\pi} \cos(3\pi)\right) - (0) = -\frac{L^2}{3\pi} (-1) = \frac{L^2}{3\pi}$
Evaluating the integral term:
$\frac{L}{3\pi} \int_0^L \cos\left(\frac{3\pi x}{L}\right) \, dx = \frac{L}{3\pi} \left[ \frac{L}{3\pi} \sin\left(\frac{3\pi x}{L}\right) \right]_0^L = \frac{L}{3\pi} \left( \frac{L}{3\pi} \sin(3\pi) - \frac{L}{3\pi} \sin(0) \right) = \frac{L}{3\pi} (0 - 0) = 0$
So, the total value of the integral is:
$\int_0^L x \sin\left(\frac{3\pi x}{L}\right) \, dx = \frac{L^2}{3\pi} + 0 = \frac{L^2}{3\pi}$
Substitute this back into our moment equilibrium equation:
$R_A \times L = \frac{L^2}{3\pi}$
Solving for $R_A$ :
$R_A = \frac{1}{L} \left( \frac{L^2}{3\pi} \right) = \frac{L}{3\pi}$
The vertical reaction force at the left support is $\frac{L}{3\pi}$ N.

Q45GATE 2013MCQ2MEngineering Mathematics

The probability that a student knows the correct answer to a multiple choice question is $\frac{2}{3}$ . If the student does not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is $\frac{1}{4}$ . Given that the student has answered the question correctly, the conditional probability that the student knows the correct answer is

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📖 Explanation

This problem requires us to find a conditional probability using Bayes' Theorem. We want to find the probability that the student *knew* the answer, *given* that they answered correctly. Let's define our events:
* $K$ : Student Knows the answer.
* $G$ : Student Guesses the answer (meaning they don't know).
* $C$ : Student answers Correctly.

We are given:
* Probability of knowing the answer: $P(K) = \frac{2}{3}$
* Probability of not knowing the answer (and thus guessing): $P(G) = 1 - P(K) = 1 - \frac{2}{3} = \frac{1}{3}$
* Probability of guessing correctly given that the student guessed: $P(C|G) = \frac{1}{4}$
* If a student *knows* the answer, they will certainly answer correctly: $P(C|K) = 1$

We need to find $P(K|C)$ , the probability that the student knew the answer given that they answered correctly. Bayes' Theorem states:
$P(K|C) = \frac{P(C|K) \cdot P(K)}{P(C)}$

First, let's find the total probability of answering correctly, $P(C)$ , using the Law of Total Probability:
$P(C) = P(C|K) \cdot P(K) + P(C|G) \cdot P(G)$
$P(C) = 1 \cdot \frac{2}{3} + \frac{1}{4} \cdot \frac{1}{3}$
$P(C) = \frac{2}{3} + \frac{1}{12}$
To add these fractions, find a common denominator (12):
$P(C) = \frac{8}{12} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4}$

Now substitute $P(C)$ back into Bayes' Theorem:
$P(K|C) = \frac{1 \cdot \frac{2}{3}}{\frac{3}{4}}$
$P(K|C) = \frac{2}{3} \times \frac{4}{3}$
$P(K|C) = \frac{8}{9}$

Q46GATE 2013MCQ2MEngineering Mathematics

The solution to the differential equation $\frac{d^2u}{dx^2} - k \frac{du}{dx} = 0$ where $k$ is a constant, subjected to the boundary conditions $u(0) = 0$ and $u(L) = U$ , is

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📖 Explanation

To solve the given second-order linear homogeneous differential equation, $\frac{d^2u}{dx^2} - k \frac{du}{dx} = 0$ , we start by finding its characteristic equation. Assume a solution of the form $u = e^{rx}$ , which transforms the differential equation into the algebraic equation $r^2 - kr = 0$ .

Factoring the characteristic equation gives $r(r - k) = 0$ , yielding two distinct roots: $r_1 = 0$ and $r_2 = k$ . With distinct roots, the general solution for $u(x)$ is a linear combination of exponential terms: $u(x) = C_1 e^{0x} + C_2 e^{kx}$ , which simplifies to $u(x) = C_1 + C_2 e^{kx}$ . Here, $C_1$ and $C_2$ are constants we need to determine using the boundary conditions.

Now, we apply the boundary conditions $u(0) = 0$ and $u(L) = U$ .
For the first boundary condition, $u(0) = 0$ :
$C_1 + C_2 e^{k \cdot 0} = 0 \implies C_1 + C_2 = 0 \implies C_1 = -C_2$ .

For the second boundary condition, $u(L) = U$ :
$C_1 + C_2 e^{kL} = U$ .

Substitute $C_1 = -C_2$ into the second boundary condition equation:
$-C_2 + C_2 e^{kL} = U$ .
Factor out $C_2$ : $C_2 (e^{kL} - 1) = U$ .
Solving for $C_2$ (assuming $k \neq 0$ and $e^{kL} \neq 1$ ):
$C_2 = \frac{U}{e^{kL} - 1}$ .

Now find $C_1$ using $C_1 = -C_2$ :
$C_1 = -\frac{U}{e^{kL} - 1} = \frac{U}{1 - e^{kL}}$ .

Finally, substitute the values of $C_1$ and $C_2$ back into the general solution $u(x) = C_1 + C_2 e^{kx}$ :
$u(x) = \frac{U}{1 - e^{kL}} + \left( \frac{U}{e^{kL} - 1} \right) e^{kx}$ .
To simplify this expression and match the options, rewrite $\frac{1}{e^{kL} - 1}$ as $-\frac{1}{1 - e^{kL}}$ :
$u(x) = \frac{U}{1 - e^{kL}} - \frac{U}{1 - e^{kL}} e^{kx}$ .
Factor out the common term $\frac{U}{1 - e^{kL}}$ :
$u(x) = U \left( \frac{1 - e^{kx}}{1 - e^{kL}} \right)$ .

Q47GATE 2013MCQ2MEngineering Mathematics

The value of the definite integral $\int_1^e \sqrt{x} \ln(x) dx$ is

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📖 Explanation

To evaluate the definite integral $\int_1^e \sqrt{x} \ln(x) dx$ , we use integration by parts, given by $\int u \, dv = uv - \int v \, du$ .

First, we choose our parts:
Let $u = \ln(x)$ and $dv = \sqrt{x} \, dx = x^{1/2} \, dx$ .
Then, we find $du$ and $v$ :
$du = \frac{1}{x} \, dx$
$v = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}$ .

Now, substitute these into the integration by parts formula with the limits:
$\int_1^e \sqrt{x} \ln(x) dx = \left[ \ln(x) \cdot \frac{2}{3} x^{3/2} \right]_1^e - \int_1^e \frac{2}{3} x^{3/2} \cdot \frac{1}{x} dx$

Let's evaluate the first term:
$\left[ \frac{2}{3} x^{3/2} \ln(x) \right]_1^e = \left( \frac{2}{3} e^{3/2} \ln(e) \right) - \left( \frac{2}{3} (1)^{3/2} \ln(1) \right)$
Since $\ln(e) = 1$ and $\ln(1) = 0$ , this simplifies to:
$= \left( \frac{2}{3} e^{3/2} \cdot 1 \right) - \left( \frac{2}{3} \cdot 1 \cdot 0 \right) = \frac{2}{3} e^{3/2}$

Next, we evaluate the remaining integral:
$\int_1^e \frac{2}{3} x^{3/2} \cdot \frac{1}{x} dx = \int_1^e \frac{2}{3} x^{3/2 - 1} dx = \int_1^e \frac{2}{3} x^{1/2} dx$
Integrate this term:
$= \frac{2}{3} \left[ \frac{x^{3/2}}{3/2} \right]_1^e = \frac{2}{3} \left[ \frac{2}{3} x^{3/2} \right]_1^e = \frac{4}{9} \left[ x^{3/2} \right]_1^e$
Evaluate at the limits:
$= \frac{4}{9} (e^{3/2} - 1^{3/2}) = \frac{4}{9} (e^{3/2} - 1) = \frac{4}{9} e^{3/2} - \frac{4}{9}$

Finally, combine both parts to get the total integral value:
$\int_1^e \sqrt{x} \ln(x) dx = \left( \frac{2}{3} e^{3/2} \right) - \left( \frac{4}{9} e^{3/2} - \frac{4}{9} \right)$
$= \frac{2}{3} e^{3/2} - \frac{4}{9} e^{3/2} + \frac{4}{9}$
To combine the $e^{3/2}$ terms, use a common denominator of 9:
$= \frac{6}{9} e^{3/2} - \frac{4}{9} e^{3/2} + \frac{4}{9} = \frac{2}{9} e^{3/2} + \frac{4}{9}$
Express $e^{3/2}$ as $\sqrt{e^3}$ :
$= \frac{2}{9} \sqrt{e^3} + \frac{4}{9}$

Q48GATE 2013MCQ2MEngineering Mathematics

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $F = \xi + yj + zk$ defined with respect to a Cartesian coordinate system having $i, j$ and $k$ as unit base vectors. $\int\int_S \frac{1}{4}(F \cdot n) dA$ where $S$ is the sphere, $x^2 + y^2 + z^2 =1$ and $n$ is the outward unit normal vector to the sphere. The value of the surface integral is

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📖 Explanation

This problem asks us to evaluate a surface integral over a sphere. The integral is $\int\int_S \frac{1}{4}(F \cdot n) dA$ , where $F = \xi + yj + zk$ and $S$ is the unit sphere $x^2 + y^2 + z^2 = 1$ .

First, let's determine the outward unit normal vector $n$ and the dot product $F \cdot n$ . For a sphere centered at the origin with radius $R$ , the outward unit normal vector $n$ at any point $(x, y, z)$ on its surface is given by the position vector divided by its magnitude: $n = \frac{\xi + yj + zk}{\sqrt{x^2 + y^2 + z^2}}$ . Since $S$ is the unit sphere, $x^2 + y^2 + z^2 = 1$ , which means $\sqrt{x^2 + y^2 + z^2} = 1$ . So, the outward unit normal vector simplifies to $n = \xi + yj + zk$ .

Now, we calculate the dot product $F \cdot n$ :
$F \cdot n = (\xi + yj + zk) \cdot (\xi + yj + zk) = x(x) + y(y) + z(z) = x^2 + y^2 + z^2$
On the surface $S$ , we know that $x^2 + y^2 + z^2 = 1$ . Therefore, $F \cdot n = 1$ .

Next, we substitute this value back into the surface integral:
$\int\int_S \frac{1}{4}(F \cdot n) dA = \int\int_S \frac{1}{4}(1) dA$
We can factor out the constant $\frac{1}{4}$ :
$= \frac{1}{4} \int\int_S dA$
The integral $\int\int_S dA$ represents the surface area of the sphere $S$ . The surface area of a sphere with radius $R$ is $A = 4\pi R^2$ . For our unit sphere, $R=1$ , so the surface area is $A = 4\pi (1)^2 = 4\pi$ .

Finally, substitute the surface area back into the expression:
$= \frac{1}{4} \times (4\pi) = \pi$
The value of the surface integral is $\pi$ .

Q49GATE 2013MCQ2MEngineering Mathematics

The function $f(t)$ satisfies the differential equation $\frac{d^2f}{dt^2} + f = 0$ and the auxiliary conditions, $f(0) = 0, \frac{df}{dt}(0) = 4$ . The Laplace transform of $f(t)$ is given by

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📖 Explanation

We need to find the Laplace transform, $F(s) = \mathcal{L}\{f(t)\}$ , of a function $f(t)$ that satisfies the differential equation $\frac{d^2f}{dt^2} + f = 0$ with initial conditions $f(0) = 0$ and $\frac{df}{dt}(0) = 4$ .

First, we apply the Laplace transform to both sides of the given differential equation:
$\mathcal{L}\left\{\frac{d^2f}{dt^2} + f\right\} = \mathcal{L}\{0\}$
Using the linearity property of the Laplace transform, this becomes:
$\mathcal{L}\left\{\frac{d^2f}{dt^2}\right\} + \mathcal{L}\{f(t)\} = 0$
Next, we use the Laplace transform property for derivatives: $\mathcal{L}\left\{\frac{d^2f}{dt^2}\right\} = s^2 F(s) - s f(0) - f'(0)$ . Substituting this into our equation, along with $F(s) = \mathcal{L}\{f(t)\}$ , we get:
$(s^2 F(s) - s f(0) - f'(0)) + F(s) = 0$
Now, we substitute the given initial conditions, $f(0) = 0$ and $f'(0) = 4$ :
$s^2 F(s) - s(0) - 4 + F(s) = 0$
Simplifying the equation:
$s^2 F(s) - 4 + F(s) = 0$
To solve for $F(s)$ , we group the terms containing $F(s)$ :
$F(s) (s^2 + 1) - 4 = 0$
Moving the constant term to the other side:
$F(s) (s^2 + 1) = 4$
Finally, divide by $(s^2 + 1)$ to find $F(s)$ :
$F(s) = \frac{4}{s^2 + 1}$
The Laplace transform of $f(t)$ is $\frac{4}{s^2 + 1}$ .

Q50GATE 2013MCQ2MGeneral Engineering

A flywheel connected to a punching machine has to supply energy of 400 Nm while running at a mean angular speed of 20 rad/s. If the total fluctuation of speed is not to exceed $\pm2\%$ , the mass moment of inertia of the flywheel in kg-m $^2$ is

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📖 Explanation

A flywheel helps to smooth out energy fluctuations in machines. Its ability to do this is measured by its *mass moment of inertia* ( $J$ ). We need to find $J$ for a punching machine's flywheel.

The key formula relating the energy supplied ( $\Delta E$ ), mass moment of inertia ( $J$ ), mean angular speed ( $\omega_{mean}$ ), and the fluctuation in angular speed ( $\Delta \omega$ ) is:
$\Delta E = J \times \omega_{mean} \times \Delta \omega$

Given:
* Energy supplied, $\Delta E = 400$ Nm
* Mean angular speed, $\omega_{mean} = 20$ rad/s
* Total fluctuation of speed = $\pm 2\%$

First, let's calculate the absolute fluctuation in angular speed ( $\Delta \omega$ ). A fluctuation of $\pm 2\%$ means the speed varies $2\%$ below the mean to $2\%$ above the mean. So, the total percentage fluctuation is $2\% + 2\% = 4\%$ .
Therefore, the absolute fluctuation is:
$\Delta \omega = 0.04 \times \omega_{mean}$
$\Delta \omega = 0.04 \times 20 \, \text{rad/s} = 0.8 \, \text{rad/s}$

Now, rearrange the main formula to solve for $J$ :
$J = \frac{\Delta E}{\omega_{mean} \times \Delta \omega}$
Substitute the known values into this equation:
$J = \frac{400 \, \text{Nm}}{20 \, \text{rad/s} \times 0.8 \, \text{rad/s}}$
$J = \frac{400}{16} \, \text{kg-m}^2$
$J = 25 \, \text{kg-m}^2$

Thus, the mass moment of inertia of the flywheel is $25$ kg-m $^2$ .

Q51GATE 2013MCQ2MGeneral Engineering

A single riveted lap joint of two similar plates as shown in the figure below has the following geometrical and material details. plate width $w$ = 200 mm plate thickness $t$ = 5 mm number of rivets $n$ = 3 rivet diameter $d_r$ = 10 mm rivet hole diameter $d_h$ = 11 mm allowable tensile stress of plate $\sigma_p$ = 200 MPa allowable bearing stress of rivet $\sigma_c$ = 150 MPa If the plates are to be designed to avoid tearing failure, the maximum permissible load $P$ in kN is

🚀 Solve in Practice Mode

📖 Explanation

To prevent the plate from tearing, we need to calculate its tearing resistance. This is determined by the formula: $P = \sigma_p \cdot (w - n \cdot d_h) \cdot t$ . Given $\sigma_p = 200 \, \text{MPa} = 200 \, \text{N/mm}^2$ , $w = 200 \, \text{mm}$ , $n = 3$ , $d_h = 11 \, \text{mm}$ , and $t = 5 \, \text{mm}$ . Substituting these values: $P = 200 \times (200 - 3 \times 11) \times 5 = 200 \times (200 - 33) \times 5 = 200 \times 167 \times 5 = 167000 \, \text{N}$ . Converting to kN, $P = \frac{167000}{1000} = 167 \, \text{kN}$ . Therefore, the maximum permissible load $P$ to avoid tearing failure is $167 \, \text{kN}$ .

Q52GATE 2013MCQ2MManufacturing Processes II

Two cutting tools are being compared for a machining operation. The tool life equations are: Carbide tool: $VT^{1.6} = 3000$ HSS tool: $VT^{0.6} = 200$ where $V$ is the cutting speed in m/min and $T$ is the tool life in min. The carbide tool will provide higher tool life if the cutting speed in m/min exceeds

🚀 Solve in Practice Mode

📖 Explanation

Here's how to figure out when the carbide tool is better:

First, let's express tool life ( $T$ ) for each tool in terms of cutting speed ( $V$ ) using the given equations:
For the Carbide tool: $VT^{1.6} = 3000 \implies T_{carbide} = \left(\frac{3000}{V}\right)^{\frac{1}{1.6}}$
For the HSS tool: $VT^{0.6} = 200 \implies T_{HSS} = \left(\frac{200}{V}\right)^{\frac{1}{0.6}}$

We want to find the cutting speed ( $V$ ) where the carbide tool gives a longer tool life, meaning $T_{carbide} > T_{HSS}$ :
$\left(\frac{3000}{V}\right)^{\frac{1}{1.6}} > \left(\frac{200}{V}\right)^{\frac{1}{0.6}}$
To solve for $V$ , we take the natural logarithm of both sides. This allows us to bring the exponents down:
$\frac{1}{1.6} \ln\left(\frac{3000}{V}\right) > \frac{1}{0.6} \ln\left(\frac{200}{V}\right)$
Substitute the decimal values for the exponents ( $1/1.6 = 0.625$ and $1/0.6 \approx 1.667$ ) and expand the logarithms:
$0.625 (\ln(3000) - \ln(V)) > 1.667 (\ln(200) - \ln(V))$
Rearrange the terms to isolate $\ln(V)$ :
$0.625 \ln(3000) - 0.625 \ln(V) > 1.667 \ln(200) - 1.667 \ln(V)$
$(1.667 - 0.625) \ln(V) > 1.667 \ln(200) - 0.625 \ln(3000)$
$1.042 \ln(V) > 1.667 \times 5.2983 - 0.625 \times 8.0064$
$1.042 \ln(V) > 8.837 - 5.004$
$1.042 \ln(V) > 3.833$
$\ln(V) > \frac{3.833}{1.042} \approx 3.678$
Finally, to find $V$ , we exponentiate both sides:
$V > e^{3.678}$
$V > 39.55 \, \text{m/min}$
Therefore, the carbide tool provides higher tool life when the cutting speed exceeds approximately $39.55$ m/min, which is closest to option B.

Q53GATE 2013MCQ2MManufacturing Processes II

In water jet machining, the water jet is issued through a 0.3 mm diameter orifice at a pressure of 400 MPa. The density of water is 1000 kg/m $^3$ . The coefficient of discharge is 1.0. Neglecting all losses during water jet formation through the orifice, the power of the water jet in kW is

🚀 Solve in Practice Mode

📖 Explanation

We need to calculate the power of the water jet in a Water Jet Machining (WJM) process. We are given the orifice diameter ( $d$ ), pressure ( $P$ ), water density ( $\rho$ ), and coefficient of discharge ( $C_d$ ). Since losses are neglected, $C_d = 1.0$ .

Given Data:
* Orifice Diameter, $d = 0.3 \text{ mm} = 0.3 \times 10^{-3} \text{ m}$
* Pressure, $P = 400 \text{ MPa} = 400 \times 10^6 \text{ Pa}$
* Density of water, $\rho = 1000 \text{ kg/m}^3$
* Coefficient of Discharge, $C_d = 1.0$

**Step 1: Calculate the Jet Velocity ( $v$ )**
The kinetic energy per unit volume of the jet is equal to the pressure energy. This gives us the velocity of the water jet.
$\frac{1}{2} \rho v^2 = P$
Rearranging for $v$ :
$v = \sqrt{\frac{2P}{\rho}}$
Substitute the given values:
$v = \sqrt{\frac{2 \times (400 \times 10^6 \text{ Pa})}{1000 \text{ kg/m}^3}} = \sqrt{800 \times 10^3} \text{ m/s} = \sqrt{800000} \text{ m/s} \approx 894.43 \text{ m/s}$

**Step 2: Calculate the Orifice Area ( $A$ )**
The area of the circular orifice is calculated from its diameter.
$A = \frac{\pi d^2}{4}$
Substitute the orifice diameter:
$A = \frac{\pi (0.3 \times 10^{-3} \text{ m})^2}{4} = \frac{\pi \times 0.09 \times 10^{-6}}{4} \text{ m}^2 \approx 7.0686 \times 10^{-8} \text{ m}^2$

**Step 3: Calculate the Volumetric Flow Rate ( $Q$ )**
The volumetric flow rate is the product of the orifice area, jet velocity, and the coefficient of discharge.
$Q = C_d \times A \times v$
Substitute the calculated values:
$Q = 1.0 \times (7.0686 \times 10^{-8} \text{ m}^2) \times (894.43 \text{ m/s}) \approx 6.324 \times 10^{-5} \text{ m}^3/\text{s}$

**Step 4: Calculate the Jet Power ( $P_{jet}$ )**
The power of the water jet is the rate at which energy is delivered, which can be found by multiplying the pressure by the volumetric flow rate.
$P_{jet} = P \times Q$
Substitute the pressure and flow rate:
$P_{jet} = (400 \times 10^6 \text{ Pa}) \times (6.324 \times 10^{-5} \text{ m}^3/\text{s}) \approx 25296 \text{ W}$

**Step 5: Convert Power to Kilowatts (kW)**
To express the power in kilowatts, divide by 1000.
$P_{jet} \approx \frac{25296}{1000} \text{ kW} = 25.296 \text{ kW}$
Rounding to one decimal place, the power of the water jet is approximately $25.3 \text{ kW}$ .

Q54GATE 2013MCQ2MOperations Research and Operations Management

A linear programming problem is shown below. Maximize $3x + 7y$ Subject to $3x + 7y \le 10$ $4x + 6y \le 8$ $x, y \ge 0$ It has

🚀 Solve in Practice Mode

📖 Explanation

To find the optimal solution, we first identify the feasible region and its corner points (vertices). The given linear programming problem is:
Maximize $Z = 3x + 7y$
Subject to:

$3x + 7y \le 10$
$4x + 6y \le 8 \implies 2x + 3y \le 4$
$x \ge 0, y \ge 0$

We find the vertices of the feasible region:
* **Origin:** The intersection of $x=0$ and $y=0$ is $(0, 0)$ .
* **X-axis intercept:** Set $y=0$ in the simplified Constraint 2 ( $2x + 3y = 4$ ). This gives $2x = 4 \implies x = 2$ . So, the vertex is $(2, 0)$ . Checking with Constraint 1: $3(2) + 7(0) = 6 \le 10$ , which means it's feasible.
* **Y-axis intercept:** Set $x=0$ in Constraint 2 ( $2x + 3y = 4$ ). This gives $3y = 4 \implies y = 4/3$ . So, the vertex is $(0, 4/3)$ . Checking with Constraint 1: $3(0) + 7(4/3) = 28/3 \approx 9.33 \le 10$ , which means it's feasible.
* **Intersection of constraint lines:** Solve $3x + 7y = 10$ and $2x + 3y = 4$ . This system yields $x = -2/5$ and $y = 8/5$ . Since $x < 0$ , this point is outside the feasible region (as $x \ge 0$ is a constraint).

The vertices of the feasible region are $(0, 0)$ , $(2, 0)$ , and $(0, 4/3)$ . Notice that the constraint $3x + 7y \le 10$ is redundant because the feasible region defined by $x \ge 0, y \ge 0, 2x + 3y \le 4$ already satisfies it.

Next, we evaluate the objective function $Z = 3x + 7y$ at each vertex:
* At $(0, 0)$ : $Z = 3(0) + 7(0) = 0$ .
* At $(2, 0)$ : $Z = 3(2) + 7(0) = 6$ .
* At $(0, 4/3)$ : $Z = 3(0) + 7(4/3) = 28/3$ .

The maximum value of the objective function is $28/3$ , which occurs uniquely at the vertex $(0, 4/3)$ . Therefore, the linear programming problem has exactly one optimal solution.

Q55GATE 2013MCQ2MOperations Research and Operations Management

Consider a two machine flow shop where jobs are first processed in Machine X and then in Machine Y, in the same sequence. The processing times of four jobs (1, 2, 3 and 4) on the machines are: Job Processing time (in min) Machine X Machine Y 1 6 5 2 3 4 3 7 6 4 5 4 The sequence of jobs on the machines that minimizes make span is

🚀 Solve in Practice Mode

📖 Explanation

To minimize the makespan (total completion time) in a two-machine flow shop, we apply Johnson's Algorithm. First, categorize jobs into two groups: Group 1 where processing time on Machine X ( $P_x$ ) is less than or equal to processing time on Machine Y ( $P_y$ ), and Group 2 where $P_x > P_y$ . Then, sort jobs in Group 1 in increasing order of $P_x$ , and jobs in Group 2 in decreasing order of $P_y$ . Finally, combine the sorted Group 1 jobs followed by the sorted Group 2 jobs to get the optimal sequence.

Given data:
| Job | Machine X ( $P_x$ ) (min) | Machine Y ( $P_y$ ) (min) | Condition ( $P_x$ vs $P_y$ ) |
|-----|-------------------------|-------------------------|--------------------------|
| 1 | 6 | 5 | $P_x > P_y$ |
| 2 | 3 | 4 | $P_x \le P_y$ |
| 3 | 7 | 6 | $P_x > P_y$ |
| 4 | 5 | 4 | $P_x > P_y$ |

Based on the conditions:
* Group 1 ( $P_x \le P_y$ ): Job 2 ( $P_x=3, P_y=4$ )
* Group 2 ( $P_x > P_y$ ): Job 1 ( $P_x=6, P_y=5$ ), Job 3 ( $P_x=7, P_y=6$ ), Job 4 ( $P_x=5, P_y=4$ )

Sorting within groups:
* Sorted Group 1 (increasing $P_x$ ): [2]
* Sorted Group 2 (decreasing $P_y$ ): Job 3 ( $P_y=6$ ), Job 1 ( $P_y=5$ ), Job 4 ( $P_y=4$ ) $\rightarrow$ [3, 1, 4]

Combining these sequences yields the optimal sequence: 2 - 3 - 1 - 4.

Q56GATE 2013MCQ2MIndustrial Engineering

Match the CORRECT pairs. Group I Group II P. Man-machine chart 1. Determines standard time of jobs Q. Learning curve 2. Finds the preferred method of doing work R. Time study 3. Measures work improvement S. Motion study 4. Shows idle times

🚀 Solve in Practice Mode

Q57GATE 2013MCQ2MIndustrial Engineering

A firm produces 120 units of product in every 8 hour shift. Four operations as given below are needed to manufacture each unit of product. Operation Precedence Processing time (in min) P none 1 Q P 2 R P 4 S Q, R 3 The above operations are to be assigned to workstations, such that one or more operations are performed in each workstation. Only one unit of product will be processed in each workstation at a time. The minimum number of workstations that will achieve the production target, without violating the precedence constraints, is

🚀 Solve in Practice Mode

📖 Explanation

To find the minimum number of workstations, we first determine the production pace.
The production target is 120 units in an 8-hour shift.
Shift duration in minutes: $8 \times 60 = 480$ minutes.
Takt time, which is the available production time divided by the required output, is:
$\text{Takt Time} = \frac{480 \text{ minutes}}{120 \text{ units}} = 4 \text{ minutes/unit}$
This means each workstation must complete its assigned tasks within 4 minutes per unit to meet the production target. Now, let's assign operations while respecting precedence and the takt time:
\begin{itemize}
\item \textbf{Workstation 1:} Operation P (1 min) and Operation Q (2 min). Total time: $1+2=3$ minutes. This is $\le 4$ minutes.
\item \textbf{Workstation 2:} Operation R (4 min). Total time: 4 minutes. This is $\le 4$ minutes.
\item \textbf{Workstation 3:} Operation S (3 min). Total time: 3 minutes. This is $\le 4$ minutes.
\end{itemize}
All operations are assigned, precedence constraints are met (P before Q and R; Q and R before S), and each workstation's total time is within the 4-minute takt time. Thus, the minimum number of workstations required is 3.

Q58GATE 2013MCQ2MManufacturing Processes II

Neglecting the contribution of the feed force towards cutting power, the specific cutting energy in $J/mm^3$ is

🚀 Solve in Practice Mode

📖 Explanation

The specific cutting energy ( $u$ ) tells us how much energy is needed to remove a tiny amount (unit volume) of material during machining. We find it by dividing the cutting power ( $P_c$ ) by the material removal rate (MRR): $u = \frac{P_c}{MRR}$ .

First, let's gather our given parameters:
* Outer Diameter, $D_o = 200 \text{ mm}$
* Inner Diameter, $D_i = 80 \text{ mm}$
* Feed, $f = 0.1 \text{ mm/rev}$
* Depth of Cut, $d = 1 \text{ mm}$
* Cutting Speed, $v = 90 \text{ m/min}$
* Main Cutting Force, $F_c = 200 \text{ N}$

**Step 1: Calculate Average Diameter ( $D_{avg}$ )**
For these calculations, we use the average diameter:
$D_{avg} = \frac{D_o + D_i}{2} = \frac{200 \text{ mm} + 80 \text{ mm}}{2} = 140 \text{ mm} = 0.14 \text{ m}$

**Step 2: Calculate Cutting Power ( $P_c$ )**
Cutting power is the product of cutting force and cutting speed.
$P_c = F_c \times v = 200 \text{ N} \times 90 \text{ m/min} = 18000 \text{ N·m/min} = 18000 \text{ J/min}$

**Step 3: Calculate Material Removal Rate (MRR)**
The MRR can be calculated using the formula $MRR = f \times d \times v$ . We need to ensure consistent units.
Convert feed and depth of cut to meters: $f = 0.1 \text{ mm/rev} = 0.0001 \text{ m/rev}$ and $d = 1 \text{ mm} = 0.001 \text{ m}$ .
$MRR = (0.0001 \text{ m/rev}) \times (0.001 \text{ m}) \times (90 \text{ m/min}) = 0.000009 \text{ m}^3/\text{min}$
Now, convert MRR to $mm^3/min$ :
$MRR = 0.000009 \text{ m}^3/\text{min} \times (10^9 \text{ mm}^3 / 1 \text{ m}^3) = 9000 \text{ mm}^3/\text{min}$

**Step 4: Calculate Specific Cutting Energy ( $u$ )**
Finally, we use the calculated cutting power and MRR:
$u = \frac{P_c}{MRR} = \frac{18000 \text{ J/min}}{9000 \text{ mm}^3/\text{min}} = 2 \text{ J/mm}^3$
The specific cutting energy is $2 \text{ J/mm}^3$ .

Q59GATE 2013MCQ2MManufacturing Processes II

Assuming approach and over-travel of the cutting tool to be zero, the machining time in min is

🚀 Solve in Practice Mode

📖 Explanation

This problem asks us to calculate the machining time for a facing operation on a CNC lathe with constant cutting speed.

Given Information:
Outer Diameter ( $D_{outer}$ ) = 200 mm
Inner Diameter ( $D_{inner}$ ) = 80 mm
Feed rate ( $f$ ) = 0.1 mm/rev
Depth of cut ( $d$ ) = 1 mm
Constant cutting speed ( $V$ ) = 90 m/min
Tool approach and over-travel = 0 mm

First, convert all dimensions to consistent units (meters):
Outer Radius ( $R$ ): $R = D_{outer} / 2 = 200 \text{ mm} / 2 = 100 \text{ mm} = 0.1 \text{ m}$
Inner Radius ( $r$ ): $r = D_{inner} / 2 = 80 \text{ mm} / 2 = 40 \text{ mm} = 0.04 \text{ m}$
Feed rate ( $f$ ): $f = 0.1 \text{ mm/rev} = 0.0001 \text{ m/rev}$
Cutting speed ( $V$ ): $V = 90 \text{ m/min}$

For a facing operation with constant cutting speed, the machining time ( $T_m$ ) is given by the integral formula:
$T_m = \frac{\pi (R^2 - r^2)}{f V}$
Note that the depth of cut (1 mm) determines the number of passes ( $200 \text{ mm} / 1 \text{ mm} = 200$ passes, if it were radial, but here depth of cut does not affect the time calculation by this formula).

Substitute the values into the formula:
$T_m = \frac{\pi ((0.1 \text{ m})^2 - (0.04 \text{ m})^2)}{(0.0001 \text{ m/rev}) \times (90 \text{ m/min})}$
$T_m = \frac{\pi (0.01 \text{ m}^2 - 0.0016 \text{ m}^2)}{0.009 \text{ m}^2/\text{rev-min}}$
$T_m = \frac{\pi (0.0084)}{0.009} \text{ min}$
$T_m = \pi \times \frac{84}{90} \text{ min} = \pi \times \frac{14}{15} \text{ min}$
$T_m \approx 2.932 \text{ min}$
The calculated machining time is approximately 2.93 minutes.

Q60GATE 2013MCQ2MOperations Research and Operations Management

To avoid stock out situations, the retailer needs to place orders when the inventory level (in kg) drops to

🚀 Solve in Practice Mode

📖 Explanation

To avoid running out of stock (stockouts), the retailer must place a new order when the inventory reaches a specific level, known as the Reorder Point (ROP). The ROP is calculated as the demand expected during the lead time.
$ROP = \text{Daily Demand} \times \text{Lead Time in days}$
Given a Daily Demand of $40 \, \text{kg/day}$ and a Lead Time of $3 \, \text{days}$ , we calculate the ROP:
$ROP = 40 \, \text{kg/day} \times 3 \, \text{days} = 120 \, \text{kg}$
Therefore, the retailer needs to place an order when the inventory level drops to $120 \, \text{kg}$ .

Q61GATE 2013MCQ2MOperations Research and Operations Management

If the retailer uses an optimum order policy to minimize the total cost, the saving in Rs. in the total cost as compared to the current policy will be

🚀 Solve in Practice Mode

📖 Explanation

This problem asks us to find the cost savings when a retailer switches from their current ordering policy to an optimal policy, known as the Economic Order Quantity (EOQ). We'll calculate all costs on a daily basis.

First, let's find the **Optimum Order Quantity (EOQ)** that minimizes total inventory costs using the formula:
$Q^* = \sqrt{\frac{2DS}{H}}$
Here, $D$ (Daily Demand) $= 40 \text{ kg/day}$ , $S$ (Ordering Cost per order) $= \text{Rs. } 200$ , and $H$ (Holding Cost per kg per day) $= \text{Rs. } 0.1$ .
Plugging in the values:
$Q^* = \sqrt{\frac{2 \times 40 \times 200}{0.1}} = \sqrt{\frac{16000}{0.1}} = \sqrt{160000} = 400 \text{ kg}$

Next, we calculate the **Daily Cost for the Optimum Policy (EOQ)**. This includes purchase, ordering, and holding costs.
Daily Purchase Cost $= \text{Daily Demand} \times \text{Cost per kg} = 40 \times 50 = 2000 \text{ Rs.}$
Daily Ordering Cost $= (\text{Number of orders per day}) \times \text{Ordering Cost per order}$
Number of orders per day $= D / Q^* = 40 / 400 = 0.1$
So, Daily Ordering Cost $= 0.1 \times 200 = 20 \text{ Rs.}$
Daily Holding Cost $= (\text{Average Inventory}) \times \text{Holding Cost per kg per day}$
Average Inventory $= Q^* / 2 = 400 / 2 = 200 \text{ kg}$
So, Daily Holding Cost $= 200 \times 0.1 = 20 \text{ Rs.}$
Total Daily Cost (EOQ) $= 2000 + 20 + 20 = 2040 \text{ Rs.}$

Now, let's calculate the **Daily Cost for the Current Policy**. The current policy involves ordering $200 \text{ kg}$ every $5$ days.
Daily Purchase Cost $= 40 \times 50 = 2000 \text{ Rs.}$
Daily Ordering Cost $= \text{Ordering Cost per order} / \text{Order Frequency (days)} = 200 / 5 = 40 \text{ Rs.}$
Daily Holding Cost $= (\text{Average Inventory}) \times \text{Holding Cost per kg per day}$
Average Inventory $= \text{Current Order Quantity} / 2 = 200 / 2 = 100 \text{ kg}$
So, Daily Holding Cost $= 100 \times 0.1 = 10 \text{ Rs.}$
Total Daily Cost (Current) $= 2000 + 40 + 10 = 2050 \text{ Rs.}$

Finally, we find the **Daily Saving**. This is the difference between the current daily cost and the optimal daily cost.
Daily Saving $= \text{Total Daily Cost (Current)} - \text{Total Daily Cost (EOQ)}$
Daily Saving $= 2050 - 2040 = 10 \text{ Rs.}$
The saving in total cost per day is $\text{Rs. } 10$ .

Q62GATE 2013MCQ2MOperations Research and Operations Management

The critical path of the project, based on the mean activity duration, is

🚀 Solve in Practice Mode

📖 Explanation

To find the critical path, we must identify all possible paths from the project start (Node 1) to the end (Node 5) and sum the mean durations ( $\mu$ ) for each activity along these paths.

The activity durations are:
* $\mu_{1-2} = 6$ days
* $\mu_{1-3} = 9$ days
* $\mu_{2-3} = 2$ days
* $\mu_{2-4} = 8$ days
* $\mu_{3-4} = 7$ days
* $\mu_{3-5} = 8$ days
* $\mu_{4-5} = 4$ days

Let's calculate the total duration for each possible path:
* Path 1: $1 \to 2 \to 3 \to 4 \to 5$
Duration $= \mu_{1-2} + \mu_{2-3} + \mu_{3-4} + \mu_{4-5} = 6 + 2 + 7 + 4 = 19$ days
* Path 2: $1 \to 2 \to 3 \to 5$
Duration $= \mu_{1-2} + \mu_{2-3} + \mu_{3-5} = 6 + 2 + 8 = 16$ days
* Path 3: $1 \to 2 \to 4 \to 5$
Duration $= \mu_{1-2} + \mu_{2-4} + \mu_{4-5} = 6 + 8 + 4 = 18$ days
* Path 4: $1 \to 3 \to 4 \to 5$
Duration $= \mu_{1-3} + \mu_{3-4} + \mu_{4-5} = 9 + 7 + 4 = 20$ days
* Path 5: $1 \to 3 \to 5$
Duration $= \mu_{1-3} + \mu_{3-5} = 9 + 8 = 17$ days

The critical path is the path with the longest total mean duration. Comparing the path durations: $19, 16, 18, 20, 17$ days. The maximum duration is $20$ days, corresponding to Path 4: $1 \to 3 \to 4 \to 5$ .

Q63GATE 2013MCQ2MOperations Research and Operations Management

Let $\Phi$ denote the cumulative distribution function of the standard normal random variable. The probability that all activities on the critical path, based on the mean activity duration, are completed in 22 days is

🚀 Solve in Practice Mode

📖 Explanation

This problem asks for the probability that a project's critical path finishes within a specific timeframe, using the standard normal distribution. We'll find the critical path, calculate its mean and standard deviation, and then use the Z-score to determine the probability.

**Identify Project Paths and Durations:**
We list all possible paths from the start (node 1) to the end (node 5) and sum their mean activity durations:
* Path 1 (1-2-3-4-5): Mean = $6 + 2 + 7 + 4 = 19$ days
* Path 2 (1-2-4-5): Mean = $6 + 8 + 4 = 18$ days
* Path 3 (1-3-4-5): Mean = $9 + 7 + 4 = 20$ days
* Path 4 (1-3-5): Mean = $9 + 8 = 17$ days
**Determine Critical Path:**
The critical path is the one with the longest mean duration.
Critical Path: 1-3-4-5, with a mean duration $\mu = 20$ days.
**Calculate Critical Path Statistics:**
Next, we sum the variances of activities on the critical path (1-3, 3-4, 4-5).
* Variances: $\sigma^2_{1-3} = 2^2 = 4$ , $\sigma^2_{3-4} = 1^2 = 1$ , $\sigma^2_{4-5} = 1^2 = 1$ .
* Total Variance: $\sigma^2_{total} = 4 + 1 + 1 = 6$ .
* Total Standard Deviation: $\sigma_{total} = \sqrt{6}$ days.
**Calculate Probability for Completion Time:**
We need to find $P(T \le 22)$ , where $T$ is the total duration of the critical path. We use the $z$ -score formula for a normal distribution: $z = \frac{X - \mu}{\sigma}$ .
$z = \frac{22 - 20}{\sqrt{6}} = \frac{2}{\sqrt{6}}$
Rationalizing the denominator:
$z = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3} \approx 0.816496$
The probability is given by the cumulative distribution function (CDF) of the standard normal distribution, $\Phi(z)$ .
$P(T \le 22) = \Phi\left(\frac{\sqrt{6}}{3}\right) \approx \Phi(0.816496)$
**Relate to Answer Options:**
The problem asks for an option in the format $\Phi^{-1}(\cdot)$ . The calculated $z$ -score is approximately $0.816$ . The correct option provided is $\Phi^{-1}(0.816)$ , which directly corresponds to this $z$ -score.

Q64GATE 2013MCQ2MManufacturing Processes II

The orthogonal rake angle of the cutting tool in degree is

🚀 Solve in Practice Mode

📖 Explanation

In orthogonal turning, the main cutting force ( $F_c$ ) acts tangentially to the workpiece. The friction force ( $F_{friction}$ ) acts along the rake face of the cutting tool. The angle between the rake face direction and the direction of $F_c$ is $(90^\circ - \alpha)$ , where $\alpha$ is the orthogonal rake angle. Since $F_{friction}$ acts along the rake face, the angle between $F_c$ and $F_{friction}$ is also $(90^\circ - \alpha)$ . The problem states that $F_c$ is perpendicular to $F_{friction}$ , meaning the angle between them is $90^\circ$ .
$90^\circ - \alpha = 90^\circ$
Solving for $\alpha$ :
$\alpha = 90^\circ - 90^\circ$
$\alpha = 0^\circ$
The orthogonal rake angle is $0^\circ$ . Other parameters like diameter, feed, depth of cut, cutting velocity, and the magnitude of $F_c$ are not needed for this calculation.

Q65GATE 2013MCQ2MManufacturing Processes II

The normal force acting at the chip-tool interface in N is

🚀 Solve in Practice Mode

📖 Explanation

This problem asks us to find the normal force ( $F_n$ ) at the chip-tool interface using the main cutting force ( $F_c$ ) and a special condition.

In orthogonal turning, forces are analyzed in two main ways:

**Workpiece Forces:** The main cutting force ( $F_c$ ) acts in the direction of cutting, and the thrust force ( $F_t$ ) acts perpendicular to $F_c$ .
**Chip-Tool Interface Forces:** The friction force ( $F_f$ ) acts along the interface, opposing chip flow, and the normal force ( $F_n$ ) acts perpendicular to the interface.

Both sets of forces result in the same overall resultant force, $R$ .

We are given a crucial condition: the main cutting force ( $F_c$ ) is perpendicular to the friction force ( $F_f$ ), i.e., $F_c \perp F_f$ .

Let's set up a coordinate system:
* Assume $F_c$ acts along the positive x-axis.
* Since $F_c \perp F_f$ , $F_f$ must act along the y-axis.
* By definition, $F_t$ is perpendicular to $F_c$ , so $F_t$ also acts along the y-axis.
* By definition, $F_n$ is perpendicular to $F_f$ , so $F_n$ must act along the x-axis.

Now, we can express the resultant force $R$ using unit vectors $\hat{i}$ (x-axis) and $\hat{j}$ (y-axis):
* From the workpiece forces: $R = F_c \hat{i} + F_t \hat{j}$
* From the chip-tool interface forces: $R = F_n \hat{i} + F_f \hat{j}$

By equating the components along the x-axis from both expressions for $R$ , we get:
$F_c = F_n$
Equating the components along the y-axis gives:
$F_t = F_f$
We are given $F_c = 1500 \text{ N}$ .
Using the derived relationship $F_n = F_c$ :
$F_n = 1500 \text{ N}$
The other parameters like diameter, feed, depth of cut, and cutting velocity are not needed for this specific calculation.

The normal force acting at the chip-tool interface is $1500 \text{ N}$ .

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