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GATE Mechanical 2024 | Question: 32
If the value of the double integral \[ \int_{x=3}^{4} \int_{y=1}^{2} \frac{d y d x}{(x+y)^{2}} \] is $\log _{e}(a / 24)$, then $a$ is __________ (answer in integer).
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GATE Mechanical 2024 | Question: 33
If $x(t)$ satisfies the differential equation \[ t \frac{d x}{d t}+(t-x)=0 \] subject to the condition $x(1)=0$, then the value of $x(2)$ is __________ (rounded off to 2 decimal places).
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GATE Mechanical 2024 | Question: 34
Let $\text{X}$ be a continuous random variable defined on $[0,1]$ such that its probability density function $f(x)=1$ for $0 \leq x \leq 1$ and $0$ otherwise. Let $Y=\log _{\mathrm{e}}(X+1)$. Then the expected value of $\text{Y}$ is ___________. (rounded off to $2$ decimal places)
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GATE Mechanical 2024 | Question: 35
Consider an air-standard Brayton cycle with adiabatic compressor and turbine, and a regenerator, as shown in the figure. Air enters the compressor at $100 \mathrm{kPa}$ and $300 \mathrm{~K}$ and exits the compressor at $600 \mathrm{kPa}$ ... volume is $c_{p} / c_{v}=1.4$. The thermal efficiency of the cycle is $\%$ ________ (answer in integer).
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GATE Mechanical 2024 | Question: 36
A piston-cylinder arrangement shown in the figure has a stop located $2 \mathrm{~m}$ above the base. The cylinder initially contains air at $140 \mathrm{kPa}$ and $350^{\circ} \mathrm{C}$ and the piston is resting in equilibrium at a ... of specific work done during the process is ___________ $\mathrm{kJ} / \mathrm{kg}$ (rounded off to $1$ decimal place)
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GATE Mechanical 2024 | Question: 37
A heat pump $\text{(H.P.)}$ is driven by the work output of a heat engine $\text{(H.E.)}$ as shown in the figure. The heat engine extracts $150 \mathrm{~kJ}$ of heat from the source at $1000 \mathrm{~K}$. The heat pump ... , the total amount of heat delivered to the room together by the heat engine and heat pump is ___________ $\mathrm{kJ}$ (answer in integer).
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GATE Mechanical 2024 | Question: 38
Consider a slab of $20 \mathrm{~mm}$ thickness. There is a uniform heat generation of $\dot{q}=100 \mathrm{MW} / \mathrm{m}^{3}$ ... state heat conduction, the location of the maximum temperature from the left face will be at ________ $\mathrm{mm}$ (answer in integer).
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GATE Mechanical 2024 | Question: 39
A condenser is used as a heat exchanger in a large steam power plant in which steam is condensed to liquid water. The condenser is a shell and tube heat exchanger which consists of $1$ shell and $20,000$ tubes. Water flows through each of the ... $3$ decimal places).
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GATE Mechanical 2024 | Question: 40
Consider a hemispherical furnace of diameter $\text{D}=6 \mathrm{~m}$ with a flat base. The dome of the furnace has an emissivity of $0.7$ and the flat base is a blackbody. The base and the dome are maintained at uniform temperature of $300 \mathrm{~K}$ ... -Boltzmann constant $=5.67 \times 10^{-8} \mathrm{~W} /\left(\mathrm{m}^{2} \mathrm{~K}^{4}\right)$
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GATE Mechanical 2024 | Question: 41
A liquid fills a horizontal capillary tube whose one end is dipped in a large pool of the liquid. Experiments show that the distance $L$ travelled by the liquid meniscus inside the capillary in time $t$ is given by \[ L=k \gamma^{a} R^{b} \ ... of the liquid. If $k$ is a dimensionless constant, then the exponent $a$ is ________ (rounded off to $1$ decimal place).
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GATE Mechanical 2024 | Question: 42
The Levai type-$\text{A}$ train illustrated in the figure has gears with module $m=8 \mathrm{~mm} /$ tooth. Gears $2$ and $3$ have $19$ and $24$ teeth respectively. Gear $2$ is fixed and internal gear $4$ ... $\text{rev/min}$. (rounded off to $2$ decimal places)
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GATE Mechanical 2024 | Question: 43
A horizontal beam of length $1200 \mathrm{~mm}$ is pinned at the left end and is resting on a roller at the other end as shown in the figure. A linearly varying distributed load is applied on the beam. The magnitude of maximum bending moment acting on the beam is ________ $\text{N.m}$. (round off to $1$ decimal place)
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GATE Mechanical 2024 | Question: 44
At the instant when $\text{OP}$ is vertical and $\text{AP}$ is horizontal, the link $\text{OD}$ is rotating counter clockwise at a constant rate $\omega=7 \mathrm{rad} / \mathrm{s}$. Pin $\mathrm{P}$ on link $\text{OD}$ slides in ... velocity of link $\text{ABC}$ for this instant is ______________$\mathrm{rad} / \mathrm{s}$ (rounded off to $2$ decimal places).
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GATE Mechanical 2024 | Question: 45
A vibratory system consists of mass $m$, a vertical spring of stiffness $2 k$ and a horizontal spring of stiffness $k$. The end $\mathrm{A}$ of the horizontal spring is given a horizontal motion $x_{A}=a \sin \omega t$. The other end ... the oscillations of mass $m$ tend to become excessively large is ___________ $\mathrm{rad} / \mathrm{s}$ (answer in integer).
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GATE Mechanical 2024 | Question: 46
A solid massless cylindrical member of $50 \mathrm{~mm}$ diameter is rigidly attached at one end, and is subjected to an axial force $P=100 \mathrm{kN}$ and a torque $T=600 \mathrm{~N}$. $\mathrm{m}$ at the other end as shown. Assume that ... yield stress as $300$ $\mathrm{MPa}$, the factor of safety in the design is ________ (rounded off to $1$ decimal place).
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GATE Mechanical 2024 | Question: 47
The figure shows a thin cylindrical pressure vessel constructed by welding plates together along a line that makes an angle $\alpha=60^{\circ}$ with the horizontal. The closed vessel has a wall thickness of $10 \mathrm{~mm}$ ... magnitude of the normal stress acting on the weld is __________ $\mathrm{MPa}$ (rounded off to $1$ decimal place).
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GATE Mechanical 2024 | Question: 48
A three-hinge arch $\text{ABC}$ in the form of a semi-circle is shown in the figure. The arch is in static equilibrium under vertical loads of $\text{P}=100 \mathrm{kN}$ and $\text{Q}=50 \mathrm{kN}$. Neglect friction at all the hinges. The magnitude of the horizontal reaction at $\text{B}$ is __________ $\mathrm{kN}$ (rounded off to $1$ decimal place).
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GATE Mechanical 2024 | Question: 49
A band brake shown in the figure has a coefficient of friction of $0.3$. The band can take a maximum force of $1.5 \mathrm{kN}$. The maximum braking force $\text{(F)}$ that can be safely applied is ___________ $\mathrm{N}$ (rounded off to the nearest integer).
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GATE Mechanical 2024 | Question: 50
A cutting tool provides a tool life of $60$ minutes while machining with the cutting speed of $60 \mathrm{~m} / \mathrm{min}$. When the same tool is used for machining the same material, it provides a tool life of $10$ minutes ... material combination, the tool life computed using Taylor's tool life model is ________ minutes (rounded off to $2$ decimal places).
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GATE Mechanical 2024 | Question: 51
Aluminium is casted in a cube-shaped mold having dimensions as $20 \mathrm{~mm} \times 20 \mathrm{~mm} \times 20 \mathrm{~mm}$ ... is the same. The ratio of the solidification times of the cube-shaped mold to the spherical mold is __________ (answer in integer).
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GATE Mechanical 2024 | Question: 52
A blanking operation is performed on $\text{C}20$ steel sheet to obtain a circular disc having a diameter of $20 \mathrm{~mm}$ and a thickness of $2 \mathrm{~mm}$. An allowance of $0.04$ is provided. The punch size used for the operation is _________ $\mathrm{mm}$ (rounded off to $2$ decimal places).
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GATE Mechanical 2024 | Question: 53
In an arc welding process, the voltage and current are $30 \mathrm{~V}$ and $200 \mathrm{~A}$, respectively. The cross-sectional area of the joint is $20 \mathrm{~mm}^{2}$ ... $2$ decimal places).
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GATE Mechanical 2024 | Question: 54
A flat surface of a $\text{C}60$ steel having dimensions of $100 \mathrm{~mm}$ (length) $\times 200 \mathrm{~mm}$ (width) is produced by a $\text{HSS}$ slab mill cutter. The $8$ -toothed cutter has $100 \mathrm{~mm}$ diameter ... is $2 \mathrm{~mm}$. The machining time required to remove the entire stock is ________ minutes (rounded off to $2$ decimal places).
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GATE Mechanical 2024 | Question: 55
In a supplier-retailer supply chain, the demand of each retailer, the capacity of each supplier, and the unit cost in rupees of material supply from each supplier to each retailer are tabulated below. The supply chain manager wishes to ... \end{array} The optimal cost of satisfying the total demand from all retailers is __________ rupees (answer in integer).
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GATE Mechanical 2022 Set 2 | Question: 20
A rope with two mass-less platforms at its two ends passes over a fixed pulley as shown in the figure. Discs with narrow slots and having equal weight of $20\: N$ each can be placed on the platforms. The number of discs placed ... $\text{(ii)}$ of the figure) required to prevent downward motion of the left side platform is _____________________(in integer).
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GATE Mechanical 2022 Set 2 | Question: 21
For a dynamical system governed by the equation. $\ddot{x}\left ( t \right ) + 2\zeta \omega _{n}\dot{x}\left ( t \right )+\omega _{n}^{2}x\left ( t \right ) = 0$ the damping ratio $\zeta$ ... $(>1)$ of the damping ratio, the displacement at the next peak in the positive direction will be __________________ $\text{mm}$ (in integer).
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GATE Mechanical 2022 Set 2 | Question: 22
An electric car manufacturer underestimated the January sales of car by $20$ units, while the actual sales was $120$ units. If the manufacturer uses exponential smoothing method with a smoothing constant of $\alpha = 0.2$, then the sales forecast for the month of February of the same year is ____________ units (in integer).
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GATE Mechanical 2022 Set 2 | Question: 23
The demand of a certain part is $1000$ parts/year and its cost is ₹$1000$/part. The orders are placed based on the economic order quantity $\text{(EOQ)}$. The cost of ordering is ₹$100$/order and the lead time for receiving ... cost is ₹$20$/part/year, the inventory level for placing the orders is ________________ parts (round off to the nearest integer).
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GATE Mechanical 2022 Set 2 | Question: 24
Consider $1$ $\text{kg}$ of an ideal gas at $1$ bar and $300$ $K$ contained in a rigid and perfectly insulated container. The specific heat of the gas at constant volume $c_{v}$ ... Assume that the container does not participate in the thermodynamic interaction. The final pressure of the gas will be _____________ bar (in integer).
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GATE Mechanical 2022 Set 2 | Question: 25
Wien's law is stated as follows: $\lambda _{m}T = C$, where $C$ is $2898$ $\mu m \cdot K$ and $\lambda_{m}$ is the wavelength at which the emissive power of a black body is maximum for a given ... $K$ (round off to the nearest integer).
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GATE Mechanical 2022 Set 2 | Question: 38
If the sum and product of eigenvalues of a $2 \times 2$ real matrix $\begin{bmatrix} 3 & p\\ p & q \end{bmatrix}$ are $4$ and $-1$ respectively, then $\left | p \right |$ is ______________ (in integer).
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GATE Mechanical 2022 Set 2 | Question: 39
Given $z = x + iy, i = \sqrt{-1}$. $C$ is a circle of radius $2$ with the centre at the origin. If the contour $C$ is traversed anticlockwise, then the value of the integral $\dfrac{1}{2\pi}\int _{C}\dfrac{1}{\left ( z-i \right )\left ( z+4i \right )}dz$ is __________________ (round off to one decimal place).
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GATE Mechanical 2022 Set 2 | Question: 40
A shaft of length $L$ is made of two materials, one in the inner core and the other in the outer rim, and the two are perfectly joined together (no slip at the interface) along the entire length of the shaft. The diameter of the inner ... strain relations are linear. Then the ratio $\tau_{i}/ \tau_{o}$ is ______________ (round off to $2$ decimal places).
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GATE Mechanical 2022 Set 2 | Question: 41
A rigid beam $\text{AD}$ of length $3a = 6\:m$ is hinged at frictionless pin joint $A$ and supported by two strings as shown in the figure. String $\text{BC}$ passes over two small frictionless pulleys of negligible radius. All the ... deflections, the tension developed in the string at $C$ is _______________ $\text{kN}$ (round off to $2$ decimal places).
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GATE Mechanical 2022 Set 2 | Question: 42
In the configuration of the planar four-bar mechanism at a certain instant as shown in the figure, the angular velocity of the $2$ $\text{cm}$ long link is $w_{2} = 5$ $\text{rad/s}$. Given the dimensions as shown, the magnitude of the angular ... of the $4$ $\text{cm}$ long link is given by ______________ $\text{rad/s}$ (round off to $2$ decimal places).
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GATE Mechanical 2022 Set 2 | Question: 43
A shaft $\text{AC}$ rotating at a constant speed carries a thin pulley of radius $r= 0.4\:m$ at the end $C$ which drives a belt. A motor is coupled at the end $A$ ... and assuming maximum shear stress theory, the minimum required shaft diameter is _____________ $\text{mm}$ (round off to $2$ decimal places).
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GATE Mechanical 2022 Set 2 | Question: 44
A straight-teeth horizontal slab milling cutter is shown in the figure. It has $4$ teeth and diameter $(D)$ of $200$ $\text{mm}$. The rotational speed of the cutter is $100$ $\text{rpm}$ and the linear feed given to the workpiece ... $\text{kN}$ (round off to one decimal place).
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GATE Mechanical 2022 Set 2 | Question: 45
ln an orthogonal machining operation, the cutting and thrust forces are equal in magnitude. The uncut chip thickness is $0.5$ $\text{mm}$ and the shear angle is $15^{\circ}$. The orthogonal rake angle of the tool is $0^{\circ}$ and ... strength is $500$ $\text{MPa}$. The cutting force is ___________________ $\text{N}$ (round off to the nearest integer).
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GATE Mechanical 2022 Set 2 | Question: 46
The best size wire is fitted in a groove of a metric screw such that the wire touches the flanks of the thread on the pitch line as shown in the figure. The pitch $(p)$ and included angle of the thread are $4$ $\text{mm}$ and $60^{\circ}$, respectively. The diameter of the best size wire is _______________ $\text{mm}$ (round off to $2$ decimal places).
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GATE Mechanical 2022 Set 2 | Question: 47
In a direct current arc welding process, the power source has an open circuit voltage of $100\: V$ and short circuit current of $1000 \:A$. Assume a linear relationship between voltage and current. The arc voltage $(V)$ varies with the ... is in $\text{mm}$. The maximum available arc power during the process is ________________ $\text{kVA}$ (in integer).
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