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GATE Mechanical 2024 | Question: 25
A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Poisson distribution with a mean of $5$ jobs/hour. The service time of the server is exponentially distributed ... probability that the server is not busy at any point in time is $0.20$ $0.17$ $0.50$ $0.83$

A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Po...

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GATE Mechanical 2024 | Question: 26
The matrix $\left[\begin{array}{ll}1 & a \\ 8 & 3\end{array}\right]$ (where $a>0$ ) has a negative eigenvalue if $a$ is greater than $\frac{3}{8}$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{1}{5}$

The matrix $\left[\begin{array}{ll}1 & a \\ 8 & 3\end{array}\right]$ (where $a>0$ ) has a negative eigenvalue if $a$ is greater than$\frac{3}{8}$$\frac{1}{8}$...

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GATE Mechanical 2024 | Question: 27
In the pipe network shown in the figure, all pipes have the same cross-section and can be assumed to have the same friction factor. The pipes connecting points $W, N$, and $S$ with point $J$ have an equal length $L$. The pipe connecting points $J$ ... $\text{I}$ and $\text{II}$ are True Both $\text{I}$ and $\text{II}$ are False

In the pipe network shown in the figure, all pipes have the same cross-section and can be assumed to have the same friction factor. The pipes connecting po...

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GATE Mechanical 2024 | Question: 28
A company orders gears in conditions identical to those considered in the economic order quantity $\text{(EOQ)}$ model in inventory control. The annual demand is $8000$ gears, the cost per order is $300$ rupees, and the holding cost is $12$ ... cost of ordering and holding inventory from that associated with the optimal order quantity is $2.5$ $5$ $0$ $12.5$

A company orders gears in conditions identical to those considered in the economic order quantity $\text{(EOQ)}$ model in inventory control. The annual demand...

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GATE Mechanical 2024 | Question: 29
At the current basic feasible solution $\text{(bfs)}$ $v_{0}\left(v_{0} \in \mathbb{R}^{5}\right)$, the simplex method yields the following form of a linear programming problem in standard form. \[ \begin{array}{ll} \text { minimize } & z=-x_{1}-2 x_{2} \\ \text { s.t. } ... $z=-3+x_{5}-2 x_{2}$ $z=-4-5 x_{1}+2 x_{4}$ $z=-6-5 x_{1}+2 x_{3}$

At the current basic feasible solution $\text{(bfs)}$ $v_{0}\left(v_{0} \in \mathbb{R}^{5}\right)$, the simplex method yields the following form of a linear p...

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GATE Mechanical 2024 | Question: 30
Steady, compressible flow of air takes place through an adiabatic convergingdiverging nozzle, as shown in the figure. For a particular value of pressure difference across the nozzle, a stationary normal shock wave forms in the diverging section of the nozzle. If ... at $\text{F}$ Specific entropy at $\text{E}$ is lower than the specific entropy at $\text{F}$

Steady, compressible flow of air takes place through an adiabatic convergingdiverging nozzle, as shown in the figure. For a particular value of pressure differen...

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GATE Mechanical 2024 | Question: 31
Which of the following beam(s) is/are statically indeterminate?

Which of the following beam(s) is/are statically indeterminate?

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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).

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).

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 deci...

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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)

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...

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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).

Consider an air-standard Brayton cycle with adiabatic compressor and turbine, and a regenerator, as shown in the figure. Air enters the compressor at $100 \mathrm{kPa}$ a...

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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)

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^{\...

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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).

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...

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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).

Consider a slab of $20 \mathrm{~mm}$ thickness. There is a uniform heat generation of $\dot{q}=100 \mathrm{MW} / \mathrm{m}^{3}$ inside the slab. The left and right faces...

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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).

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 c...

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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 $=5.67 \times 10^{-8} \mathrm{~W} /\left(\mathrm{m}^{2} \mathrm{~K}^{4}\right)$

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 blackbod...

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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).

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 ...

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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)

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 ...

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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)

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 distribut...

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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).

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} / ...

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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).

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 g...

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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).

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=6...

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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).

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 c...

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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).

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 $\text{P}=100 \mathrm{kN}$ ...

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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).

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 $\text{(F)}$ th...

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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 $10$ ... material combination, the tool life computed using Taylor's tool life model is ________ minutes (rounded off to $2$ decimal places).

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 th...

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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).

Aluminium is casted in a cube-shaped mold having dimensions as $20 \mathrm{~mm} \times 20 \mathrm{~mm} \times 20 \mathrm{~mm}$. Another mold of the same mold material is ...

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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).

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 all...

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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).

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}...

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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).

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. ...

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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).

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...

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GATE Mechanical 2023 | Question: 2
The figure shows the plot of a function over the interval $[-4,4]$. Which one of the options given $\text{CORRECTLY}$ identifies the function? $|2-x|$ $|2-| x||$ $|2+| x||$ $2-|x|$

The figure shows the plot of a function over the interval $[-4,4]$. Which one of the options given $\text{CORRECTLY}$$ identifies the function?$|2-x|$$|2-| x||$$|2+| x||$...

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GATE Mechanical 2023 | Question: 3
With reference to the Economic Order Quantity $\text{(EOQ)}$ model, which one of the options given is correct? $\text{Curve P1:}$ Total cost, $\text{Curve P2:}$ Holding cost, $\text{Curve P3:}$ Setup cost, and $\text{Curve P4:}$ Production cost. ... Total cost, $\text{Curve P2:}$ Production cost, $\text{Curve P3:}$ Holding cost, and $\text{Curve P4:}$ Setup cost.

With reference to the Economic Order Quantity $\text{(EOQ)}$ model, which one of the options given is correct?$\text{Curve P1:}$ Total cost, $\text{Curve P2:}$ Holding co...

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GATE Mechanical 2023 | Question: 4
Which one of the options given represents the feasible region of the linear programming model: $ \begin{gathered} \text { Maximize } 45 X_1+60 X_2 \\ X_1 \leq 45 \\ X_2 \leq 50 \\ 10 X_1+10 X_2 \geq 600 \\ 25 X_1+5 X_2 \leq 750 \end{gathered} $ Region $\mathrm{P}$ Region $\mathrm{Q}$ Region $\mathrm{R}$ Region $\mathrm{S}$

Which one of the options given represents the feasible region of the linear programming model:$$\begin{gathered}\text { Maximize } 45 X_1+60 X_2 \\X_1 \leq 45 \\X_2 \leq ...

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GATE Mechanical 2023 | Question: 5
A cuboidal part has to be accurately positioned first, arresting six degrees of freedom and then clamped in a fixture, to be used for machining. Locating pins in the form of cylinders with hemi-spherical tips are to be placed on the fixture for ... $\mathrm{P}1$ and $\mathrm{P}3$ are over-constrained and Configuration $\mathrm{P}2$ is under-constrained.

A cuboidal part has to be accurately positioned first, arresting six degrees of freedom and then clamped in a fixture, to be used for machining. Locating pins in the form...

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GATE Mechanical 2023 | Question: 7
The options show frames consisting of rigid bars connected by pin joints. Which one of the frames is non-rigid?

The options show frames consisting of rigid bars connected by pin joints. Which one of the frames is non-rigid?

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GATE Mechanical 2023 | Question: 9
Air (density $=1.2 \mathrm{~kg} / \mathrm{m}^3$, kinematic viscosity $=1.5 \times 10^{-5} \mathrm{~m}^2 / \mathrm{s}$ ) flows over a flat plate with a free-stream velocity of $2 \mathrm{~m} / \mathrm{s}$ ... at a location $30 \mathrm{~mm}$ from the leading edge? $\tau_w / 2$ $\sqrt{2} \tau_w$ $2 \tau_w$ $\tau_w / \sqrt{2}$

Air (density $=1.2 \mathrm{~kg} / \mathrm{m}^3$, kinematic viscosity $=1.5 \times 10^{-5} \mathrm{~m}^2 / \mathrm{s}$ ) flows over a flat plate with a free-stream velocit...

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GATE Mechanical 2023 | Question: 10
Consider an isentropic flow of air (ratio of specific heats $=1.4$ ) through a duct as shown in the figure. The variations in the flow across the cross-section are negligible. The flow conditions at Location $1$ ... $u_2<u_1, P_2<P_1$ $u_2<u_1, P_2>P_1$ $u_2>u_1, P_2<P_1$ $u_2>u_1, P_2>P_1$

Consider an isentropic flow of air (ratio of specific heats $=1.4$ ) through a duct as shown in the figure.The variations in the flow across the cross-section are negligi...

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GATE Mechanical 2023 | Question: 11
Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on neither the Reynolds ... and Prandtl numbers the Reynolds number but not the Prandtl number the Prandtl number but not the Reynolds number

Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and veloci...

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GATE Mechanical 2023 | Question: 12
A heat engine extracts heat $\left(Q_{\mathrm{H}}\right)$ from a thermal reservoir at a temperature of $1000 \mathrm{~K}$ and rejects heat $\left(Q_{\mathrm{L}}\right)$ to a thermal reservoir at a temperature of $100 \mathrm{~K}$, while producing work $(W)$. ... $Q_{\mathrm{H}}=6000 \mathrm{~J}, Q_{\mathrm{L}}=600 \mathrm{~J}, W=5500 \mathrm{~J}$

A heat engine extracts heat $\left(Q_{\mathrm{H}}\right)$ from a thermal reservoir at a temperature of $1000 \mathrm{~K}$ and rejects heat $\left(Q_{\mathrm{L}}\right)$ t...

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