GATE Mechanical Engineering 2022 SET-1

 Question 1
The limit
$p=\lim_{x \to \pi}\left ( \frac{x^2+\alpha x+2\pi ^2}{x-\pi+2 \sin x } \right )$
has a finite value for a real $\alpha$. The value of $\alpha$ and the corresponding limit $p$ are
 A $\alpha =-3\pi, \text{ and }p= \pi$ B $\alpha =-2\pi, \text{ and }p= 2\pi$ C $\alpha =\pi, \text{ and }p= \pi$ D $\alpha =2\pi, \text{ and }p= 3\pi$
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} p&=\lim_{x \to \pi}\left ( \frac{x^2+\alpha x+2 \pi ^2}{x-\pi+2 \sin x} \right ) \\ p&=\left ( \frac{\pi ^2+\alpha \pi +2 \pi ^2}{\pi-\pi+2 \sin \pi } \right ) \\ &= \frac{2 \pi ^2+\alpha \pi}{0}\\ \therefore \;\; \alpha &= -3 \pi\\ p&=\lim_{x \to \pi}\left ( \frac{x^2- 3 \pi x+2 \pi ^2}{x-\pi+2 \sin x} \right ) \\ &=\lim_{x \to \pi}\left ( \frac{2x- 3 \pi }{1+2 \cos \pi} \right ) \\ &= \frac{2 \pi-3 \pi}{1-2}=\frac{-\pi}{-1}=\pi\\ \therefore \; \alpha &=-3 \pi \text{ and }p= \pi \end{aligned}
 Question 2
Solution of $\triangledown^2T=0$ in a square domain ($0 \lt x \lt 1$ and $0 \lt y \lt 1$) with boundary conditions:
$T(x, 0) = x; T(0, y) = y; T(x, 1) = 1 + x; T(1, y) = 1 + y$ is
 A $T(x,y)=x-xy+y$ B $T(x,y)=x+y$ C $T(x,y)=-x+y$ D $T(x,y)=x+xy+y$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
T(x, 0) = x $\Rightarrow$ option (c) is not correct.
T(0, y) = y $\Rightarrow$ all options satisfied.
T(x, 1) = 1 + x; $\Rightarrow$ only option (b) is satisfied.
T(1, y) = 1 + y is $\Rightarrow$ only option (b) is satisfied.
 Question 3
Given a function $\varphi =\frac{1}{2}(x^2+y^2+z^2)$ in threedimensional Cartesian space, the value of the surface integral
$\oiint_{S}{\hat{n}.\triangledown \varphi dS}$
where S is the surface of a sphere of unit radius and $\hat{n}$ is the outward unit normal vector on S, is
 A $4 \pi$ B $3 \pi$ C $4 \pi/3$ D $0$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \varphi &=\frac{1}{2}(x^2+y^2+z^2)\\ \triangledown \varphi &=(x\hat{i}+y\hat{j}+z\hat{k})=\bar{F}\\ \oiint_{S}(\triangledown \varphi\cdot \bar{n})dS&=\int \int _v\int Div\; \bar{F} dv\\ &=\int \int \int 3dv\\ &=3v\\ &=3\left ( \frac{4}{3} \pi \right )=4\pi \end{aligned}
 Question 4
The Fourier series expansion of $x^3$ in the interval $-1\leq x\leq 1$ with periodic continuation has
 A only sine terms B only cosine terms C both sine and cosine terms D only sine terms and a non-zero constant
Engineering Mathematics   Calculus
Question 4 Explanation:
$f(x)=x^3, \;\; -1 \leq x \leq 1$
It is an odd function
Fourier series contains only sine terms.
 Question 5
If $A=\begin{bmatrix} 10 &2k+5 \\ 3k-3 & k+5 \end{bmatrix}$ is a symmetric matrix, the value of $k$ is ___________.
 A 8 B 5 C -0.4 D $\frac{1+\sqrt{1561}}{12}$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
$A=\begin{bmatrix} 10 & 2k+5\\ 3k-3 & k+5 \end{bmatrix}$
$(2k + 5) = (3k - 3)$
$k=8$
 Question 6
A uniform light slender beam AB of section modulus EI is pinned by a frictionless joint A to the ground and supported by a light inextensible cable CB to hang a weight W as shown. If the maximum value of W to avoid buckling of the beam AB is obtained as $\beta \pi ^2 EI$, where $\pi$ is the ratio of circumference to diameter of a circle, then the value of $\beta$ is

 A $0.0924\; m^{-2}$ B $0.0713\; m^{-2}$ C $0.1261\; m^{-2}$ D $0.1417\; m^{-2}$
Strength of Materials   Euler's Theory of Column
Question 6 Explanation:
Draw FBD of AB

$\Sigma M_A=0$
$W \times 2.5=T \sin 30^{\circ} \times 2.5$
$T=2W$
Compressive load acting on AB $=T \cos 30^{\circ}=2W \times \frac{\sqrt{3}}{2}=\sqrt{3}W$
Buckling happens when $\sqrt{3}W=P_{cr}=\frac{\pi ^2 EI}{L_e^2}$
$\sqrt{3}W=\frac{\pi ^2 EI}{L^2} \;\;(\because L_e=L \text{as both ends hinged})$
$W=\frac{1 \times \pi ^2 EI}{\sqrt{3} \times (2.5)^2}=0.0924 \pi^2 EI$
$W0.0924 \pi^2 EI=\beta \pi ^2 EI$
$\beta =0.0924 m^{-2}$
 Question 7
The figure shows a schematic of a simple Watt governor mechanism with the spindle $O_1O_2$ rotating at an angular velocity $\omega$ about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot $O_1$ is denoted by $h$. The value of $h=400$ mm at a certain $\omega$. If $\omega$ is doubled, the value of $h$ will be _________ mm.

 A 50 B 100 C 150 D 200
Theory of Machine   Gyroscope
Question 7 Explanation:
$h_1 = 400 mm, h_2 = ?$
$\omega _1=\omega \;\;\;\omega _2=2\omega$
For Watt governor,
\begin{aligned} h&=\frac{g}{\omega ^2}\\ h\propto \frac{1}{\omega ^2}\\ \Rightarrow h_1\omega _1^2&=h_2\omega _2^2\\ 400 \times \omega ^2&=h_2 \times (2\omega )^2\\ h_2&=100mm \end{aligned}
 Question 8
A square threaded screw is used to lift a load W by applying a force F. Efficiency of square threaded screw is expressed as
 A The ratio of work done by W per revolution to work done by F per revolution B W/F C F/W D The ratio of work done by F per revolution to work done by W per revolution
Machine Design   Bolted, Riveted and Welded Joint
Question 8 Explanation:
$\text{Screw efficiency}=\frac{\text{Work done by the applied force/rev}}{\text{Work done in lifting the load/rev}}$
Efficiency of screw jack $\eta =\frac{\tan \alpha }{\tan(\alpha +\phi )}$
Efficiency depends on helix angle and friction angle.
 Question 9
A CNC worktable is driven in a linear direction by a lead screw connected directly to a stepper motor. The pitch of the lead screw is 5 mm. The stepper motor completes one full revolution upon receiving 600 pulses. If the worktable speed is 5 m/minute and there is no missed pulse, then the pulse rate being received by the stepper motor is
 A 20 KHz B 10 kHz C 3 kHz D 15 kHz
Manufacturing Engineering   Machining and Machine Tool Operation
Question 9 Explanation:
No. of steps required for one full revolution of stepper motor shaft or lead screws $n_S= 600$
Pitch $(p) = 5 mm$
Linear table speed $V_{table}= 5 m/min = 5000 mm/min$
RPM of lead Screw $(N_S)= \frac{V_{table}}{p}=1000 rpm$
We have equation of frequency of pulse generator
\begin{aligned} f_p&= N_s \times n_S\\ f_p&= 1000 \times 600=600,000 pulses/min\\ f_p&=\frac{600000}{60}pulses/sec\\ f_p&=10000 pulses/sec \text{ or }Hz\\ f_p&=10kHz \end{aligned}
 Question 10
The type of fit between a mating shaft of diameter $25.0^{\begin{matrix} +0.010\\ -0.010 \end{matrix}}$mm and a hole of diameter $25.015^{\begin{matrix} +0.015\\ -0.015 \end{matrix}}$mm is __________.
 A Clearance B Transition C Interference D Linear
Manufacturing Engineering   Metrology and Inspection
Question 10 Explanation:

If,
$D_{hole} = 25.00 mm,$
$D_{shaft} = 25.01 mm$ ( Interference fit.)
$D_{hole} = 25.03 mm,$
$D_{shaft} = 24.99 mm$
 (Clearance fit)
Some of the assemblies provide clearance fit and some provides interference fit.
Hence, It is transition fit
There are 10 questions to complete.