## GATE Electrical Engineering 2022

 Question 1
The transfer function of a real system, $H(s)$, is given as:
$H(s)=\frac{As+B}{s^2+Cs+D}$
where A, B, C and D are positive constants. This system cannot operate as
 A low pass filter. B high pass filter C band pass filter. D an integrator.
Electric Circuits   Magnetically Coupled Circuits, Network Topology and Filters
Question 1 Explanation:
Put $s=0, H(0)=\frac{A \times 0+B}{0+C \times 0+D}=\frac{B}{D}$
So, the system pass low frequency component. Put $s=\infty , H(\infty )=0$
For high pass filter, high frequency component should be non zero. Hence this system cannot be operated as high pass filter.
 Question 2
For an ideal MOSFET biased in saturation, the magnitude of the small signal current gain for a common drain amplifier is
 A 0 B 1 C 100 D infinite
Analog Electronics   Small Signal Analysis
Question 2 Explanation:
For ideal MOSFET, $i_G=0$
Therefore, Current gain, $A_I=\frac{i_s}{i_G}=\infty$
 Question 3
The most commonly used relay, for the protection of an alternator against loss of excitation, is
 A offset Mho relay. B over current relay. C differential relay D Buchholz relay.
Power Systems   Switch Gear and Protection
 Question 4
The geometric mean radius of a conductor, having four equal strands with each strand of radius $'r'$, as shown in the figure below, is

 A $4r$ B $1.414r$ C $2r$ D $1.723r$
Power Systems   Performance of Transmission Lines, Line Parameters and Corona
Question 4 Explanation:
Redraw the configuration:

$\therefore \; GMR=(r' \times 2r\times 2r\times 2\sqrt{2}r)^{1/4}$
Where, $r'=0.7788r$
Hence, $GMR=1.723r$
 Question 5
The valid positive, negative and zero sequence impedances (in p.u.), respectively, for a 220 kV, fully transposed three-phase transmission line, from the given choices are
 A 1.1, 0.15 and 0.08 B 0.15, 0.15 and 0.35 C 0.2, 0.2 and 0.2 D 0.1, 0.3 and 0.1
Power Systems   Fault Analysis
Question 5 Explanation:
We have,
$X_0 \gt X_1=X_2$
(for $3-\phi$ transposed transmission line)
 Question 6
The steady state output ($V_{out}$), of the circuit shown below, will

 A saturate to $+V_{DD}$ B saturate to $-V_{EE}$ C become equal to 0.1 V D become equal to -0.1 V
Analog Electronics   Operational Amplifiers
Question 6 Explanation:
Redraw the circuit:

From circuit,
\begin{aligned} V_{out} &=-\frac{1}{C_1}\int I\cdot dt \\ &= -\frac{1}{R_1C_1}\int 0 \cdot 1dt \\ \\ &=-\frac{0.1}{R_1C_1}\int dt \\ \\ &= -\frac{0.1}{R_1C_1}t \end{aligned}

Hence, $V_{out}=-V_{EE}$
 Question 7
The Bode magnitude plot of a first order stable system is constant with frequency. The asymptotic value of the high frequency phase, for the system, is $-180^{\circ}$. This system has

 A one LHP pole and one RHP zero at the same frequency B one LHP pole and one LHP zero at the same frequency C two LHP poles and one RHP zero D two RHP poles and one LHP zero.
Control Systems   Frequency Response Analysis
Question 7 Explanation:
The given system is non-minimum phase system Therefore, transfer function, $T.F=\frac{s-1}{s+1}$
Hence, one LHP pole and one RHP zero at the same frequency.
 Question 8
A balanced Wheatstone bridge $ABCD$ has the following arm resistances:
$R_{AB}=1k\Omega \pm 2.1%; R_{BC}=100\Omega \pm 0.5%, R_{CD}$ is an unknown resistance; $R_{DA}=300\Omega \pm 0.4%;$. The value of $R_{CD}$ and its accuracy is
 A $30\Omega \pm 3\Omega$ B $30\Omega \pm 0.9\Omega$ C $3000\Omega \pm 90\Omega$ D $3000\Omega \pm 3\Omega$
Electrical and Electronic Measurements   A.C. Bridges
Question 8 Explanation:
The condition for balanced bridge
\begin{aligned} R_{AB}R_{CD}&=R_{DA}R_{BC} \\ R_{CD} &=\frac{300 \times 100}{1000}=30\Omega \\ %Error &=\pm (2.1+0.5+0.4)=\pm 3% \\ \therefore \; R_{CD}&=30\pm 30 \times \frac{3}{100}=30\pm 0.9\Omega \end{aligned}
 Question 9
The open loop transfer function of a unity gain negative feedback system is given by $G(s)=\frac{k}{s^2+4s-5}$.
The range of $k$ for which the system is stable, is
 A $k \gt 3$ B $k \lt 3$ C $k \gt 5$ D $k \lt 5$
Control Systems   Root Locus Techniques
Question 9 Explanation:
Characteristic equation:
\begin{aligned} 1+G(s)H(s)&=0\\ 1+\frac{k}{s^2+4s-5}&=0\\ s^2+4s+k-5&=0 \end{aligned}
R-H criteria:
$\left.\begin{matrix} s^2\\ s^1\\ s^0 \end{matrix}\right| \begin{matrix} 1 & k-5\\ 4 & 0\\ k-5 & \end{matrix}$
Hence, for stable system,
$k-5 \gt 0 \;\; \Rightarrow \; k \gt 5$
 Question 10
Consider a 3 x 3 matrix A whose (i,j)-th element, $a_{i,j}=(i-j)^3$. Then the matrix A will be
 A symmetric. B skew-symmetric. C unitary D null.
Engineering Mathematics   Linear Algebra
Question 10 Explanation:
$for \; i=j\Rightarrow a_{ij}=(i-i)^3=0\forall i$
$for \; i\neq j\Rightarrow a_{ij}=(i-j)^3=(-(j-i))^3=-(j-i)^3=-a_{ji}$
$\therefore \; A_{3 \times 3 }$ is skew symmetric matrix.
There are 10 questions to complete.

## GATE Electronics and Communication 2022

 Question 1
Consider the two-dimensional vector field $\vec{F}(x,y)=x\vec{i}+y\vec{j}$, where $\vec{i}$ and $\vec{j}$ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral
$\oint _c \vec{F}(x,y)\cdot (dx\vec{i}+dy\vec{j})$

 A 0 B 1 C $8+2 \pi$ D -1
Engineering Mathematics   Calculus
Question 1 Explanation:
$\oint \vec{F} (x,y)\cdot [dx\vec{i}+dy\vec{j}]$
Given $\vec{F} (x,y)=x\vec{i}+y\vec{j}$
$\therefore \int_{c}xdx+ydy=0$
Because here vector is conservative.
If the integral function is the total derivative over the closed contoure then it will be zero
 Question 2
Consider a system of linear equations $Ax=b$, where
$A=\begin{bmatrix} 1 & -\sqrt{2} & 3\\ -1& \sqrt{2}& -3 \end{bmatrix},b=\begin{bmatrix} 1\\ 3 \end{bmatrix}$
This system of equations admits ______.
 A a unique solution for x B infinitely many solutions for x C no solutions for x D exactly two solutions for x
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
Here equation will be
$x-\sqrt{2}y+3z=1$
$-x+\sqrt{2}y-3z=3$
therefore inconsistant solution i.e. there will not be any solution.
 Question 3
The current $I$ in the circuit shown is ________

 A $1.25 \times 10^{-3}A$ B $0.75 \times 10^{-3}A$ C $-0.5 \times 10^{-3}A$ D $1.16 \times 10^{-3}A$
Network Theory   Basics of Network Analysis
Question 3 Explanation:

Applying Nodal equation at Node-A
\begin{aligned} \frac{V_A}{2k}+\frac{V_A-5}{2k}&=10^{-3}\\ \Rightarrow 2V_A-5&=2k \times 10^{-3}\\ V_A&=3.5V\\ Again,&\\ I&=\frac{5-V_A}{2k}\\ &=\frac{5-3.5}{2k}\\ &=0.75 \times 10^{-3}A \end{aligned}
 Question 4
Consider the circuit shown in the figure. The current $I$ flowing through the $10\Omega$ resistor is _________.

 A 1A B 0A C 0.1A D -0.1A
Network Theory   Basics of Network Analysis
Question 4 Explanation:
Here, there is no any return closed path for Current (I) . Hence I=0
Current always flow in loop.
 Question 5
The Fourier transform $X(j\omega )$ of the signal $x(t)=\frac{t}{(1+t^2)^2}$ is _________.
 A $\frac{\pi}{2j}\omega e^{-|\omega|}$ B $\frac{\pi}{2}\omega e^{-|\omega|}$ C $\frac{\pi}{2j} e^{-|\omega|}$ D $\frac{\pi}{2} e^{-|\omega|}$
Signals and Systems   DTFS, DTFT and DFT
Question 5 Explanation:
$x(t)=\frac{t}{(1+t^2)^2}$
As we know that FT of $te^{-|t|} \; \underleftrightarrow{FT} \;\frac{-j4\omega }{(1+\omega ^2)^2}$
Duality $\frac{-j4\omega }{(1+t ^2)^2} \leftrightarrow 2 \pi(-\omega )e^{-|-\omega |}$
$\Rightarrow \frac{t}{(1+t^2)^2} \underrightarrow{FT} \frac{-2\pi}{-j4}\omega e^{-|\omega |}$
$\Rightarrow \;\;\;\rightarrow\frac{\pi}{j2} \omega e^{-|\omega |}$
 Question 6
Consider a long rectangular bar of direct bandgap p-type semiconductor. The equilibrium hole density is $10^{17}cm^{-3}$ and the intrinsic carrier concentration is $10^{10}cm^{-3}$. Electron and hole diffusion lengths are $2\mu m$and $1\mu m$, respectively. The left side of the bar ($x=0$) is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated ONLY at $x=0$ because of the laser. The steady state electron density at $x=0$ is $10^{14}cm^{-3}$ due to laser illumination. Under these conditions and ignoring electric field, the closest approximation (among the given options) of the steady state electron density at $x=2 \mu m$, is _____
 A $0.37 \times 10^{14} cm^{-3}$ B $0.63 \times 10^{13} cm^{-3}$ C $3.7 \times 10^{14} cm^{-3}$ D $0^{3} cm^{-3}$
Electronic Devices   Basic Semiconductor Physics
Question 6 Explanation:

From continuity equation of electrons
$\frac{dn}{dt}=n\mu _n\frac{dE}{dx}+\mu _nE\frac{dn}{dx}+G_n-R_n+x_n\frac{d^2x}{dx^2} \;\;\;...(i)$
[Because $\vec{E}$ is not mentioned hence
$\frac{dE}{dx}=0$
For $x \gt 0, G_n$ is also zero
$n=\frac{n_i^2}{N_A}=\frac{10^{20}}{10^{17}}=10^3$
$n=n_0+\delta n=10^3+10^{14}=10^{14}$
at steady state, $\frac{db}{dt}=0$
Hence equation (i) becomes:
$O=D_n\frac{d^2\delta n}{dx^2}-\frac{\delta n}{\tau _n}$
$\frac{d^2\delta n}{dx^2}=\frac{\delta n}{L_n^2} \;\;\;...(ii)$
From solving equation (ii)
$\delta _n(x)=\delta _n(0)e^{-x/L_n}$
at $x=2\mu m$
$\delta _n(2\mu m)=10^{14}e^{-2/2}=10^{14}e^{-1}=0.37 \times 10^{14}$
 Question 7
In a non-degenerate bulk semiconductor with electron density $n=10^{16}cm^{-3}$, the value of $E_C-E_{Fn}=200meV$, where $E_C$ and $E_{Fn}$ denote the bottom of the conduction band energy and electron Fermi level energy, respectively. Assume thermal voltage as 26 meV and the intrinsic carrier concentration is $10^{10}cm^{-3}$. For $n=0.5 \times 10^{16}cm^{-3}$, the closest approximation of the value of ($E_C-E_{Fn}$), among the given options, is ______.
 A 226 meV B 174 meV C 218 meV D 182 meV
Electronic Devices   Basic Semiconductor Physics
Question 7 Explanation:
Here we have to find the value of $E_c-E_{fn}$
As we know,
$E_C-E_F=kT \ln\left ( \frac{N_c}{n} \right ) \;\;\;...(i)$
$E_C-E_{F1}=kT \ln\left ( \frac{N_c}{n_1} \right ) \;\;\;...(ii)$
$E_C-E_{F2}=kT \ln\left ( \frac{N_c}{n_2} \right ) \;\;\;...(iii)$
Equation (ii) - Equation (iii)
$(E_C-E_{F1})-(E_C-E_{F2})=kT \ln \left ( \frac{\frac{N_c}{n_1}}{\frac{N_c}{n_2}} \right )=kT \ln \frac{n_2}{n_1}$
$\Rightarrow 200meV-(E_C-E_{F2})=26meV \times \ln \left ( \frac{0.5 \times 10^{16}}{1 \times 10^{16}} \right )$
$200meV-(E_C-E_{F2})=+26meV \ln (0.5)=-18$
$(E_C-E_{F2})=200+8=218meV$
 Question 8
Consider the CMOS circuit shown in the figure (substrates are connected to their respective sources). The gate width ($W$) to gate length ($L$) ratios $\frac{W}{L}$ of the transistors are as shown. Both the transistors have the same gate oxide capacitance per unit area. For the pMOSFET, the threshold voltage is -1 V and the mobility of holes is $40\frac{cm^2}{V.s}$. For the nMOSFET, the threshold voltage is 1 V and the mobility of electrons is $300\frac{cm^2}{V.s}$. The steady state output voltage $V_o$ is ________.

 A equal to 0 V B more than 2 V C less than 2 V D equal to 2 V
Analog Circuits   FET and MOSFET Analysis
Question 8 Explanation:

\begin{aligned} \mu _PCO_x\left ( \frac{\omega }{L} \right )_1[4-V_0-1]^2&=\mu _nCO_x\left ( \frac{\omega }{L} \right )_2[V_0-0-1]^2\\ \Rightarrow \frac{300}{40}\times \frac{1}{5}(V_0-1)^2&=(3-V_0)^2\\ \Rightarrow \sqrt{1.5} (V_0-1)&=3-V_0\\ \Rightarrow V_0&=\frac{3+\sqrt{1.5}}{\sqrt{1.5}+1} \lt 2V \end{aligned}
 Question 9
Consider the 2-bit multiplexer (MUX) shown in the figure. For OUTPUT to be the XOR of C and D, the values for $A_0,A_1,A_2 \text{ and }A_3$ are _______

 A $A_0=0,A_1=0,A_2=1,A_3=1$ B $A_0=1,A_1=0,A_2=1,A_3=0$ C $A_0=0,A_1=1,A_2=1,A_3=0$ D $A_0=1,A_1=1,A_2=0,A_3=0$
Digital Circuits   Combinational Circuits
Question 9 Explanation:

$f=\bar{C}\bar{D}I_0+\bar{C}DI_1+C\bar{D}I_2+CDI_3$
For this
$A_0=A_3=0$
$A_1=A_2=1$
 Question 10
The ideal long channel nMOSFET and pMOSFET devices shown in the circuits have threshold voltages of 1 V and -1 V, respectively. The MOSFET substrates are connected to their respective sources. Ignore leakage currents and assume that the capacitors are initially discharged. For the applied voltages as shown, the steady state voltages are ______

 A $V_1=5 V, V_2=5 V$ B $V_1=5 V, V_2=4 V$ C $V_1=4 V, V_2=5 V$ D $V_1=4V, V_2=-5 V$
Analog Circuits   FET and MOSFET Analysis
Question 10 Explanation:

There are 10 questions to complete.

## Concrete Technology

 Question 1
It is given that an aggregate mix has 260 grams of coarse aggregates and 240 grams of fine aggregates. The specific gravities of the coarse and fine aggregates are 2.6 and 2.4, respectively. The bulk specific gravity of the mix is 2.3.
The percentage air voids in the mix is ____________. (round off to the nearest integer)
 A 2 B 4 C 8 D 16
GATE CE 2022 SET-2   RCC Structures
Question 1 Explanation:
Given that,
Coarse aggregate = 260 gms
Fine aggregate = 240 gms
$G_{CA}=2.6$
$G_{FA}=2.4$
Bulk specific gravity $G_m=2.3$
Percentage air voids in the mix = ?
$G_t$ (Theoretical specific gravity)

\begin{aligned} &=\frac{\Sigma W}{\Sigma \frac{W}{G}}\\ &=\frac{260+240}{\frac{260}{2.6}+\frac{240}{2.4}}\\ &=2.5 \end{aligned}
\begin{aligned} % \text{ air voids} (V_V)&=\frac{G_t-G_m}{G_t} \times 100\\ &=\frac{2.5-2.3}{2.3}\times 100\\ V_V&=8% \end{aligned}
 Question 2
Match all the possible combinations between Column X (Cement compounds) and Column Y (Cement properties):
$\begin{array}{|c|l|}\hline \text{Column X}&\text{Column Y} \\ \hline (i) C_3S & \text{(P) Early age strength} \\ \hline (ii) C_2S & \text{(Q) Later age strength}\\ \hline (iii) C_3A& \text{(R) Flash setting}\\ \hline & \text{(S) Highest heat of hydration}\\ \hline & \text{(T) Lowest heat of hydration}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (i) - (P), (ii) - (Q) and (T), (iii) - (R) and (S) B (i) - (Q) and (T), (ii) - (P) and (S), (iii) - (R) C (i) - (P), (ii) - (Q) and (R), (iii) - (T) D (i) - (T), (ii) - (S), (iii) - (P) and (Q)
GATE CE 2022 SET-2   RCC Structures
Question 2 Explanation:
$C_3S-$ Responsible for early age strength
$C_2S -$ Responsible for later age strength and lowest heat of hydration
$C_3A-$ Flash setting and highest heat of hydration
 Question 3
Which of the following equations is correct for the Pozzolanic reaction?
 A $Ca(OH)_2$ + Reactive Superplasticiser + $H_2O \rightarrow C-S-H$ B $Ca(OH)_2$ + Reactive Silicon dioxide + $H_2O \rightarrow C-S-H$ C $Ca(OH)_2$ + Reactive Sulphates + $H_2O \rightarrow C-S-H$ D $Ca(OH)_2$ + Reactive Sulphur + $H_2O \rightarrow C-S-H$
GATE CE 2022 SET-1   RCC Structures
Question 3 Explanation:
Pozzolanic materials have no cementing properties itself but have the property of combining with lime to produce stable compound.
Pozzolana is considered as siliceous and aluminous materials and when added in cement it have $SiO_2$ and $Al_2O_3$ form.
So, pozzolanic reaction :
$H_2O$ + Reactive slilica-di-oxide + $H_2O \rightarrow C-S-H$ gel or tobermonite gel
There are 3 questions to complete.

## Friction

 Question 1
A horizontal force of P kN is applied to a homogeneous body of weight 25 kN, as shown in the figure. The coefficient of friction between the body and the floor is 0.3. Which of the following statement(s) is/are correct?

 A The motion of the body will occur by overturning. B Sliding of the body never occurs. C No motion occurs for $P \leq 6 kN$. D The motion of the body will occur by sliding only.
GATE CE 2022 SET-1   Solid Mechanics
Question 1 Explanation:

Minimum force for sliding
$(P_{min})_{sliding}=(f_s)_{max}$ ...(i)
Applying equilibrium equation in vertical direction
Normal reaction = Weight
N=mg=25 kN ...(ii)
Using equation (i) and (ii)
$(P_{min})_{sliding}=\mu N =0.3 \times 25=7.5 kN$
Minimum force for overturning

At the verge of overturning
$(P_{min})_{oberturning} \times 2=W \times 2$
$(P_{min})_{oberturning}=\frac{25 \times 0.5} {2}=6.25 kN$
Here, $(P_{min})_{oberturning} \lt (P_{min})_{sliding}$
First overtuning will take place.
Sliding will not take place.
There is 1 question to complete.

## GATE Civil Engineering 2022 SET-2

 Question 1
The function $f(x, y)$ satisfies the Laplace equation
$\triangledown ^2f(x,y)=0$
on a circular domain of radius $r = 1$ with its center at point P with coordinates $x = 0, y = 0$. The value of this function on the circular boundary of this domain is equal to 3.
The numerical value of $f(0, 0)$ is:
 A 0 B 2 C 3 D 1
Engineering Mathematics   Partial Differential Equation
Question 1 Explanation:
According to given condition given function f(x,y) is nothing but constant function i.e. f(x,y)=3 because this is the only function whose value is 3 at any point on the boundary of unit circle and it is also satisfying Laplace equation, so
f(0,0)=3
 Question 2
$\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+... \right )dx$ is equil to
 A $\frac{1}{1+x}+constant$ B $\frac{1}{1+x^2}+constant$ C $-\frac{1}{1-x}+constant$ D $-\frac{1}{1-x^2}+constant$
Engineering Mathematics   Calculus
Question 2 Explanation:
MTA- Marks to All
$I=\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...\infty \right )dx$
$I=\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{12}-\frac{x^5}{20}+...$
Option (A)
$\frac{1}{1+x}=(1+x)^{-1}=1-x+x^2-x^3...\infty$
So, its incorrect.
Option (B)
$\frac{1}{1+x^2}=(1+x^2)^{-1}=1-x^2+x^4-x^6...\infty$
So, its incorrect.
Similarly option (C) and (D) both are incorrect.
No-correct choice given.
 Question 3
For a linear elastic and isotropic material, the correct relationship among Young's modulus of elasticity ($E$), Poisson's ratio ($v$), and shear modulus ($G$) is
 A $G=\frac{E}{2(1+v)}$ B $G=\frac{E}{(1+2v)}$ C $E=\frac{G}{2(1+v)}$ D $E=\frac{G}{(1+2v)}$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 3 Explanation:
$E=2G(1+\mu )$
$G=$ Shear modulas
$\mu =$Poission's ratio
$E=$ Young's modulus
 Question 4
Read the following statements relating to flexure of reinforced concrete beams:

I. In over-reinforced sections, the failure strain in concrete reaches earlier than the yield strain in steel.
II. In under-reinforced sections, steel reaches yielding at a load lower than the load at which the concrete reaches failure strain.
III. Over-reinforced beams are recommended in practice as compared to the under-reinforced beams.
IV. In balanced sections, the concrete reaches failure strain earlier than the yield strain in tensile steel.

Each of the above statements is either True or False.
Which one of the following combinations is correct?
 A I (True), II (True), III (False), IV (False) B I (True), II (True), III (False), IV (True) C I (False), II (False), III (True), IV (False) D I (False), II (True), III (True), IV (False)
RCC Structures   Footing, Columns, Beams and Slabs
Question 4 Explanation:
The question is based on LSM design principle as it is describing different conditions related to strain
Depending on amount of reinforcement in a cross- section, here ca be three types of sections viz. balanced, under reinforced and over reinforced.
Balanced section is a section that is expected to result in a balanced failure. It means at the ultimate limit state in flexure, the concrete will attain a limiting compressive strain of 0.0035 and steel will attain minimum specified tensile strain of $0.002+\frac{0.87f_y}{E_s}$
Under reinforced section is a section in which steel yield before collapse. Over reinforced section is a section in which crushing of concrete in compression i.e. attainment of compressive strain of 0.0035 occurs prior to yielding of steel.
In case of over reinforced section the deflection, crack width remain relatively low and failure occurs without any sign of warning and hence over reinforced flexural members are not recommended by IS code.
Based on the above information:
Statement I is true.
Statement II is true.
Statement III is false.
Statement IV is false.
 Question 5
Match all the possible combinations between Column X (Cement compounds) and Column Y (Cement properties):
$\begin{array}{|c|l|}\hline \text{Column X}&\text{Column Y} \\ \hline (i) C_3S & \text{(P) Early age strength} \\ \hline (ii) C_2S & \text{(Q) Later age strength}\\ \hline (iii) C_3A& \text{(R) Flash setting}\\ \hline & \text{(S) Highest heat of hydration}\\ \hline & \text{(T) Lowest heat of hydration}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (i) - (P), (ii) - (Q) and (T), (iii) - (R) and (S) B (i) - (Q) and (T), (ii) - (P) and (S), (iii) - (R) C (i) - (P), (ii) - (Q) and (R), (iii) - (T) D (i) - (T), (ii) - (S), (iii) - (P) and (Q)
RCC Structures   Concreate Technology
Question 5 Explanation:
$C_3S-$ Responsible for early age strength
$C_2S -$ Responsible for later age strength and lowest heat of hydration
$C_3A-$ Flash setting and highest heat of hydration
 Question 6
Consider a beam PQ fixed at P, hinged at Q, and subjected to a load F as shown in figure (not drawn to scale). The static and kinematic degrees of indeterminacy, respectively, are

 A 2 and 1 B 2 and 0 C 1 and 2 D 2 and 2
Structural Analysis   Determinacy and Indeterminacy
Question 6 Explanation:

Static indeterminacy, SI=r-3=(3+2)-3=2
Kinematic indeterminacy=0+1=1
 Question 7
Read the following statements:

(P) While designing a shallow footing in sandy soil, monsoon season is considered for critical design in terms of bearing capacity.
(Q) For slope stability of an earthen dam, sudden drawdown is never a critical condition.
(R) In a sandy sea beach, quicksand condition can arise only if the critical hydraulic gradient exceeds the existing hydraulic gradient.
(S) The active earth thrust on a rigid retaining wall supporting homogeneous cohesionless backfill will reduce with the lowering of water table in the backfill.

Which one of the following combinations is correct?
 A (P)-True, (Q)-False, (R)-False, (S)-False B (P)-False, (Q)-True, (R)-True, (S)-True C (P)-True, (Q)-False, (R)-True, (S)-True D (P)-False, (Q)-True, (R)-False, (S)-False
Geotechnical Engineering   Shallow Foundation and Bearing Capacity
Question 7 Explanation:
In monsoon season sand will be fully saturated hence this will be critical condition in designing of shallow foundation.
In case of sudden drawdown flow direction reverses hence for slope stability, it will be critical condition.
In sandy sea beach, quicksand condition can arise only if existing hydraulic gradient exceeds the critical hydraulic gradient.
 Question 8
Stresses acting on an infinitesimal soil element are shown in the figure (with $\sigma _z \gt \sigma _x$). The major and minor principal stresses are $\sigma _1$ and $\sigma _3$, respectively. Considering the compressive stresses as positive, which one of the following expressions correctly represents the angle between the major principal stress plane and the horizontal plane?

 A $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1-\sigma _x} \right )$ B $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _3-\sigma _x} \right )$ C $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1+\sigma _x} \right )$ D $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1+\sigma _3} \right )$
Solid Mechanics   Principal Stress and Principal Strain
Question 8 Explanation:

\begin{aligned} \Sigma F_x &=0 \\ \sigma _x(BC)-\tau _Z \times (AB)\sigma _1 \sin \theta &= 0\\ \sigma _x\left ( \frac{AC \sin \theta }{\cos \theta } \right )+\tau _{zx}\left ( \frac{AC \cos \alpha }{\cos \theta } \right ) &=\sigma _1 \frac{AC \sin \theta }{\cos \theta }\\ \sigma _x \tan \theta +\tau _{zx} &=\sigma _1 \tan \theta \\ \tan \theta(\sigma _1-\sigma _2) &= \tau _{zx}\\ \tan \theta &= \left ( \frac{\tau _{zx}}{\sigma _1-\sigma _x} \right ) \end{aligned}
 Question 9
Match Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}&\text{Column Y} \\ \hline \text{(P) Horton equation} & \text{((I) Design of alluvial channel} \\ \hline \text{(Q) Penman method} & \text{(II) Maximum flood discharge}\\ \hline \text{(R) Chezys formula}& \text{(III) Evapotranspiration}\\ \hline \text{(S) Lacey's theory}& \text{(IV) Infiltration}\\ \hline \text{(T) Dicken's formula}& \text{(V) Flow velocity}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P)-(IV), (Q)-(III), (R)-(V), (S)-(I), (T)-(II) B (P)-(III), (Q)-(IV), (R)-(V), (S)-(I), (T)-(II) C (P)-(IV), (Q)-(III), (R)-(II), (S)-(I), (T)-(V) D (P)-(III), (Q)-(IV), (R)-(I), (S)-(V), (T)-(II)
Fluid Mechanics and Hydraulics   Fluid Dynamics and Flow Measurements
 Question 10
In a certain month, the reference crop evapotranspiration at a location is 6 mm/day. If the crop coefficient and soil coefficient are 1.2 and 0.8, respectively, the actual evapotranspiration in mm/day is
 A 5.76 B 7.2 C 6.8 D 8
Engineering Hydrology   Evaporation, Transpiration and Stream Flow Measurement
Question 10 Explanation:
Actual evapotranspiration $(ET_C)$
$=K_S \times K_C \times$ Reference evapotranspiration $(ET_0)$
$=0.8 \times 1.2 \times 6=5.76mm$
There are 10 questions to complete.

## GATE Civil Engineering 2022 SET-1

 Question 1
Consider the following expression:
$z=\sin(y+it)+\cos(y-it)$
where $z, y,$ and $t$ are variables, and $i=\sqrt{-1}$ is a complex number. The partial differential equation derived from the above expression is
 A $\frac{\partial^2 z}{\partial t^2}+\frac{\partial^2 z}{\partial y^2}=0$ B $\frac{\partial^2 z}{\partial t^2}-\frac{\partial^2 z}{\partial y^2}=0$ C $\frac{\partial z}{\partial t}-i\frac{\partial z}{\partial y}=0$ D $\frac{\partial z}{\partial t}+i\frac{\partial z}{\partial y}=0$
Engineering Mathematics   Partial Differential Equation
Question 1 Explanation:
\begin{aligned} z&=\sin(y+it)+ \cos (y-it)\\ \frac{\partial z}{\partial y}&=\cos (y+it)-\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-\sin(y+it)- \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-z \;\;...(i)\\ \frac{\partial z}{\partial t}&=i \cos (y+it)+i\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=+\sin(y+it)+ \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=z\;\;...(ii)\\ &\text{Adding (i) and (ii)}\\ &\frac{\partial ^2 z}{\partial ^2 y^2}+\frac{\partial ^2 z}{\partial ^2 t^2}=0 \end{aligned}
 Question 2
For the equation
$\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{\frac{3}{2}}+x^2y=0$
the correct description is
 A an ordinary differential equation of order 3 and degree 2. B an ordinary differential equation of order 3 and degree 3. C an ordinary differential equation of order 2 and degree 3. D an ordinary differential equation of order 3 and degree 3/2.
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
$\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{3/2}+x^2y=0$
Power of $\left ( \frac{dy}{dx} \right )$ is fractional, make it integer.
$\frac{d^3y}{dx^3}+x^2y=-x\left ( \frac{dy}{dx} \right )^{3/2}$
$\left (\frac{d^3y}{dx^3}+x^2y \right )^2=x^2\left ( \frac{dy}{dx} \right )^{3}$
Now order = 3 and degree = 2
 Question 3
The hoop stress at a point on the surface of a thin cylindrical pressure vessel is computed to be 30.0 MPa. The value of maximum shear stress at this point is
 A 7.5 MPa B 15.0 MPa C 30.0 MPa D 22.5 MPa
Solid Mechanics   Bending and Shear Stresses
Question 3 Explanation:
Given,
Hoop stress $(\sigma _h)=\frac{pd}{2t}=30MPa$
Maximum shear stress in plane $(\tau _{max})_{\text{in plane}}=\frac{\frac{pd}{2t}-\frac{pd}{4t}}{2}=7.5MPa$
 Question 4
In the context of elastic theory of reinforced concrete, the modular ratio is defined as the ratio of
 A Young's modulus of elasticity of reinforcement material to Young?s modulus of elasticity of concrete. B Youngs modulus of elasticity of concrete to Young?s modulus of elasticity of reinforcement material. C shear modulus of reinforcement material to the shear modulus of concrete. D Young's modulus of elasticity of reinforcement material to the shear modulus of concrete.
RCC Structures   Working Stress and Limit State Method
Question 4 Explanation:
This is a question of working stress method i.e. elastic theory.
Modular ratio
$=\frac{E_s}{E_c}=\frac{\text{Young's modulus of steel}}{\text{Young's modulus of concrete}}$
 Question 5
Which of the following equations is correct for the Pozzolanic reaction?
 A $Ca(OH)_2$ + Reactive Superplasticiser + $H_2O \rightarrow C-S-H$ B $Ca(OH)_2$ + Reactive Silicon dioxide + $H_2O \rightarrow C-S-H$ C $Ca(OH)_2$ + Reactive Sulphates + $H_2O \rightarrow C-S-H$ D $Ca(OH)_2$ + Reactive Sulphur + $H_2O \rightarrow C-S-H$
RCC Structures   Concrete Technology
Question 5 Explanation:
Pozzolanic materials have no cementing properties itself but have the property of combining with lime to produce stable compound.
Pozzolana is considered as siliceous and aluminous materials and when added in cement it have $SiO_2$ and $Al_2O_3$ form.
So, pozzolanic reaction :
$H_2O$ + Reactive slilica-di-oxide + $H_2O \rightarrow C-S-H$ gel or tobermonite gel
 Question 6
Consider the cross-section of a beam made up of thin uniform elements having thickness $t(t \lt \lt a)$ shown in the figure. The $(x,y)$ coordinates of the points along the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are:
 A x = 0, y = 3a B x = 2a, y = 2a C x = -a, y = 2a D x = -2a, y = a
Solid Mechanics   Theory of Columns and Shear Centre
Question 6 Explanation:
Shear centre of section consisting of two intersecting narrow rectangles always lies at the intersection of centrelines of two rectangles.

Coordinate of shear centre (0, 3a).
 Question 7
Four different soils are classified as CH, ML, SP, and SW, as per the Unified Soil Classification System. Which one of the following options correctly represents their arrangement in the decreasing order of hydraulic conductivity?
 A SW, SP, ML, CH B SW, SP, ML, CH C SP, SW, CH, ML D ML, SP, CH, SW
Geotechnical Engineering   Classification of Soils and Clay Minerals
Question 7 Explanation:
Hydraulic conductivity Order.
Gravel $\gt$ Sand $\gt$ silt $\gt$ lay
 Question 8
Let $\sigma _v'$ and $\sigma _h'$ denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil element to reach the failure state under Rankine?s passive earth pressure condition?
 A $\sigma ' _v \lt\sigma ' _h$ B $\sigma ' _v \gt\sigma ' _h$ C $\sigma ' _v = \sigma ' _h$ D $\sigma ' _v + \sigma ' _h =0$
Geotechnical Engineering   Retaining Wall-Earth Pressure Theories
Question 8 Explanation:
We know, $\sigma _h'=K\sigma _v'$
For passive earth pressure,
\begin{aligned} k&=K_P \gt 1\\ \Rightarrow \frac{\sigma _h'}{\sigma _v'}&=K_P \gt 1\\ \Rightarrow \sigma _h' \gt \sigma _v' \end{aligned}
 Question 9
With respect to fluid flow, match the following in Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}& \text{Column Y}\\ \hline \text{(P) Viscosity} & \text{(I) Mach number}\\ \hline \text{(Q) Gravity}&\text{(II) Reynolds number}\\ \hline \text{(R) Compressibility}&\text{(III) Euler number}\\ \hline \text{(S) Pressure} &\text{(IV) Froude number}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P) - (II), (Q) - (IV), (R) - (I), (S) - (III) B (P) - (III), (Q) - (IV), (R) - (I), (S) - (II) C (P) - (IV), (Q) - (II), (R) - (I), (S) - (III) D (P) - (II), (Q) - (IV), (R) - (III), (S) - (I)
Fluid Mechanics and Hydraulics   Flow Through Pipes
Question 9 Explanation:
Reynold's number ($R_e$) is defined when apart from inertial force, viscous forces are dominant.
$R_e=\frac{\text{Inertial force}}{\text{Viscous force}}$
Froude?s number ($F_e$): It is used when in addition to inertial force, gravity forces are important.
$F_e=\frac{\text{Inertial force}}{\text{Gravity force}}$
Euler number ($E_u$): It is used when apart from inertial force, only pressure forces are dominant.
$E_u=\frac{\text{Inertial force}}{\text{Pressure force}}$
Mach number ($M$): It is used when in addition to inertial force, compressibility forces are dominant
$M=\frac{\text{Inertial force}}{\text{Elastic force}}$
 Question 10
Let $\psi$ represent soil suction head and $K$ represent hydraulic conductivity of the soil. If the soil moisture content $\theta$ increases, which one of the following statements is TRUE?
 A $\psi$ decreases and K increases. B $\psi$ increases and K decreases. C Both $\psi$ and K decrease. D Both $\psi$ and K increase.
Geotechnical Engineering   Properties of Soils
Question 10 Explanation:
$h_c\propto \frac{1}{R}$
$K\propto S$
Water content $\uparrow \rightarrow R\uparrow \rightarrow h_c \downarrow \rightarrow \psi \downarrow$
Water content $\uparrow \rightarrow S\uparrow \rightarrow K \uparrow$
There are 10 questions to complete.

## GATE Mechanical Engineering 2022 SET-2

 Question 1
$F(t)$ is a periodic square wave function as shown. It takes only two values, 4 and 0, and stays at each of these values for 1 second before changing. What is the constant term in the Fourier series expansion of $F(t)$?

 A 1 B 2 C 3 D 4
Engineering Mathematics   Calculus
Question 1 Explanation:
The constant term in the Fourier series expansion of $F(t)$ is the average value of $F(t)$ in one fundamental period i.e.,
$\frac{\int_{0}^{1}4dt+\int_{1}^{2}0dt}{2}=\frac{4}{2}=2$
 Question 2
Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral $\int_{A}^{}\vec{F}.d\vec{A}$ of a vector field $\vec{F}=3x\hat{i}+5y\hat{j}+6z\hat{k}$ over the entire surface $A$ of the cube is ______.
 A 14 B 27 C 28 D 31
Engineering Mathematics   Calculus
Question 2 Explanation:
Given,
\begin{aligned} \bar{F} &=3x\bar{i}+5y\bar{j}+6z\bar{k} \\ \triangledown \cdot \bar{F}&= \frac{\partial }{\partial x}(3x)+\frac{\partial }{\partial x}(5y)+\frac{\partial }{\partial x}(6z)\\ &= 3+5+6=14 \end{aligned}
By gauss divers once Theorem
\begin{aligned} \int_{A}^{}\bar{F}\cdot dA &=\int \int \int (\triangledown \cdot F)dV =\int \int \int 14\; dv\\ &=14 \times \text{volume of a cube of side 1 unit } \\ &=14 \times (1)^3=14\end{aligned}
 Question 3
Consider the definite integral
$\int_{1}^{2}(4x^2+2x+6)dx$
Let $I_e$ be the exact value of the integral. If the same integral is estimated using Simpson's rule with 10 equal subintervals, the value is $I_s$. The percentage error is defined as $e=100 \times (I_e-I_s)/I_e$. The value of $e$ is
 A 2.5 B 3.5 C 1.2 D 0
Engineering Mathematics   Numerical Methods
Question 3 Explanation:
Exact value
\begin{aligned} &=\int_{1}^{2} (4x^2+2x+6)dx\\ &=\frac{4x^3}{3}+\frac{2x^2}{2}+6x\\ &=\frac{4}{3}(\pi) \times (3)+6\\ &=\frac{28}{3}+9=\frac{55}{3} \end{aligned}
Approximate value
Here f(x) is a polynomial of degree 2, so Simpsons rule gives exact value with zero error
\begin{aligned} \therefore \;\; \text{Approx value}&=\frac{55}{3}\\ \frac{I_e-I_s}{I_e}&=0\\ \therefore \;\;e=\left (\frac{I_e-I_s}{I_e} \right )\times 100&=0 \end{aligned}
 Question 4
Given $\int_{-\infty }^{\infty }e^{-x^2}dx=\sqrt{\pi}$
If a and b are positive integers, the value of $\int_{-\infty }^{\infty }e^{-a(x+b)^2}dx$ is ___.
 A $\sqrt{\pi a}$ B $\sqrt{\frac{\pi}{a}}$ C $b \sqrt{\pi a}$ D $b \sqrt{\frac{\pi}{a}}$
Engineering Mathematics   Calculus
Question 4 Explanation:
\begin{aligned} &\text{ Let }(x+b)=t\\ &\Rightarrow \; dx=dt\\ &\text{When ,} x=-\infty ;t=-\infty \\ &\int_{-\infty }^{-\infty }e^{-n(x+b)^2}dx=\int_{-\infty }^{-\infty }e^{-at^2}dt\\ &\text{Let, }at^2=y^2\Rightarrow t=\frac{y}{\sqrt{a}}\\ &2at\;dt=3y\;dy\\ &dt=\frac{ydy}{at}=\frac{ydy}{a\frac{y}{\sqrt{a}}}=\frac{y}{\sqrt{a}}\\ &\int_{-\infty }^{-\infty }e^{-at^2}dt=\int_{-\infty }^{-\infty }e^{-y^2}\cdot \frac{dy}{\sqrt{a}}=\sqrt{\frac{\pi}{a}} \end{aligned}
 Question 5
A polynomial $\phi (s)=a_{n}s^{n}+a_{n-1}s^{n-1}+...+a_{1}s+a_0$ of degree $n \gt 3$ with constant real coefficients $a_n, a_{n-1},...a_0$ has triple roots at $s=-\sigma$. Which one of the following conditions must be satisfied?
 A $\phi (s)=0$ at all the three values of s satisfying $s^3+\sigma ^3=0$ B $\phi (s)=0, \frac{d \phi (s)}{ds}=0, \text{ and }\frac{d^2 \phi (s)}{ds^2}=0 \text{ at }s=-\sigma$ C $\phi (s)=0, \frac{d \phi (s)}{ds}=0, \text{ and }\frac{d^4 \phi (s)}{ds^4}=0 \text{ at }s=-\sigma$ D $\phi (s)=0, \text{ and }\frac{d^3 \phi (s)}{ds^3}=0 \text{ at }s=-\sigma$
Engineering Mathematics   Differential Equations
Question 5 Explanation:
Since $\varphi (s)$ has a triple roots at $s=-\sigma$
Therefore, $\varphi (s)=(s+\sigma )^3\psi (s)$
It satisfies all the conditions in option (B) is correct.
 Question 6
Which one of the following is the definition of ultimate tensile strength (UTS) obtained from a stress-strain test on a metal specimen?
 A Stress value where the stress-strain curve transitions from elastic to plastic behavior B The maximum load attained divided by the original cross-sectional area C The maximum load attained divided by the corresponding instantaneous crosssectional area D Stress where the specimen fractures
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 6 Explanation:
Tensile Strength: The tensile strength, or ultimate tensile strength (UTS), is the maximum load obtained in a tensile test, divided by the original cross-sectional area of the specimen.
$\sigma _u=\frac{P_{max}}{A_o}$
where, $\sigma _u=$ Ultimate tensile strength, $kg/mm^2$
$P_{max}=$ Maximum load obtained in a tensile test, kg
$A_o=$ Original cross-sectional area of gauge length of the test piece, $mm^2$
The tensile strength is a very familiar property and widely used for identification of a material. It is very easy to determine and is a quite reproducible property. It is used for the purposes of specifications and for quality control of a product. Tensile strength can be empirically correlated to other properties such as hardness and fatigue strength. For brittle materials, the tensile strength is a valid criterion for design.
 Question 7
A massive uniform rigid circular disc is mounted on a frictionless bearing at the end E of a massive uniform rigid shaft AE which is suspended horizontally in a uniform gravitational field by two identical light inextensible strings AB and CD as shown, where G is the center of mass of the shaft-disc assembly and $g$ is the acceleration due to gravity. The disc is then given a rapid spin $\omega$ about its axis in the positive x-axis direction as shown, while the shaft remains at rest. The direction of rotation is defined by using the right-hand thumb rule. If the string AB is suddenly cut, assuming negligible energy dissipation, the shaft AE will

 A rotate slowly (compared to $\omega$) about the negative z-axis direction B rotate slowly (compared to $\omega$) about the positive z-axis direction C rotate slowly (compared to $\omega$) about the negative y-axis direction D rotate slowly (compared to $\omega$) about the positive y-axis direction
Theory of Machine   Gyroscope
Question 7 Explanation:

The spin vector will chase the couple on torque vector and produce precision in system.
Hence precision will be $-y$ direction. Rotate slowly (compared to $\omega$) about negative $z-$axis direction.
 Question 8
A structural member under loading has a uniform state of plane stress which in usual notations is given by $\sigma _x=3P,\sigma _y=-2P,\tau _{xy}=\sqrt{2}P$, where $P \gt 0$. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of $P$ at which yielding starts (according to the maximum distortion energy theory) is
 A 70 Mpa B 90 Mpa C 120 Mpa D 75 Mpa
Machine Design   Satic Dynamic Loading and Failure Theories
Question 8 Explanation:
Given,
$\sigma _x=3P$
$\sigma _y=-2P$
$\tau =\sqrt{2}P$
According to maximum distortion energy theory,
\begin{aligned} \sqrt{\sigma _x^2-\sigma _x\sigma _y+\sigma _y^2+3\tau _{xy}^2} &=\frac{S_{yt}}{FOS} \\ P\sqrt{3^2-3(-2)+(-2)^2+3(\sqrt{2})^2}&= \frac{350}{1}\\ P \times 5&=350 \\ P&=70\; MPa \end{aligned}
 Question 9
Fluidity of a molten alloy during sand casting depends on its solidification range. The phase diagram of a hypothetical binary alloy of components A and B is shown in the figure with its eutectic composition and temperature. All the lines in this phase diagram, including the solidus and liquidus lines, are straight lines. If this binary alloy with 15 weight % of B is poured into a mould at a pouring temperature of $800^{\circ}C$, then the solidification range is

 A $400 \; ^{\circ}C$ B $250 \; ^{\circ}C$ C $800 \; ^{\circ}C$ D $150 \; ^{\circ}C$
Manufacturing Engineering   Engineering Materials
Question 9 Explanation:

Solidification range $=A'B$
$\triangle ABC \text{ and } \triangle MB'C$ is similar
\begin{aligned} \frac{MA}{MB'}&=\frac{BC}{BC'}\\ \frac{700-T_A}{700-400}&=\frac{15}{30}\\ 700-T_A&=\frac{1}{2} \times 300\\ T_A&=550^{\circ}C \end{aligned}
Solidification range $=T_A-T_B=550-400=150 ^{\circ}C$
 Question 10
A shaft of diameter $25^{^{-0.04}}_{-0.07}$mm is assembled in a hole of diameter $25^{^{+0.02}}_{0.00}$mm.
Match the allowance and limit parameter in Column I with its corresponding quantitative value in Column II for this shaft-hole assembly.

Allowance and limit parameter (Column I)
P. Allowance
Q. Maximum clearance
R. Maximum material limit for hole

Quantitative value (Column II)
1. 0.09 mm
2. 24.96 mm
3. 0.04 mm
4. 25.0 mm
 A P-3, Q-1, R-4 B P-1, Q-3, R-2 C P-1, Q-3, R-4 D P-3, Q-1, R-2
Manufacturing Engineering   Metrology and Inspection
Question 10 Explanation:

(1) Allowance = Lower limit of hole - upper limit of shaft
Allowance = 25.00 - 24.96 = 0.04 mm

(2) Maximum clearance $C_{max}$ = Upper limit of hole - lower limit of shaft
$C_{max}$ = 25.02 - 24.93 = 0.09 mm

(3) Maximum material limit for hole = minimum size of hole = 25.00
There are 10 questions to complete.

## 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.

## GATE CSE 2022

 Question 1
Which one of the following statements is TRUE for all positive functions $f(n)$?
 A $f(n^2)=\theta (f(n)^2)$, where $f(n)$ is a polynomial B $f(n^2)=o (f(n)^2)$ C $f(n^2)=O (f(n)^2)$, where $f(n)$ is an exponential function D $f(n^2)=\Omega (f(n)^2)$
Algorithm   Asymptotic Notation
Question 1 Explanation:
 Question 2
Which one of the following regular expressions correctly represents the language of the finite automaton given below?

 A ab*bab* + ba*aba* B (ab*b)*ab* + (ba*a)*ba* C (ab*b + ba*a)*(a* + b*) D (ba*a + ab*b)*(ab* + ba*)
Theory of Computation   Finite Automata
Question 2 Explanation:
 Question 3
Which one of the following statements is TRUE?
 A The LALR(1) parser for a grammar G cannot have reduce-reduce conflict if the LR(1) parser for G does not have reduce-reduce conflict. B Symbol table is accessed only during the lexical analysis phase. C Data flow analysis is necessary for run-time memory management. D LR(1) parsing is sufficient for deterministic context-free languages.
Compiler Design   Parsing
Question 3 Explanation:
 Question 4
In a relational data model, which one of the following statements is TRUE?
 A A relation with only two attributes is always in BCNF. B If all attributes of a relation are prime attributes, then the relation is in BCNF. C Every relation has at least one non-prime attribute. D BCNF decompositions preserve functional dependencies.
Database Management System   Normal Form
Question 4 Explanation:
 Question 5
Consider the problem of reversing a singly linked list. To take an example, given the linked list below,

the reversed linked list should look like

Which one of the following statements is TRUE about the time complexity of algorithms that solve the above problem in O(1) space?
 A The best algorithm for the problem takes $\theta(n)$ time in the worst case. B The best algorithm for the problem takes $\theta(n \log n)$ time in the worst case. C The best algorithm for the problem takes $\theta(n^2)$ time in the worst case. D It is not possible to reverse a singly linked list in O(1) space.
Data Structure   Link List
Question 5 Explanation:
 Question 6
Suppose we are given $n$ keys, $m$ hash table slots, and two simple uniform hash functions $h_1$ and $h_2$. Further suppose our hashing scheme uses $h_1$ for the odd keys and $h_2$ for the even keys. What is the expected number of keys in a slot?
 A $\frac{m}{n}$ B $\frac{n}{m}$ C $\frac{2n}{m}$ D $\frac{n}{2m}$
Data Structure   Hashing
Question 6 Explanation:
 Question 7
Which one of the following facilitates transfer of bulk data from hard disk to main memory with the highest throughput?
 A DMA based I/O transfer B Interrupt driven I/O transfer C Polling based I/O transfer D Programmed I/O transfer
Computer Organization   IO Interface
Question 7 Explanation:
 Question 8
Let R1 and R2 be two 4-bit registers that store numbers in 2's complement form. For the operation R1+R2, which one of the following values of R1 and R2 gives an arithmetic overflow?
 A R1 = 1011 and R2 = 1110 B R1 = 1100 and R2 = 1010 C R1 = 0011 and R2 = 0100 D R1 = 1001 and R2 = 1111
Digital Logic   Number System
Question 8 Explanation:
 Question 9
Consider the following threads, $T_1, T_2, \text{ and }T_3$ executing on a single processor, synchronized using three binary semaphore variables, $S_1, S_2, \text{ and }S_3$, operated upon using standard $wait()$ and $signal()$. The threads can be context switched in any order and at any time.

Which initialization of the semaphores would print the sequence BCABCABCA ...?
 A $S_1 = 1; S_2 = 1; S_3 = 1$ B $S_1 = 1; S_2 = 1; S_3 = 0$ C $S_1 = 1; S_2 = 0; S_3 = 0$ D $S_1 = 0; S_2 = 1; S_3 = 1$
Operating System   Process Synchronization
Question 9 Explanation:
 Question 10
Consider the following two statements with respect to the matrices $A_{m \times n},B_{n \times m},C_{n \times n} \text{ and }D_{n \times n},$

Statement 1: $tr(AB) = tr(BA)$
Statement 2: $tr(CD) = tr(DC)$

where$tr()$ represents the trace of a matrix. Which one of the following holds?
 A Statement 1 is correct and Statement 2 is wrong. B Statement 1 is wrong and Statement 2 is correct. C Both Statement 1 and Statement 2 are correct. D Both Statement 1 and Statement 2 are wrong.
Engineering Mathematics   Linear Algebra
Question 10 Explanation:
There are 10 questions to complete.

Gate 2022 CSE question paper and solution.

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## AVLTree Mock Test-1

 Question 1
An AVL tree $T$ contains $n$ integers, all distinct. For a given integer $k$, what is time comlexity of an algorithm to find the element $x$ in $T$ such that $|k-x|$ is minimized?
 A $\Theta ( n)$ B $\Theta (\log n)$ C $\Theta (n \log n)$ D $\Theta (n^2)$
Data Structure
Question 1 Explanation:
INSERT $k$, then find the PREDECESSOR and SUCCESSOR of $k$. Return the one whose difference with $k$ is smaller. All three methods take $\Theta (\log n)$ time.
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 Question 2
Given a binary search tree containing N integers, time complexity of creating an AVL tree containing the same values without destroying the original BST in the process is
 A O(N) B O(log N) C O(N log N) D O(N log log N)
Data Structure
Question 2 Explanation:
Traverse the BST (in any order) as you visit a node, insert that value into the AVL Tree. Each AVL Tree insert takes O(log N). You have to perform such N insert. So, total time is O(N log N).
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 Question 3
We have n distinct values stored in a height balanced (AVL) binary search tree. Which of the following statements is always true?
 A The value at each node is the median of the values in the subtree rooted at that node. B The shortest path between any pair of nodes is at most O(log n). C For any node, the difference between the size of the left subtree and the right subtree is at most 3. D The number of leaf nodes is greater than or equal to the number of internal nodes.
Data Structure
Question 3 Explanation:
A height balanced tree has overall height at most O(log n), so the shortest path between any pair of nodes is always at most O(log n).
The following AVL tree is a counterexample for all other statements.

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 Question 4
The number of ways in which the numbers {1,2,3,4,5,6,7} can be inserted in an empty AVL tree, so that we don't have to perform any rotations on it and value of root node as 4, is _____.
 A 64 B 24 C 48 D 128
Data Structure
Question 4 Explanation:
One possible is insert in the order {4, 2, 6, 1, 3, 5, 7} to make an AVL tree.
The ordering of {2, 6} and the ordering of {1, 3, 5, 7} do not matter. One can see the resulting tree is perfectly balance AVL tree.
Therefore, Total possible sequence of elements = 2!*4!=2*24=48
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 Question 5
What is the running time of an efficient method to merge two balanced binary search trees with n elements each into a balanced BST.
 A O(n log n) B O(n) C O(log n) D O(n log log n)
Data Structure
Question 5 Explanation:
We can start by doing an in-order walk of both trees concurrently. At each step, we compare the two tree elements and add the smaller one into a list, L, before calling that element's successor. When we finish walking both trees, L will contain a sorted list of elements from both trees. This takes O(n + n) = O(n) total time.
Now, from the sorted list, we want to create a balanced binary tree. We can do this by setting the root as the middle element of the list, and letting the first half of the list be its left subtree and the second half be its right subtree (recursively creating the balanced subtrees as well). This also takes O(n + n) = O(n) time.
The total time for this algorithm is therefore O(n)
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