## GATE Electronics and Communication 2023

 Question 1
Let $v_{1}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]$ and $v_{2}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ be two vectors. The value of the coefficient $\alpha$ in the expression $v_{1}=\alpha v_{2}+e$, which minimizes the length of the error vector $e$, is
 A $\frac{7}{2}$ B $-\frac{2}{7}$ C $\frac{2}{7}$ D $-\frac{7}{2}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} e & =V_{1}-\alpha V_{2} \\ e & =(i+2 k+0 k)-\alpha(2 i+j+3 k) \\ \hat{e} & =(1-2 \alpha) \hat{i}+(2-\alpha) \hat{j}+(0-3 \alpha) \hat{k} \\ |\hat{e}| & =\sqrt{(1-2 \alpha)^{2}+(2-\alpha)^{2}+(-3 \alpha)^{2}} \\ |\hat{e}|^{2} & =5+14 \alpha^{2}-8 \alpha \text { to be minimum at } \frac{\partial e^{2}}{\partial \alpha}=28 \alpha-8=0 \\ \alpha & =\frac{2}{7} \text { stationary point } \end{aligned}
 Question 2
The rate of increase, of a scalar field $f(x, y, z)=x y z$ in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is
 A $\frac{2}{3}$ B $\frac{4}{3}$ C $2$ D $4$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} f(x, y, z) & =x y z \\ \overline{\nabla f} & =\hat{i} f_{x}+\hat{j} f_{y}+\hat{k} f_{z} \\ & =\hat{i}(y z)+\hat{j}(x z)+\hat{k}(x y) \\ \overline{\nabla f}_{(0,2,1)} & =\hat{i}(2)+0 \hat{j}+0 \hat{k} \end{aligned}

Directional derivative,
\begin{aligned} D \cdot D & =\overline{\nabla f} \cdot \frac{\bar{a}}{|\bar{a}|} \\ & =(2 \hat{i}+0 \hat{j}+0 \hat{k}) \cdot \frac{(2 \hat{i}+\hat{j}+2 \hat{k})}{\sqrt{2^{2}+1^{2}+2^{2}}}=\frac{4}{\sqrt{9}}=\frac{4}{3} \end{aligned}

 Question 3
Let $w^{4}=16 j$. Which of the following cannot be a value of $w$ ?
 A $2 e^{\frac{j 2 \pi}{8}}$ B $2 e^{\frac{j \pi}{8}}$ C $2 e^{\frac{j 5 \pi}{8}}$ D $2 e^{\frac{j 9 \pi}{8}}$
Engineering Mathematics   Complex Variables
Question 3 Explanation:
$w=(2) j^{1 / 4}$
$w=2(0+j)^{1 / 4}$
$w=2\left[e^{j(2 n+1) \pi / 2}\right]^{1 / 4}$ $=2\left[e^{j(2 n+1) \pi / 8}\right]$

For $n=0$, $w=e^{j \pi / 8}$
For $n=2$, $w=2 e^{5 \pi j / 8}$
For $n=4$, $w=2 e^{9 \pi j / 8}$
 Question 4
The value of the contour integral, $\oint_{c}\left(\frac{z+2}{z^{2}+2 z+2}\right) d z$, where the contour $C$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is
 A $-\pi(1+j)$ B $\pi(1+j)$ C $\pi(1-j)$ D $-\pi(1-j)$
Engineering Mathematics   Calculus
Question 4 Explanation:
$I=\oint_{c} \frac{z+2}{z^{2}+2 z+2} d z ; \quad c=\left|z+1-\frac{3}{2} i\right|=1$

Poles are given $(z+1)^{2}+1=0$
\begin{aligned} & z+1= \pm \sqrt{-1} \\ & z=-1+j,-1-j \end{aligned}
where $-1-i$ lies outside ' $c$ '
$z=(-1,1) \text { lies inside } 'c'$.

by $\mathrm{CRT}$
\begin{aligned} \oint_{c} f(z) d z & =2 \pi i \operatorname{Res}(f(z), z=-1+j) \\ & =2 \pi i\left(\frac{z+2}{2(z+1)}\right)_{z=-1+i}\\ & =2 \pi i\left(\frac{-1+j+2}{2(-1+j+1)}\right) \\ & =\pi(1+j) \end{aligned}
 Question 5
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1} A B$, respectively, are
 A $\left\{\lambda_{k} \operatorname{det}\mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ B $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$ C $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ D $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P^{-1} v_{k} \mid 1 \leq k \leq n\right\}$
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
\begin{aligned} & B & =P^{-1} A P \\ A & =P B P^{-1}\end{aligned}

$\Rightarrow A, B$ are called matrices similar.
$\Rightarrow$ Both $A, B$ have same set 7 eigen values
But eigen vectors of $A, B$ are different.

Let $B X=\lambda X$
$\Rightarrow \quad\left(P^{-1} A P\right) X=\lambda X$
$\Rightarrow \quad A(P X)=\lambda(P X)$

$\therefore$ Eigen vectors of $A$ are $P X$.

There are 5 questions to complete.

## GATE Electrical Engineering 2023

 Question 1
For a given vector $w=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]^{\top}$, the vector normal to the plane defined by $\mathbf{w}^{\top} x=1$ is
 A $\left[\begin{array}{lll}-2 & -2 & 2\end{array}\right]^{T}$ B $\left[\begin{array}{lll}3 & 0 & -1\end{array}\right]^{T}$ C $\left[\begin{array}{lll}3 & 2 & 1\end{array}\right]^{T}$ D $\left[\begin{array}{llll}1 & 2 & 3\end{array}\right]^{T}$
Engineering Mathematics   Calculus
Question 1 Explanation:
Given, $W^{T}=1$

$\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=1$

We have, vector normal to the plane $=\nabla F$
\begin{aligned} & =i \frac{\partial F}{\partial x}+\hat{j} \frac{\partial F}{\partial y}+\hat{k} \frac{\partial F}{\partial z} \\ & =\hat{i}+2 \hat{j}+3 \hat{k} \end{aligned}

$\therefore$ Normal vector $=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]^{T}$
 Question 2
For the block diagrm shown in the figure, the transfer function $\frac{Y(s)}{R(s)}$ is

 A $\frac{2 s+3}{s+1}$ B $\frac{3 s+2}{s-1}$ C $\frac{s+1}{3 \mathrm{~s}+2}$ D $\frac{3 s+2}{s+1}$
Control Systems   Mathematical Models of Physical Systems
Question 2 Explanation:
Signal flow graph:

Forward paths,

\begin{aligned} & \mathrm{P}_{1}=3, \quad \Delta_{1}=1 \\ & \mathrm{P}_{2}=\frac{2}{\mathrm{~S}}, \Delta_{2}=1 \end{aligned}

Loops: $L_1=\frac{1}{S}$

Using Masson's graph formula,

\begin{aligned} \frac{Y(s)}{R(s)} & =\frac{P_{1} \Delta_{1}+P_{1} \Delta_{2}}{1-L_{1}} \\ & =\frac{3+\frac{2}{S}}{1-\frac{1}{S}} \\ & =\frac{3 S+2}{S-1} \end{aligned}

 Question 3
In the Nyquist plot of the open-loop transfer function

$\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=\frac{3 \mathrm{~s}+5}{\mathrm{~s}-1}$

corresponding to the feedback loop shown in the figure, the infinite semi-circular arc of the Nyquist contour in s-plane is mapped into a point at

 A $\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=\infty$ B $\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=0$ C $\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=3$ D $\mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s})=-5$
Control Systems   Frequency Response Analysis
Question 3 Explanation:
Nyquist Contour :

Given:
\begin{aligned} \mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s}) & =\frac{3 \mathrm{~s}+5}{\mathrm{~s}-1} \\ \text { Put } \quad \mathrm{s} & =R e^{j \theta} \\ \mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s}) & =\operatorname{Lim}_{R \rightarrow \infty} \frac{3 R e^{j \theta}+5}{\operatorname{Re}^{j \theta}-1} \\ \mathrm{G}(\mathrm{s}) \mathrm{H}(\mathrm{s}) & =3 \end{aligned}
 Question 4
Consider a unity-gain negative feedback system consisting of the plant $G(s)$ (given below) and a proportional-integral controller. Let the proportional gain and integral gain be 3 and 1, respectively. For a unit step reference input, the final values of the controller output and the plant output, respectively, are

$G(s)=\frac{1}{s-1}$
 A $\infty, \infty$ B $1,0$ C $1,-1$ D $-1,1$
Control Systems   Feedback Characteristics of Control Systems
Question 4 Explanation:
Given plant:

So, $\quad$ OLTF $=\frac{(3 s+1)}{s(s-1)}$

Closed loop transfer function,
\begin{aligned} \frac{Y(s)}{R(s)} & =\frac{3 s+1}{s^{2}+2 s+1} \\ Y(s) & =\frac{3 s+1}{s\left(s^{2}+2 s+1\right)} \quad\left[\because R(s)=\frac{1}{s}\right] \end{aligned}

Final value of plant,
$Y(\infty)=\operatorname{Lims}_{s \rightarrow 0} Y(s)=1$

From plant,
\begin{aligned} X(\mathrm{~s}) & =\left[\mathrm{R}(\mathrm{s})-\frac{\mathrm{X}(\mathrm{s})}{\mathrm{s}-1}\right]\left(3+\frac{1}{\mathrm{~s}}\right) \\ X(\mathrm{~s})\left[1+\frac{3 \mathrm{~s}+1}{\mathrm{~s}(\mathrm{~s}-1)}\right] & =\left(\frac{3 \mathrm{~s}+1}{\mathrm{~s}^{2}}\right)\left[\because \mathrm{R}(\mathrm{s})=\frac{1}{\mathrm{~s}}\right] \\ X(\mathrm{~s})\left[\frac{\mathrm{s}^{2}+2 s+1}{\mathrm{~s}(\mathrm{~s}-1)}\right] & =\left(\frac{3 \mathrm{~s}+1}{\mathrm{~s}^{2}}\right) \\ \Rightarrow \quad X(\mathrm{~s}) & =\frac{(3 \mathrm{~s}+1)(\mathrm{s}-1)}{\mathrm{s}\left(\mathrm{s}^{2}+2 \mathrm{~s}+1\right)} \end{aligned}

$\therefore$ Final value of controller,
$x(\infty)=\operatorname{LimsX}_{s \rightarrow 0} \mathrm{~s}(\mathrm{~s})=-1$
 Question 5
The following columns present various modes of induction machine operation and the ranges of slip

$\begin{array}{ll} \textbf{A (Mode of operation)}& \textbf{B (Range of slip)}\\\\ \text{a. Running in generator mode}&\text{p) From 0.0 to 1.0}\\\\ \text{b. Running in motor mode} & \text{q) From 1.0 to 2.0}\\\\ \text{c. Plugging in motor mode} & \text{r) From - 1.0 to 0.0} \end{array}$
The correct matching between the elements in column A with those of column B is
 A a-r, b-p, and c-q B a-r, b-q, and c-p C a-p, b-r, and c-q D a-q, b-p, and c-r
Electrical Machines   Single Phase Induction Motors, Special Purpose Machines and Electromechanical Energy Conversion System
Question 5 Explanation:
Torque speed characteristic of $3-\phi$ IM :

$\mathrm{S} \gt 1 \Rightarrow$ Plugging mode
$0 \lt \mathrm{S} \lt 1 \Rightarrow$ Motoring mode
$\mathrm{S} \lt 0 \Rightarrow$ Generating Mode

There are 5 questions to complete.

## GATE Civil Engineering 2023 SET-2

 Question 1
Let $\phi$ be a scalar field, and $\mathbf{u}$ be a vector field. Which of the following identities is true for $div(\phi \mathbf{u})$ ?
 A $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)$ B $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)$ C $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)$ D $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)$
Engineering Mathematics   Calculus
Question 1 Explanation:
$div(\phi \mathrm{u})=\phi div(\mu)+ugrad(\phi)$
 Question 2
Which of the following probability distribution functions (PDFs) has the mean greater than the median?

 A Function 1 B Function 2 C Function 3 D Function 4
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:

Option (B) is correct.

 Question 3
A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000 . Out of the total vehicles, $30 \%$ are without pollution clearance certificate. Further, even- and oddnumbered vehicles are operated on even- and oddnumbered dates, respectively.
If 100 vehicles are chosen at random on an evennumbered date, the number of vehicles expected without pollution clearance certificate is.
 A 15 B 30 C 50 D 70
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
Since $30 \%$ of the total vehicles are without pollution clearance certificate.
Out of the 100 chosen vehicle, $30 \%$ i.e. $100 \times 0.3=30$ vehicle are expected to be without pollution clearance certificate.
 Question 4
A circular solid shaft of span $L=5 \mathrm{~m}$ is fixed at one end and free at the other end. $A$ torque $T=$ $100 \mathrm{kN} . \mathrm{m}$ is applied at the free end. The shear modulus and polar moment of inertia of the section are denoted as $\mathrm{G}$ and $\mathrm{J}$, respectively. The torsional rigidity $\mathrm{GJ}$ is $50,000 \mathrm{kN} . \mathrm{m}^ 2 / \mathrm{rad}$. The following are reported for this shaft:

Statement i) The rotation at the free end is 0.01 $\mathrm{rad}$
Statement ii) The torsional strain energy is 1.0 kN.m

With reference to the above statements, which of the following is true?
 A Both the statements are correct B Statement i) is correct, but Statement ii) is wrong C Statement i) is wrong, but Statement ii) is correct D Both the statements are wrong
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 4 Explanation:

$\phi_{\mathrm{BA}}=\frac{\text { T.L. }}{\mathrm{GJ}}=\frac{(100)^{*} 5}{50000}=0.01 \mathrm{rad}$
$\Rightarrow$ Torsional strain energy $(U)$
\begin{aligned} & U=\frac{T^{2} L}{2 G J}=\frac{1}{2} \times T * \phi_{B A} \\ & U=\frac{1}{2} * 100 * 0.01=0.5 \mathrm{kN}-\mathrm{m} \end{aligned}
Hence, statement (i) correct and statement (ii) is incorrect.
 Question 5
M20 concrete as per IS 456: 2000 refers to concrete with a design mix having
 A an average cube strength of $20 \mathrm{MPa}$ B an average cylinder strength of $20 \mathrm{MPa}$ C a 5-percentile cube strength of $20 \mathrm{MPa}$ D a 5-percentile cylinder strength of $20 \mathrm{MPa}$
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
Question 5 Explanation:
In M20, M refers to mix and 20 to characteristic cube strength. As per clause no. 6.1.1, IS456: 2000 characteristic strength is defined as the strength below which not more than 5 percent of the test results are expected to fall.
Hence, correct option is (C).

There are 5 questions to complete.

## GATE Civil Engineering 2023 SET-1

 Question 1
For the integral

$\mathrm{I}=\int_{-1}^{1} \frac{1}{\mathrm{x}^{2}} \mathrm{dx}$

which of the following statements is TRUE?
 A $\quad \mathrm{I}=0$ B $\quad \mathrm{I}=2$ C $\quad \mathrm{I}=-2$ D The integral does not converge
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} I & =\int_{-1}^{1} \frac{1}{x^{2}} d x \\ & =2 \int_{0}^{1} \frac{1}{x^{2}} d x \quad(\because \quad f(-x)=f(x)) \\ & =2 \lim _{t \rightarrow 0^{+}} \int_{t}^{1} \frac{d x}{x^{2}} \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{-1}{x}\right)_{t}^{1} \\ & =-2 \lim _{t \rightarrow 0^{+}}\left(1-\frac{1}{t}\right) \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{1}{t}-1\right) \\ & =2 \lim _{h \rightarrow 0^{+}}\left(\frac{1}{0+h}-1\right)\\ & =2(\infty-1) \\ & =\infty \;\;\; (Divergent) \end{aligned}
 Question 2
A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three-point loads in the mid vertical plane as shown in the figure. Ignore the self-weight of the hanger. What is the maximum tensile stress in $\mathrm{N} / \mathrm{mm}^{2}$ anywhere in the hanger without considering stress concentration effects?

 A 15 B 25 C 35 D 45
Solid Mechanics   Principal Stress and Principal Strain
Question 2 Explanation:

$\sigma_{A B}=\frac{P_{A B}}{A_{A B}}=\frac{250 \times 10^{3}}{100 \times 100}=25 \mathrm{~N} / \mathrm{mm}^{2}$
$\sigma_{B C}=\frac{P_{B C}}{A_{B C}}=\frac{50 \times 10^3}{50 \times 50}=20 \mathrm{N} / \mathrm{mm}^{2}$
$\sigma_{\max }=\sigma_{\mathrm{AB}}=25 \mathrm{~N} / \mathrm{mm}^{2}$

 Question 3
Creep of concrete under compression is defined as the
 A increase in the magnitude of strain under constant stress B increase in the magnitude of stress under constant strain C decrease in the magnitude of strain under constant stress D decrease in the magnitude of stress under constant strain
RCC Structures   Working Stress and Limit State Method
Question 3 Explanation:
Under sustained compressive loading, deformation in concrete increases with time even through the applied stress level is not changed. The time dependent component of strain is called creep.
 Question 4
A singly reinforced concrete beam of balanced section is made of M20 grade concrete and $\mathrm{Fe} 415$ grade steel bars. The magnitudes of the maximum compressive strain in concrete and the tensile strain in the bars at ultimate state under flexure, as per IS 456: 2000 are _______ respectively. (round off to four decimal places)
 A 0.0035 and 0.0038 B 0.0020 and 0.0018 C 0.0035 and 0.0041 D 0.0020 and 0.0031
RCC Structures   Prestressed Concrete Beams
Question 4 Explanation:
Given data,
Balanced section, singly reinforced beam.
As per Clause No. 38.1, IS $456: 2000$,
Maximum strain in concrete at the outermost compression fibre $=0.0035$
and strain in the tension reinforcement for balanced section at ultimate state under flexure
\begin{aligned} & =0.002+\frac{f_{y}}{1.15 E_{s}} \\ & =0.002+\frac{415}{1.15 \times 2 \times 10^{5}}=0.0038 \end{aligned}
 Question 5
In cement concrete mix design, with the increase in water-cement ratio, which one of the following statements is TRUE?
 A Compressive strength decreases but workability increase B Compressive strength increases but workability decreases C Both compressive strength and workability decrease D Both compressive strength and workability increase
RCC Structures   Concrete Technology
Question 5 Explanation:
As the water-cement ratio increases, the porosity in the hardened concrete increases and hence the strength decreases.
Also, as water-cement ratio increases, ducts higher water availability, the workability increases.

There are 5 questions to complete.

## GATE Mechanical Engineering 2023

 Question 1
A machine produces a defective component with a probability of 0.015. The number of defective components in a packed box containing 200 components produced by the machine follows a Poisson distribution. The mean and the variance of the distribution are
 A 3 and 3, respectively B $\sqrt{3}$ and $\sqrt{3}$ , respectively C 0.015 and 0.015, respectively D 3 and 9, respectively
Industrial Engineering   Production Planning and Control
Question 1 Explanation:
P = 0.015
n = 200
mean $=\lambda =np= 200 \times 0.015 = 3$
variance $=\sigma ^2=\lambda =3$
 Question 2
The figure shows the plot of a function over the interval [-4, 4]. Which one of the options given CORRECTLY identifies the function?

 A $|2-x|$ B $|2-|x||$ C $|2+|x||$ D $2-|x|$
Engineering Mathematics   Calculus
Question 2 Explanation:
(a) Graph of y = 2 - x

(b) Graph of y = |2 - x|

(c) Graph of y = |2 - |x||

 Question 3
With reference to the Economic Order Quantity (EOQ) model, which one of the options given is correct?

 A Curve P1: Total cost, Curve P2: Holding cost, Curve P3: Setup cost, and Curve P4: Production cost. B Curve P1: Holding cost, Curve P2: Setup cost, Curve P3: Production cost, and Curve P4: Total cost. C Curve P1: Production cost, Curve P2: Holding cost, Curve P3: Total cost, and Curve P4: Setup cost. D Curve P1: Total cost, Curve P2: Production cost, Curve P3: Holding cost, and Curve P4: Setup cost.
Industrial Engineering   Inventory Control
Question 3 Explanation:

 Question 4
Which one of the options given represents the feasible region of the linear programming model:
\begin{aligned} Maximize\;\; 45X_1&+60X_2 \\ X_1&\leq 45 \\ X_2&\leq 50 \\ 10X_1+10X_2& \geq 600 \\ 25X_1+5X_2&\leq 750 \end{aligned}

 A Region P B Region Q C Region R D Region S
Industrial Engineering   Linear Programming
Question 4 Explanation:
\begin{aligned} x_1&=45 &...(i)\\ x_2&= 50&...(ii)\\ 10x_1+10x_2&=600 \\ or\; x_1+x_2&=60&...(iii) \\ 25x_1+5x_2&=750 \\ or\; 5x_1+x_2&=150&...(iv) \\ \end{aligned}
By drawing the curve we get 3 values of $x_1$ and $x_2$ as (10, 50), (20, 50), (22.5, 37.5)
So, $Z_{max}=45x_1+60x_2$ for (10,50)
$Z_{max}=450+3000=4450$
for (20,50)
$Z_{max}=45\times 20+50 \times 60=3900$
for (22.5, 37.5)
$Z_{max}=45\times 22.5+60 \times 37.5=3262.5$
So, $Z_{max}=3900\; for \; (x_1,x_2)=(20,50)$
 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 positioning. Four different configurations of locating pins are proposed as shown. Which one of the options given is correct?

 A Configuration P1 arrests 6 degrees of freedom, while Configurations P2 and P4 are over-constrained and Configuration P3 is under-constrained. B Configuration P2 arrests 6 degrees of freedom, while Configurations P1 and P3 are over-constrained and Configuration P4 is under-constrained. C Configuration P3 arrests 6 degrees of freedom, while Configurations P2 and P4 are over-constrained and Configuration P1 is under-constrained. D Configuration P4 arrests 6 degrees of freedom, while Configurations P1 and P3 are over-constrained and Configuration P2 is under-constrained.
Manufacturing Engineering   Machining and Machine Tool Operation
Question 5 Explanation:
3-2-1 principle of location
The 3-2-1 principle of location (six point location principle) is used to constrain the movement of workpiece along the three axes XX, YY and ZZ.
This is achieved by providing six locating points, 3-pins in base plate, 2-pins in vertical plane and 1-pin in a plane which is perpendicular to first two planes.

There are 5 questions to complete.

## GATE CSE 2023

 Question 1
Consider the following statements regarding the front-end and back-end of a compiler.

S1: The front-end includes phases that are independent of the target hardware.
S2: The back-end includes phases that are specific to the target hardware.
S3: The back-end includes phases that are specific to the programming language used in the source code.

Identify the CORRECT option.
 A Only S1 is TRUE. B Only S1 and S2 are TRUE. C S1, S2, and S3 are all TRUE. D Only S1 and S3 are TRUE.
Compiler Design   Intermediate Code Generation
Question 1 Explanation:
 Question 2
Which one of the following sequences when stored in an array at locations A[1], . . . , A[10] forms a max-heap?
 A 23, 17, 10, 6, 13, 14, 1, 5, 7, 12 B 23, 17, 14, 7, 13, 10, 1, 5, 6, 12 C 23, 17, 14, 6, 13, 10, 1, 5, 7, 15 D 23, 14, 17, 1, 10, 13, 16, 12, 7, 5
Data Structure   Heap Tree
Question 2 Explanation:

 Question 3
Let SLLdel be a function that deletes a node in a singly-linked list given a pointer to the node and a pointer to the head of the list. Similarly, let DLLdel be another function that deletes a node in a doubly-linked list given a pointer to the node and a pointer to the head of the list.
Let n denote the number of nodes in each of the linked lists. Which one of the following choices is TRUE about the worst-case time complexity of SLLdel and DLLdel?
 A SLLdel is O(1) and DLLdel is O(n) B Both SLLdel and DLLdel are O(log(n)) C Both SLLdel and DLLdel are O(1) D SLLdel is O(n) and DLLdel is O(1)
Question 3 Explanation:
 Question 4
Consider the Deterministic Finite-state Automaton (DFA) $A$ shown below. The DFA runs on the alphabet {0, 1}, and has the set of states {s, p, q, r}, with s being the start state and p being the only final state.

Which one of the following regular expressions correctly describes the language accepted by $A$?
 A $1(0^*11)^*$ B $0(0 + 1)^*$ C $1(0 + 11)^*$ D $1(110^*)^*$
Theory of Computation   Regular Expression
Question 4 Explanation:
 Question 5
The Lucas sequence $L_n$ is defined by the recurrence relation:
$L_n=L_{n-1}+L_{n-2}, \; for \; n\geq 3,$
with $L_1=1 \; and \; L_2=3$
Which one of the options given is TRUE?
 A $L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n$ B $L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n - \left ( \frac{1-\sqrt{5}}{3} \right )^n$ C $L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{3} \right )^n$ D $L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n- \left ( \frac{1-\sqrt{5}}{2} \right )^n$
Discrete Mathematics   Recurrence
Question 5 Explanation:

There are 5 questions to complete.

Gate 2023 CSE question paper and solution. GATE 2023 Answer Key.

Practice Previous year GATE CSE Topicwise questions with detail Solution.

## 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)

There are 5 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 $\frac{-j4\omega }{(1+t ^2)^2} \leftrightarrow 2 \pi(-\omega )e^{-|-\omega |}$
$\Rightarrow \frac{t}{(1+t^2)^2} \underrightarrow{FT} \frac{-2\pi}{-j4}\omega e^{-|\omega |}$
$\Rightarrow \;\;\;\rightarrow\frac{\pi}{j2} \omega e^{-|\omega |}$

There are 5 questions to complete.

## Concrete Technology

 Question 1
In cement concrete mix design, with the increase in water-cement ratio, which one of the following statements is TRUE?
 A Compressive strength decreases but workability increase B Compressive strength increases but workability decreases C Both compressive strength and workability decrease D Both compressive strength and workability increase
GATE CE 2023 SET-1   RCC Structures
Question 1 Explanation:
As the water-cement ratio increases, the porosity in the hardened concrete increases and hence the strength decreases.
Also, as water-cement ratio increases, ducts higher water availability, the workability increases.
 Question 2
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 2 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 3
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 3 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 4
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 4 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 4 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.