## 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 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.

There are 5 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{3 \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:
$A=\begin{bmatrix} 10 & 2k+5\\ 3k-3 & k+5 \end{bmatrix}$
$(2k + 5) = (3k - 3)$
$k=8$

There are 5 questions to complete.

## Verbal Ability

 Question 1
Which one of the sentence sequences in the given options creates a coherent narrative?

(i) I could not bring myself to knock.
(ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.
(iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.
(iv) I decided I would rather wander down the passage.
 A (iv), (i), (iii), (ii) B (iii), (i), (ii), (iv) C (ii), (i), (iv), (iii) D (i), (iii), (ii), (iv)
GATE ME 2023   General Aptitude
 Question 2
In a recently held parent-teacher meeting, the teachers had very few complaints about Ravi. After all, Ravi was a hardworking and kind student. Incidentally, almost all of Ravi's friends at school were hardworking and kind too. But the teachers drew attention to Ravi's complete lack of interest in sports. The teachers believed that, along with some of his friends who showed similar disinterest in sports, Ravi needed to engage in some sports for his overall development.
Based only on the information provided above, which one of the following statements can be logically inferred with certainty?
 A All of Ravi's friends are hardworking and kind. B No one who is not a friend of Ravi is hardworking and kind. C None of Ravi's friends are interested in sports. D Some of Ravi's friends are hardworking and kind.
GATE ME 2023   General Aptitude

 Question 3
The symbols $\bigcirc ,\ast ,\triangle ,\square$ are to be filled, one in each box, as shown below.
The rules for filling in the four symbols are as follows.

1) Every row and every column must contain each of the four symbols.
2) Every 2x2 square delineated by bold lines must contain each of the four symbols.

Which symbol will occupy the box marked with $?$ in the partially filled figure?
 A $\bigcirc$ B $\ast$ C $\triangle$ D $\square$
GATE ME 2023   General Aptitude
Question 3 Explanation:

 Question 4
Planting : Seed : : Raising : _____
(By word meaning)
 A Child B Temperature C Height D Lift
GATE ME 2023   General Aptitude
 Question 5
He did not manage to fix the car himself, so he _______ in the garage.
 A got it fixed B getting it fixed C gets fixed D got fixed
GATE ME 2023   General Aptitude

There are 5 questions to complete.

## Numerical Ability

 Question 1
An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

 A P B Q C R D S
GATE ME 2023   General Aptitude
 Question 2
How many pairs of sets (S,T) are possible among the subsets of {1, 2, 3, 4, 5, 6} that satisfy the condition that S is a subset of T?
 A 729 B 728 C 665 D 664
GATE ME 2023   General Aptitude
Question 2 Explanation:
Take one element {1}
$T=\phi ,S=\phi \rightarrow (\phi, \phi ) \Rightarrow 1\; pair$

$T=1,S=\phi \rightarrow (\phi, 1 ),(1,1) \Rightarrow 2\; pair$
For 1 element total pair $= 3^1$
Similarly
For 2 element total pair $= 3^2$
For 3 element total pair $= 3^3$
For 4 element total pair $= 3^4$
For 5 element total pair $= 3^5$
For 6 element total pair $= 3^6=729$

 Question 3
Consider the following inequalities

$p^2-4q \lt 4$
$3p+2q \lt 6$

where $p$ and $q$ are positive integers.
The value of $(p+q)$ is _______.
 A 2 B 1 C 3 D 4
GATE ME 2023   General Aptitude
Question 3 Explanation:
\begin{aligned} p^2-4q &\lt 4 \\ p^2-4&\ lt 4q \;\;...(i)\\ 3p+2Q& \lt 6 \\ 6p-12 &\lt -4q\;\;...(ii)\\ &\text{by equation (i) + (ii)} \\ p^2+6p-16 &\lt 0 \\ (p+8)(p-2)& \lt 0 \\ \therefore \; p&\in (-8,2) \end{aligned}
Given p is positive integer
$\therefore \;p=1$
Now, from equation (i), $1-4 \lt 4q$
$q \gt \frac{-3}{4}$
from equation (ii),
$q \lt 3/2$
$\therefore \; \frac{-3}{4} \lt q \lt \frac{3}{2}$
Given q is positive integer
$\therefore \;\; q=1$
Thus $p + q = 1+1 = 2$
 Question 4
The minute-hand and second-hand of a clock cross each other _______ times between 09:15:00 AM and 09:45:00 AM on a day.
 A 30 B 15 C 29 D 31
GATE ME 2023   General Aptitude
Question 4 Explanation:
After 09:15:00 AM every minute, minute and second hand cross each other once (1) times.
So, 09:16:00 to 09:45:00 minute hand and second hand cross each other 30 times.
 Question 5
A certain country has 504 universities and 25951 colleges. These are categorised into Grades I, II, and III as shown in the given pie charts.
What is the percentage, correct to one decimal place, of higher education institutions (colleges and universities) that fall into Grade III?

 A 22.7 B 23.7 C 15 D 66.8
GATE ME 2023   General Aptitude
Question 5 Explanation:
$\frac{ \text{7\% of 504 + 23 \% of 25951} }{504 + 25951}=22.7 \%$

There are 5 questions to complete.

## General Aptitude

 Question 1
An opaque pyramid (shown below), with a square base and isosceles faces, is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The pyramid can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

 A P B Q C R D S
GATE ME 2023      Numerical Ability
 Question 2
How many pairs of sets (S,T) are possible among the subsets of {1, 2, 3, 4, 5, 6} that satisfy the condition that S is a subset of T?
 A 729 B 728 C 665 D 664
GATE ME 2023      Numerical Ability
Question 2 Explanation:
Take one element {1}
$T=\phi ,S=\phi \rightarrow (\phi, \phi ) \Rightarrow 1\; pair$

$T=1,S=\phi \rightarrow (\phi, 1 ),(1,1) \Rightarrow 2\; pair$
For 1 element total pair $= 3^1$
Similarly
For 2 element total pair $= 3^2$
For 3 element total pair $= 3^3$
For 4 element total pair $= 3^4$
For 5 element total pair $= 3^5$
For 6 element total pair $= 3^6=729$

 Question 3
Which one of the sentence sequences in the given options creates a coherent narrative?

(i) I could not bring myself to knock.
(ii) There was a murmur of unfamiliar voices coming from the big drawing room and the door was firmly shut.
(iii) The passage was dark for a bit, but then it suddenly opened into a bright kitchen.
(iv) I decided I would rather wander down the passage.
 A (iv), (i), (iii), (ii) B (iii), (i), (ii), (iv) C (ii), (i), (iv), (iii) D (i), (iii), (ii), (iv)
GATE ME 2023      Verbal Ability
 Question 4
Consider the following inequalities

$p^2-4q \lt 4$
$3p+2q \lt 6$

where $p$ and $q$ are positive integers.
The value of $(p+q)$ is _______.
 A 2 B 1 C 3 D 4
GATE ME 2023      Numerical Ability
Question 4 Explanation:
\begin{aligned} p^2-4q &\lt 4 \\ p^2-4&\ lt 4q \;\;...(i)\\ 3p+2Q& \lt 6 \\ 6p-12 &\lt -4q\;\;...(ii)\\ &\text{by equation (i) + (ii)} \\ p^2+6p-16 &\lt 0 \\ (p+8)(p-2)& \lt 0 \\ \therefore \; p&\in (-8,2) \end{aligned}
Given p is positive integer
$\therefore \;p=1$
Now, from equation (i), $1-4 \lt 4q$
$q \gt \frac{-3}{4}$
from equation (ii),
$q \lt 3/2$
$\therefore \; \frac{-3}{4} \lt q \lt \frac{3}{2}$
Given q is positive integer
$\therefore \;\; q=1$
Thus $p + q = 1+1 = 2$
 Question 5
In a recently held parent-teacher meeting, the teachers had very few complaints about Ravi. After all, Ravi was a hardworking and kind student. Incidentally, almost all of Ravi's friends at school were hardworking and kind too. But the teachers drew attention to Ravi's complete lack of interest in sports. The teachers believed that, along with some of his friends who showed similar disinterest in sports, Ravi needed to engage in some sports for his overall development.
Based only on the information provided above, which one of the following statements can be logically inferred with certainty?
 A All of Ravi's friends are hardworking and kind. B No one who is not a friend of Ravi is hardworking and kind. C None of Ravi's friends are interested in sports. D Some of Ravi's friends are hardworking and kind.
GATE ME 2023      Verbal Ability

There are 5 questions to complete.

## GATE Mechanical Engineering 2021 SET-2

 Question 1
Consider an n x n matrix $A$ and a non-zero n x 1 vector $p$. Their product $Ap=\alpha ^2p$, where $\alpha \in \mathbb{R}$ and $\alpha \notin \{-1,0,1\}$. Based on the given information, the eigen value of $A^2$ is:
 A $\alpha$ B $\alpha ^2$ C $\sqrt{\alpha }$ D $\alpha ^4$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given, $A P=\alpha^{2} P$
By comparison with $A X=\lambda X \Rightarrow$
$\Rightarrow \quad \lambda=\alpha^{2}$
Hence, eigen value of A is $\alpha^{2}$, so eigen value of $A^{2}$ is $\alpha^{4}$.
 Question 2
If the Laplace transform of a function $f(t)$ is given by $\frac{s+3}{(s+1)(s+2)}$ , then $f(0)$ is
 A 0 B $\frac{1}{2}$ C 1 D $\frac{3}{2}$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
By using partial fraction concept.
\begin{aligned} f(t) &=L^{-1}\left[\frac{s+3}{(s+1)(s+2)}\right] \\ &=L^{-1}\left[\frac{2}{s+1}-\frac{1}{s+2}\right] \\ \Rightarrow \qquad f(t) &=2 e^{-t}-e^{-2 t} \\ \text { So, } \qquad f(c)&=2 e^{0}-e^{0}=2-1=1 \end{aligned}

 Question 3
The mean and variance, respectively, of a binomial distribution for $n$ independent trials with the probability of success as $p$, are
 A $\sqrt{np},np(1-2p)$ B $\sqrt{np}, \sqrt{np(1-p)}$ C $np,np$ D $np,np(1-p)$
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
Mean= np
Variance = npq = np(1 - p)
 Question 4
The Cast Iron which possesses all the carbon in the combined form as cementite is known as
 A Grey Cast Iron B Spheroidal Cast Iron C Malleable Cast Iron D White Cast Iron
Manufacturing Engineering   Engineering Materials
Question 4 Explanation:
On the basis of nature of carbon present in cast iron, it may be divided into white cast iron and gray cast iron.
In the gray cast iron, carbon is present in free form as graphite. Under very slow rate of cooling during solidification, carbon atoms get sufficient time to separate out in pure form as graphite. In addition, certain elements promote decomposition of cementite. Silicon and nickel are two commonly used graphitizing elements.
In white cast iron, carbon is present in the form of combined form as cementite. In normal conditions, carbon has a tendency to combine with iron to form cementite.
 Question 5
The size distribution of the powder particles used in Powder Metallurgy process can be determined by
 A Laser scattering B Laser reflection C Laser absorption D Laser penetration
Manufacturing Engineering   Forming Process
Question 5 Explanation:
Particle Size, Shape, and Distribution:
Particle size is generally controlled by screening, that is, by passing the metal powder through screens (sieves) of various mesh sizes. Several other methods also are available for particle-size analysis:
1. Sedimentation, which involves measuring the rate at which particles settle in a fluid.
2. Microscopic analysis, which may include the use of transmission and scanning- electron microscopy.
3. Light scattering from a laser that illuminates a sample, consisting of particles suspended in a liquid medium; the particles cause the light to be scattered, and a detector then digitizes the signals and computes the particle-size distribution.
4. Optical methods, such as particles blocking a beam of light, in which the particle is sensed by a photocell.
5. Suspending particles in a liquid and detecting particle size and distribution by electrical sensors.

There are 5 questions to complete.

## GATE Mechanical Engineering 2021 SET-1

 Question 1
If $y(x)$ satisfies the differential equation

$(\sin x)\frac{dy}{dx}+y \cos x =1$

subject to the condition $y(\pi /2)=\pi /2$, then $y(\pi /6)$ is
 A 0 B $\frac{\pi}{6}$ C $\frac{\pi}{3}$ D $\frac{\pi}{2}$
Engineering Mathematics   Differential Equations
Question 1 Explanation:
\begin{aligned} \frac{d y}{d x}+y \cot x&=\text{cosec} x\\ 1.F. \qquad&=e^{\int \cot x d x}=e^{\log \sin x}=\sin x\\ \Rightarrow \quad y(\sin x)&=\int \text{cosec} x \sin x d x+c\\ \Rightarrow \qquad y \sin x&=x+c\\ \Rightarrow \qquad \frac{\pi}{2} \sin \frac{\pi}{2} & =\frac{\pi}{2}+c \\ \Rightarrow \qquad \frac{\pi}{2} & =\frac{\pi}{2}+c \quad \Rightarrow c=0 \\ \Rightarrow \qquad y \sin x & =x\\ \Rightarrow \qquad y \sin \frac{\pi}{6}&=\frac{\pi}{6}\\ \Rightarrow \qquad y\left(\frac{1}{2}\right) &=\frac{\pi}{6} \\ \Rightarrow y &=\frac{\pi}{3} \end{aligned}
 Question 2
The value of $\lim_{x \to 0}\left ( \frac{1- \cos x}{x^2} \right )$ is
 A $\frac{1}{4}$ B $\frac{1}{3}$ C $\frac{1}{2}$ D 1
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} \lim _{x \rightarrow 0}\left(\frac{1-\cos x}{x^{2}}\right)&=? \;\;\;\;\;\;\left(\frac{0}{0} \text { form }\right) \\ \text { Applying } L \cdot H \text { rule } & =\lim _{x \rightarrow 0} \frac{\sin x}{2 x}\left(\frac{0}{0}\right)=\lim _{x \rightarrow 0} \frac{\cos x}{2}=\frac{1}{2} \end{aligned}

 Question 3
The Dirac-delta function $(\delta (t-t_0)) \text{ for }t,t_0 \in \mathbb{R},$ has the following property

$\int_{a}^{b}\varphi (t)\delta (t-t_0)dt=\left\{\begin{matrix} \varphi (t_0) & a \lt t_0 \lt b\\ 0 &\text{otherwise} \end{matrix}\right.$

The Laplace transform of the Dirac-delta function $\delta (t-a)$ for $a \gt 0$;
$\mathcal{L} (\delta (t-a))=F(s)$ is
 A 0 B $\infty$ C $e^{sa}$ D $e^{-sa}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
\begin{aligned} \because \qquad \int_{0}^{-} f(t) \delta(t-a) d t&=f(a) \\ \therefore \qquad L\{\delta(t-a)\}&=\int_{0}^{-} e^{-s t} \delta(t-a) d t=e^{-a s} \end{aligned}
 Question 4
The ordinary differential equation $\frac{dy}{dt}=-\pi y$ subject to an initial condition $y(0)=1$ is solved numerically using the following scheme:

$\frac{y(t_{n+1})-y(t_n)}{h}=-\pi y(t_n)$

where $h$ is the time step, $t_n=nh,$ and $n=0,1,2,...$. This numerical scheme is stable for all values of $h$ in the interval.
 A $0 \lt h \lt \frac{2}{\pi}$ B $0 \lt h \lt 1$ C $0 \lt h \lt \frac{\pi}{2}$ D for all $h \gt 0$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} \frac{y\left(t_{n+1}\right)-y\left(t_{n}\right)}{h} &=-\pi y\left(t_{n}\right) \\ y_{n+1} &=-\pi / y_{n}+y_{n}=(-\pi h+1) y_{n} \end{aligned}
It is recursion relation between $y_{n+1}$ and $y_{n}$
So solution will be stable if
\begin{aligned} |-\pi h+1| & \lt 1 \\ -1 \lt -\pi h+1 & \lt 1 \\ -2 \lt -\pi h & \lt 0 \\ 0 & \lt \pi h \lt 2 \\ 0 & \lt h \lt \frac{2}{\pi} \end{aligned}
Therefore option (A) is correct.
 Question 5
Consider a binomial random variable $X$. If $X_1,X_2,..., X_n$ are independent and identically distributed samples from the distribution of $X$ with sum $Y=\sum_{i=1}^{n}X_i$, then the distribution of $Y$ as $n\rightarrow \infty$ can be approximated as
 A Exponential B Bernoulli C Binomial D Normal
Engineering Mathematics   Probability and Statistics

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