# GATE Mechanical Engineering 2020 SET-2

 Question 1
The sum of two normally distributed random variables X and Y is
 A always normally distributed B normally distributed, only if X and Y are independent C normally distributed, only if X and Y have the same standard deviation D normally distributed, only if X and Y have the same mean
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:
\begin{aligned} X_{1} &\sim N\left(\mu_{1}, \sigma_{1}\right)\\ \text{and }\quad X_{2} &\sim N\left(\mu_{2}, \sigma_{2}\right)\\ \text{then }\quad X_{1}+X_{2} &\sim N\left(\mu_{1}+\mu_{2}, \sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}\right) \end{aligned}
Always normally distributed.
 Question 2
A matrix P is decomposed into its symmetric part S and skew symmetric part V. If
$S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}$, $V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix}$,
Then matrix P is
 A A B B C C D D
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
$\begin{array}{l} S=\frac{P+P^{T}}{2} \\ V=\frac{P-P^{T}}{2} \\ P=S+V=\left(\begin{array}{ccc} -4 & 2 & 5 \\ 6 & 3 & 7 \\ -1 & 0 & 2 \end{array}\right) \end{array}$
 Question 3
Let $I=\int_{x=0}^{1}\int_{y=0}^{x^2}xy^2dydx$ then, $I$ may also be expressed as
 A $I=\int_{y=0}^{1}\int_{x=0}^{\sqrt{y}}xy^2dxdy$ B $I=\int_{y=0}^{1}\int_{x=\sqrt{y}}^{1}yx^2dxdy$ C $I=\int_{y=0}^{1}\int_{x=\sqrt{y}}^{1}xy^2dxdy$ D $I=\int_{y=0}^{1}\int_{x=0}^{\sqrt{y}}yx^2dxdy$
Engineering Mathematics   Calculus
Question 3 Explanation:
$I=\int_{0}^{1} \int_{0}^{x^{2}} x y^{2} d y d x$

Change on rules, $I=\int_{y=0}^{1} \int_{x=\sqrt{y}}^{1} x y^{2} d x d y$
 Question 4
The solution of
$\frac{d^2y}{dt^2}-y=1,$
which additionally satisfies $y|_{t=0}=\left.\begin{matrix} \frac{dy}{dt} \end{matrix}\right|_{t=0}=0$ in the Laplace s-domain is
 A $\frac{1}{s(s+1)(s-1)}$ B $\frac{1}{s(s+1)}$ C $\frac{1}{s(s-1)}$ D $\frac{1}{s-1}$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} y^{\prime\prime}-y &=1 \\ y(0) &=1 \\ y^{\prime}(0) &=1 \\ L\left\{y^{\prime\prime}-y\right\} &=L\{1\} \\ s^{2} Y(s)-s y(0)-y^{\prime}(0)-y(s) &=\frac{1}{s} \\ y(s) &=\frac{1}{s\left(s^{2}-1\right)}\\ & =\frac{1}{s(s+1)(s-1)} \end{aligned}
 Question 5
An attempt is made to pull a roller of weight W over a curb (step) by applying a horizontal force F as shown in the figure.

The coefficient of static friction between the roller and the ground (including the edge of the step) is $\mu$. Identify the correct free body diagram (FBD) of the roller when the roller is just about to climb over the step.
 A A B B C C D D
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 5 Explanation:

Weigh = W
Note:
(i) When the cylinder is about to make out of the curb, it will loose its contact at point A, only contact will be at it B.
(ii) At verge of moving out of curb, Roller will be in equation under W, F and contact force from B and these three forces has to be concurrent so contact force from B will pass through C.
(iii) Even the surfaces are rough but there will be no friction at B for the said condition.
FBD

 Question 6
A circular disk of radius $r$ is confined to roll without slipping at P and Q as shown in the figure.

If the plates have velocities as shown, the magnitude of the angular velocity of the disk is
 A $\frac{v}{r}$ B $\frac{v}{2r}$ C $\frac{2v}{3r}$ D $\frac{3v}{2r}$
Engineering Mechanics   Plane Motion
Question 6 Explanation:

For pure rolling

$\begin{array}{l} v_{P}=v=(P R) \omega \quad \ldots(i)\\ v_{Q}=2 v=(Q R) \omega \quad \ldots(ii) \end{array}$
Divide by (ii) to (i),
$2=\frac{Q R}{P R} \Rightarrow Q R=2(P R)$
\begin{aligned} P R+Q R &=2 r \\ P R+2(P R) &=2 r \\ P R &=\frac{2}{3} r \\ \text{From equation(i) }\quad v &=\left(\frac{2}{3} r\right) \omega \Rightarrow \omega=\frac{3 v}{2 r} \end{aligned}
 Question 7
The equation of motion of a spring-mass-damper system is given by
$\frac{d^2x}{dt^2}+3\frac{dx}{dt}+9x=10 \sin (5t)$
The damping factor for the system is
 A 0.25 B 0.5 C 2 D 3
Theory of Machine   Vibration
Question 7 Explanation:
$\frac{d^{2} x}{d t^{2}}+3\left(\frac{d x}{d t}\right)+9 x=10 \sin 5 t$
Comparing with standard equation:
$\ddot{x}+\left(2 \xi \omega_{n}\right) \dot{x}+\left(\omega_{n}^{2}\right) x=\left(\frac{F_{o}}{m}\right) \sin \omega t$
\begin{aligned} 2 \xi \omega_{n} &=3 \\ 2 \xi \times 3 &=3 &\left[\begin{array}{l} \omega_{n}^{2}=9 \\ \omega_{n}=3 \end{array}\right] \\ \xi &=\frac{1}{2}=0.5 \end{aligned}
 Question 8
The number of qualitatively distinct kinematic inversions possible for a Grashof chain with four revolute pairs is
 A 1 B 2 C 3 D 4
Theory of Machine   Displacement, Velocity and Acceleration
Question 8 Explanation:
They are:
1. Double crank mechanism
2. Crank-rocker mechanism
3. Double rocker mechanism
 Question 9
The process, that uses a tapered horn to amplify and focus the mechanical energy for machining of glass, is
 A electrochemical machining B electrical discharge machining C ultrasonic machining D abrasive jet machining
Manufacturing Engineering   Machining and Machine Tool Operation
Question 9 Explanation:
In Ultrasonic machining, the function of horn (also called concentrator, it is a tapered metal bar) is to amplify and focus vibration of the transducer to an adequate intensity for driving the tool to fulfll the cutting operation.
 Question 10
Two plates, each of 6 mm thickness, are to be butt-welded. Consider the following processes and select the correct sequence in increasing order of size of the heat affected zone

1. Arc welding
2. MIG welding
3. Laser beam welding
4. Submerged arc welding
 A 1-4-2-3 B 3-4-2-1 C 4-3-2-1 D 3-2-4-1
Manufacturing Engineering   Joining-Welding
Question 10 Explanation:
Processes with low rate of heat input (slow heating) tend to produce high total heat constant within the metal, slow cooling rates, and large heat-affected zones. high heat input process, have low total heats, fast cooling rates and small heat affected zones
There are 10 questions to complete.