# GATE Mechanical Engineering 2020 SET-1

 Question 1
Multiplication of real valued square matrices of same dimension is
 A associative B commutative C always positive definite D not always possible to compute
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Matrix multiplication is associative.
 Question 2
The value of $\lim_{x \to 1} \left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )$ is
 A c B c+1 C $\frac{c}{c+1}$ D $\frac{c+1}{c}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Applying L Hospital rule
$\lim_{x \to 1}\left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )=\lim_{x \to 1}\left ( \frac{1-e^{-c+cx}}{-x(ce^{-c+x})-(e^{-c+cx})} \right ) =\frac{-c}{-c-1}=\frac{c}{c+1}$
 Question 3
The Laplace transform of a function f(t) is $L(f)=\frac{1}{s^2+\omega ^2}$. Then f(t) is
 A $f(t)=\frac{1}{\omega ^2}(1-\cos \omega t)$ B $f(t)=\frac{1}{\omega} \cos \omega t$ C $f(t)=\frac{1}{\omega} \sin \omega t$ D $f(t)=\frac{1}{\omega ^2}(1-\sin \omega t)$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$L(t)=\frac{1}{s^{2}+\omega ^{2}}$
$f(t)=L^{-1}\left \{ \frac{1}{s^{2}+\omega ^{2}} \right \}=\frac{1}{\omega }\sin \omega t$
 Question 4
Which of the following function f(z), of the complex variable z, is NOT analytic at all the points of the complex plane?
 A $f(z)=z^2$ B $f(z)=e^z$ C $f(z)=\sin z$ D $f(z)=\log z$
Engineering Mathematics   Complex Variables
Question 4 Explanation:
logz is not analytic at all points.
 Question 5
The members carrying zero force (i.e. zero-force members) in the truss shown in the figure, for any load $P \gt 0$ with no appreciable deformation of the truss (i.e. with no appreciable change in angles between the members), are
 A BF and DH only B BF, DH and GC only C BF, DH, GC, CD and DE only D BF, DH, GC, FG and GH only
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 5 Explanation:
If at any joint three forces are acting out of which two of them are collinear then force in third member must be zero.
For member ED look at joint E.

Similarity look for other members.
 Question 6
A single-degree-of-freedom oscillator is subjected to harmonic excitation $F(t) = F_0 \cos (\omega t)$ as shown in the figure.

The non-zero value of $\omega$, for which the amplitude of the force transmitted to the ground will be $F_0$, is
 A $\sqrt{\frac{k}{2m}}$ B $\sqrt{\frac{k}{m}}$ C $\sqrt{\frac{2k}{m}}$ D $2 \sqrt{\frac{k}{m}}$
Theory of Machine   Vibration
Question 6 Explanation:
Given,
$F_{1}=F_{0}$
then transmissibility
$\epsilon =\frac{F_{T}}{F_{0}}=\frac{F_{0}}{F_{0}}=1$
\begin{aligned} \frac{\sqrt{1+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}}{\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}=1 \\ 1+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2=\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2\\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =\pm 1 \\ \text{ Taking (-ve) sign,} \\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =-1 \\ \frac{\omega }{\omega _n} =\sqrt{2}\\ \omega=\sqrt{2}\omega_n=\sqrt{\frac{2k}{m}} \end{aligned}
 Question 7
The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of 10 MPa. If one unit on the $\sigma -\tau$ plane is 1 MPa, the Mohr's circle representation of the state-of-stress is given by
 A a circle with a radius equal to principal stress and its center at the origin of the $\sigma -\tau$ plane B a point on the $\sigma$ axis at a distance of 10 units from the origin C a circle with a radius of 10 units on the $\sigma -\tau$ plane D a point on the $\tau$ axis at a distance of 10 units from the origin
Strength of Materials   Mohr's Circle
Question 7 Explanation:

The given state of stress is represented by a point on $\sigma -\tau$ graph which is located on $\sigma$-axis at a distance of 10 units from origin.
 Question 8
A four bar mechanism is shown below

For the mechanism to be a crank-rocker mechanism, the length of the link PQ can be
 A 80 mm B 200 mm C 300 mm D 350 mm
Theory of Machine   Displacement, Velocity and Acceleration
Question 8 Explanation:

For Crank-Rocker mechanism, shortest link must be crank and adjacent to fixed as well as Grashoff's law must be satisfied.
If $l = 80 mm$ then shortest will be = 80 mm
as well as (80 + 600) $\lt$(400 + 300)
680 $\lt$ 700
Therefore law is satisfied.
$\Rightarrow$ l = 80 mm
 Question 9
A helical gear with $20^{\circ}$ pressure angle and $30^{\circ}$ helix angle mounted at the mid-span of a shaft that is supported between two bearings at the ends. The nature of the stresses induced in the shaft is
 A normal stress due to bending only B normal stress due to bending in one plane and axial loading; shear stress due to torsion C normal stress due to bending in two planes and axial loading; shear stress due to torsion D normal stress due to bending in two planes; shear stress due to torsion
Machine Design   Gears
 Question 10
The crystal structure of $\gamma$ iron (austenite phase) is
 A BCC B FCC C HCP D BCT
Manufacturing Engineering   Engineering Materials
Question 10 Explanation:
Austenite has a cubic-close packed crystal structure, also referred to as a face-centred cubic structure with an atom at each corner and in the centre of each face of the unit cell.
There are 10 questions to complete.