# GATE Mechanical Engineering 2019 SET-2

 Question 1
In matrix equation [A]{X}={R},
$[A]=\begin{bmatrix} 4 & 8 & 4\\ 8& 16 & -4\\ 4& -4 & 15 \end{bmatrix}$, $\{X\}=\begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix}$ and $\{R\}=\begin{Bmatrix} 32\\ 16\\ 64 \end{Bmatrix}$.
One of the eigenvalues of matrix [A] is
 A 4 B 8 C 15 D 16
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that AX=R
$\begin{array}{l} \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=\left[\begin{array}{l} 32 \\ 16 \\ 64 \end{array}\right] \\ \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=16\left[\begin{array}{c} 32 \\ 4 \end{array}\right](\because A X=\lambda X) \end{array}$
$\therefore$ One of eigen value of the given matrix A is given by $\lambda=16$
 Question 2
The directional derivative of the function $f(x,y)=x^2+y^2$ along aline directed from (0,0) to (1,1), evaluated at the point x=1, y=1 is
 A $\sqrt{2}$ B 2 C $2 \sqrt{2}$ D $4 \sqrt{2}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Given function is $f(x, y)=x^{2}+y^{2}$
The direction vector a is given by
$\overline{\mathrm{a}}=(1,1)-(0,0)=\overline{\mathrm{i}}+\overline{\mathrm{j}}$
Let the given point be $P=(x, y)=(1,1)$
Now, the directional directive of f(x, y) in the direction of vector a at point is given by
$\mathrm{D} . \mathrm{D}=(\nabla \mathrm{f}) \cdot \cdot \frac{\overline{\mathrm{a}}}{|\mathrm{a}|}$
$\Rightarrow D . D=(2 x \bar{i}+2 x \bar{j}) \cdot \frac{(\bar{i}+\bar{j})}{\sqrt{1+1}}$
$\therefore D . D=\frac{2+2}{\sqrt{2}}=2 \sqrt{2}$
 Question 3
The differential equation $\frac{dy}{dx}+4y=5$ is valid in the domain $0\leq x\leq 1$ with y(0)=2.25. The solution of the differential equation is
 A $y=e^{-4x}+5$ B $y=e^{-4x}+1.25$ C $y=e^{4x}+5$ D $y=e^{4x}+1.25$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$\begin{array}{l} \text { Given } \frac{\mathrm{dy}}{\mathrm{dx}}+4 \mathrm{y}=5,0 \leq \mathrm{x} \leq 1 \; \; \ldots(1) \\ \because \frac{d y}{d x}+P(x, y)=Q(x) \\ \text { With } \mathrm{y}(0)=2.25\;\;\; \ldots(2) \\ \text { Here, I.F }=\mathrm{e}^{\int 4 \mathrm{dx}}=\mathrm{e}^{4 \mathrm{x}} \end{array}$
The general solution of (1) is given by
$y \cdot e^{4 x}=\int(5)\left(e^{4 x}\right) d x+c$
$\Rightarrow \mathrm{y} \cdot \mathrm{e}^{4 \mathrm{x}}=\frac{5}{4} \mathrm{e}^{4 \mathrm{x}}+\mathrm{c}\cdots(3)$
Using (2) and (3)
$(2.25)(1)=\left(\frac{5}{4}\right)(1)+c$
$c=1 \cdots(4)$
The solution of (1) from (3) and (4) is
$y \cdot e^{4 x}=\frac{5}{4} e^{4 x}+1$
or $y=\frac{5}{4}+e^{4 x}=e^{-4 x}+1.25$
 Question 4
An analytic function f(z) of complex variable z=x+iy may be written as $f(z)=u(x,y)+iv(x,y)$. Then u(x,y) and v(x,y) must satisfy
 A $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ B $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ C $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ D $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
Engineering Mathematics   Complex Variables
Question 4 Explanation:
Given that the complex function f(z)=u(x,y)+ i v(x,y) is an analytic function.
$\Rightarrow$ the Cauchy-Riemann equation will satisfy for u(x,y) & v(x,y)
$\therefore \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=\frac{\partial \mathrm{v}}{\partial \mathrm{y}} \text{ and } \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=-\frac{\partial \mathrm{v}}{\partial \mathrm{y}}$
 Question 5
A rigid triangular body, PQR, with sides of equal length of 1 unit moves on a flat plane. At the instant shown, edge QR is parallel to the x-axis, and the body moves such that velocities of points P and R are $V_P \; and \; V_R$, in the x and y directions, respectively. The magnitude of the angular velocity of the body is
 A $2V_R$ B $2V_P$ C $V_R/\sqrt{3}$ D $V_P/\sqrt{3}$
Theory of Machine   Planar Mechanisms
Question 5 Explanation:
$\begin{array}{l} \Rightarrow \mathrm{V}_{\mathrm{R}}=(\mathrm{IR}) \omega \\ \Rightarrow \omega=\frac{\mathrm{V}_{\mathrm{R}}}{(\mathrm{IR})} \\ \Rightarrow \omega \times \frac{\mathrm{V}_{\mathrm{R}}}{\frac{1}{2}} \\ \Rightarrow \omega=2 \mathrm{V}_{\mathrm{R}} \end{array}$

 Question 6
Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of 100 MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young's modulus E=200 MPa and Poisson's ratio v=0.3 are given. The principal strains in the plane of the sheet are
 A (0.35, -0.15) B (0.5, 0.0) C (0.5, -0.15) D (0.5, -0.5)
Strength of Materials   Stress and Strain
Question 6 Explanation:
\begin{aligned} \sigma_{\mathrm{x}}&=100 \mathrm{MPa} \\ v&=\mu=0.3 \\ \sigma_{\mathrm{y}}&=0, \sigma_{\mathrm{z}}=0, \mathrm{E}=200 \mathrm{MPa} \\ &\text { Principal strain in x-direction } \\ &=\epsilon_{1}=\epsilon_{\mathrm{x}}=\frac{\sigma_{\mathrm{x}}}{\mathrm{E}}-\mu \frac{\sigma_{\mathrm{y}}}{\mathrm{E}} \\ &=\frac{100}{200}-0=0.5\\ &\text { Principal strain in y-direction } \\ &=\epsilon_{2}=\epsilon_{\mathrm{y}}=\frac{\sigma_{\mathrm{y}}}{\mathrm{E}}-\mu \frac{\sigma_{\mathrm{x}}}{\mathrm{E}}\\ &=0-(0.3)\left(\frac{100}{200}\right)=-0.15\\ &\therefore\left(\epsilon_{\mathrm{x}}, \epsilon_{\mathrm{y}}\right)=(0.5-0.15) \end{aligned}
 Question 7
A spur gear has pitch circle diameter D and number of teeth T. The circular pitch of the gear is
 A $\frac{\pi D}{T}$ B $\frac{T}{D}$ C $\frac{D}{T}$ D $\frac{2 \pi D}{T}$
Theory of Machine   Gear and Gear Train
Question 7 Explanation:
Circular pitch : It is the distance between two similar points on adjacent teeth measured along pitch
circle circumference circular pitch
$\left(P_{c}\right)=\frac{\text { Pitch circlecircum }}{\text { Number of teeth }}=\frac{\pi D}{T}$
 Question 8
Endurance limit of a beam subjected to pure bending decreases with
 A decrease in the surface roughness and decrease in the size of the beam B increase in the surface roughness and decrease in the size of the beam C increase in the surface roughness and increase in the size of the beam D decrease in the surface roughness and increase in the size of the beam
Question 8 Explanation:
Endurance limit decreases with increase in surface roughness and with increase in size of the beam.
 Question 9
A two-dimensional incompressible frictionless flow field is given by $\vec{u}=x\hat{i}-y\hat{j}$. If $\rho$ is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as
 A $\rho (x\hat{i}+y\hat{j})$ B $-\rho (x\hat{i}+y\hat{j})$ C $\rho (x\hat{i}-y\hat{j})$ D $-\rho (x^2 \hat{i} + y^2 \hat{j})$
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 9 Explanation:
Given, 2-D incompressible frictionless fluid flow.
$\overrightarrow{\mathrm{u}}=x \hat{\mathrm{i}}-y \hat{\mathrm{j}}$
Thus, velocity components in x and y directions are :
$\mathrm{u}=\mathrm{x} \text { and } \mathrm{v}=-\mathrm{y}$
Navier-Stokes equation for incompressible, frictionless fluid flow reduces to
$\rho \frac{\mathrm{DV}}{\mathrm{Dt}}=-\nabla \overrightarrow{\mathrm{P}}+\overrightarrow{\rho \mathrm{g}}$
There are no components of body force in x and y direction. Hence,
$\rho \frac{\mathrm{D} \overrightarrow{\mathrm{V}}}{\mathrm{Dt}}=-\nabla \overrightarrow{\mathrm{P}}$
where, $\nabla \overrightarrow{\mathrm{P}}$ is the pressure gradient vector
Hence,
\begin{aligned} \bigtriangledown \vec{P}&=\rho \left [ \left ( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right )\hat{i}+\left ( u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y}\right ) \hat{j}\right ]\\ &=-\rho \left [ \{ x(1)+(-y)(0)\}\hat{i}+\{ x(0)+(-y)(-1)\} \hat{j}\right ]\\ &=-\rho (x\hat{i}+y\hat{j}) \end{aligned}
 Question 10
Sphere-1 with a diameter of 0.1 m is completely enclosed by another sphere-2 of diameter 0.4 m. The view factor $F_{12}$ is
 A 0.0625 B 0.25 C 0.5 D 1
$\mathrm{d}_{1}=0.1 \mathrm{m}$
$\mathrm{d}_{2}=0.4 \mathrm{m}$
$\mathrm{F}_{1-1}=0$ (from the geometry)
$\mathrm{F}_{1-1}+\mathrm{F}_{1-2}=1$
$\mathrm{F}_{1-2}=1$