GATE ME 2007

Question 1
The minimum value of function y=x^2 in the interval [1,5] is
A
0
B
1
C
25
D
Undefined
Engineering Mathematics   Calculus
Question 1 Explanation: 
\begin{aligned} &Given \quad &y=x^{2}\\ &At\quad &x=1 ; \quad y=1\\ &at\quad &x=5 ; \quad y=25\\ \end{aligned}
Minimum value of function is 1
Question 2
If a square matrix A is real and symmetric, then the Eigenvalues
A
are always real
B
are always real and positive
C
are always real and non-negative
D
occur in complex conjugate pairs
Engineering Mathematics   Linear Algebra
Question 2 Explanation: 
The eigen values of any real and symmetric matrix is always real.
Question 3
If \varphi (x,y) and \Psi (x,y) are functions with continuous second derivatives, then \varphi (x,y) + i \Psi (x,y) can be expressed as an analytic function of x + i y \; (i=\sqrt{-1}) , when
A
\frac{\partial \varphi }{\partial x}= -\frac{\partial \Psi }{\partial x}; \frac{\partial \varphi }{\partial y}= \frac{\partial \Psi }{\partial y}
B
\frac{\partial \varphi }{\partial y}= -\frac{\partial \Psi }{\partial x}; \frac{\partial \varphi }{\partial x}= \frac{\partial \Psi }{\partial y}
C
\frac{\partial^2 \varphi }{\partial x^2}+\frac{\partial^2 \varphi }{\partial y^2}=\frac{\partial^2 \Psi }{\partial x^2}+\frac{\partial^2 \Psi }{\partial y^2}=1
D
\frac{\partial \varphi }{\partial x}+\frac{\partial \varphi }{\partial y}=\frac{\partial \Psi }{\partial x}=\frac{\partial \Psi }{\partial y}= 0
Engineering Mathematics   Complex Variables
Question 3 Explanation: 
The necessary condition for a function
f(z)=\varphi(x, y)+i \psi(x, y) to be analytic
\left.\begin{array}{l}\text { (i) } \frac{\partial \varphi}{\partial x}=\frac{\partial \psi}{\partial y} \\ \text { (ii) } \frac{\partial \varphi}{\partial y}=-\frac{\partial \psi}{\partial x}\end{array}\right\} are known as Cauchy Reiman equations
Provided \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y} exist.
Question 4
The partial differential equation \frac{\partial^2\varphi }{\partial x^2}+\frac{\partial^2\varphi }{\partial y^2}+\left ( \frac{\partial \varphi }{\partial x} \right )+\left ( \frac{\partial \varphi }{\partial y} \right )= 0 has
A
degree 1 order 2
B
degree 1 order 1
C
degree 2 order 1
D
degree 2 order 2
Engineering Mathematics   Differential Equations
Question 4 Explanation: 
Order: The order of a differential equation is the order of the highest derivative appears in the equation
Degree:The degree of a differential equation is the degree of the highest order differential coefficient or derivative, when the differential coefficient are free from radicals and fraction. The general solution of a differential equation of order 'n' must involve 'n' arbitrary constant.
Question 5
Which of the following relationships is valid only for reversible processes undergone by a closed system of simple compressible substance (neglect changes in kinetic and potential energy)?
A
\delta Q = dU + \delta W
B
T dS = dU + pdV
C
T dS = dU + \delta W
D
\delta Q = dU + p dV
Thermodynamics   First Law, Heat, Work and Energy
Question 5 Explanation: 
\delta Q=d U+p d V
This equation holds good for a closed system when only pdV work is present. This is true only for a reversible (quasistatic) process.
Question 6
Water has a critical specific volume of 0.003155 m^3/kg. A closed and rigid steel tank of volume 0.025 m^3 contains a mixture of water and steam at 0.1 MPa. The mass of the mixture is 10 kg. The tank is now slowly heated. The liquid level inside the tank
A
will rise
B
will fall
C
will remain constant
D
may rise or fall depending on the amount of heat transferred
Thermodynamics   Pure Substances
Question 6 Explanation: 
Given data:
Critical specific volume of water
v_{c}=0.003155 \mathrm{m}^{3} / \mathrm{kg}
Volume of steel tank,
V=0.025 \mathrm{m}^{3}
Pressure of mixture of water and steam,
p=0.1 \mathrm{MPa}=1 \mathrm{bar}
Mass of mixture of water and steam,
m=10 \mathrm{kg}
Specific volume of mixture,
\begin{aligned} v &=\frac{V}{m}=\frac{0.025}{10} \\ &=0.0025 \mathrm{m}^{3} / \mathrm{kg} \end{aligned}

As v \lt v_{c^{\prime}} the condition of steam lies near to saturated liquid line and the liquid level inside the tank will rise with heating.
Question 7
Consider an incompressible laminar boundary layer flow over a flat plate of length L, aligned with the direction of an oncoming uniform free stream. If F is the ratio of the drag force on the front half of the plate to the drag force on the rear half, then
A
F \lt 1/2
B
F = 1/2
C
F = 1
D
F \gt 1
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 7 Explanation: 
Drag force,
\begin{aligned} F_{D} &=C_{f} \times \frac{1}{2} \rho A V^{2}=\frac{1.328}{\sqrt{\text{Re}_{L}}} \times \frac{1}{2} \rho A V^{2} \\ &=\frac{1.328}{\sqrt{\frac{\rho L V}{\mu}}} \times \frac{1}{2} \rho \times A \times L \times V^{2}\\ \text{i.e. }F_{D} &\propto \sqrt{L} \\ \end{aligned}
Now Drag force on front half,
\begin{aligned} \quad F_{D R} &\propto \sqrt{\frac{L}{2}} \\ \therefore F_{D / 2}&=\frac{F_{D}}{\sqrt{2}} \end{aligned}
Drag force on rear half,
\begin{aligned} F_{D / 2}^{\prime}&=F_{D}-F_{D / 2}=\left(1-\frac{1}{\sqrt{2}}\right) F_{D}\\ \text{Now} F&=\frac{F_{D / 2}}{F_{D / 2}^{\prime}}=\frac{\frac{F_{D}}{\sqrt{2}}}{\left(1-\frac{1}{\sqrt{2}}\right) F_{D}}=\frac{1}{\sqrt{2}-1}>1\\ \therefore \quad F&>1 \end{aligned}
Question 8
In a steady flow through a nozzle, the flow velocity on the nozzle axis is given by v = u_{0}\left ( 1+\frac{3x}{L} \right )i, where x is the distance along the axis of the nozzle from its inlet plane and L is the length of the nozzle. The time required for a fluid particle on the axis to travel from the inlet to the exit lane of the nozzle is
A
\frac{L}{u_{0}}
B
\frac{L}{3u_{0}}ln4
C
\frac{L}{4u_{0}}
D
\frac{L}{2.5u_{0}}
Fluid Mechanics   Fluid Kinematics
Question 8 Explanation: 


v=u_{o}\left(1+\frac{3 x}{L}\right)
We know that
Velocity: \quad v=\frac{d x}{d t}
\therefore \quad \frac{d x}{d t}=u_{0}\left(1+\frac{3 x}{L}\right)
\text{or }\quad u_{0}. d t=\frac{d x}{\left(1+\frac{3 x}{L}\right)}
Integrating both sides, we get
\begin{aligned} u_{0} \int_{0}^{t} d t &=\int_{0}^{L} \frac{d x}{\left(1+\frac{3 x}{L}\right)} \\ u_{0} t &=\frac{L}{3}\left[\ln \left(1+\frac{3 x}{L}\right)\right]_{0}^{L}=\frac{L}{3} \ln 4 \\ \therefore \quad t &=\frac{L}{3 u_{0}} \ln 4 \end{aligned}
Question 9
Consider steady laminar incompressible axi-symmetric fully developed viscous flow through a straight circular pipe of constant cross - sectional area at a Reynolds number of 5. The ratio of inertia force to viscous force on a fluid particle is
A
5
B
1/5
C
0
D
\infty
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 9 Explanation: 
Reynolds number, \mathrm{Re}=\frac{\text { Inertia force }}{\text { viscous force }}=5
Question 10
In a simply - supported beam loaded as shown below, the maximum bending moment in Nm is
A
25
B
30
C
35
D
60
Strength of Materials   Bending of Beams
Question 10 Explanation: 
Resultant beam diagram

R_{A}+R_{B}=100 \quad \cdots(1)
Taking moment about A
-100 \times 0.5-10+R_{B} \times 1=0
\therefore R_{B}=60 N
\text{and }\quad R_{A}=40 N
Bending moment diagram


Maximum bending movement
=40\times0.5+1.=30\text{Nm}
There are 10 questions to complete.

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