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

$R_{A}+R_{B}=100 \quad \cdots(1)$
$-100 \times 0.5-10+R_{B} \times 1=0$
$\therefore R_{B}=60 N$
$\text{and }\quad R_{A}=40 N$
$=40\times0.5+1.=30\text{Nm}$