# 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.

There are 5 questions to complete.