Question 1 |
The minimum value of function y=x^2 in the interval [1,5] is
0 | |
1 | |
25 | |
Undefined |
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
Minimum value of function is 1
Question 2 |
If a square matrix A is real and symmetric, then the Eigenvalues
are always real | |
are always real and positive | |
are always real and non-negative | |
occur in complex conjugate pairs |
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
\frac{\partial \varphi }{\partial x}= -\frac{\partial \Psi }{\partial x}; \frac{\partial \varphi }{\partial y}= \frac{\partial \Psi }{\partial y} | |
\frac{\partial \varphi }{\partial y}= -\frac{\partial \Psi }{\partial x}; \frac{\partial \varphi }{\partial x}= \frac{\partial \Psi }{\partial y} | |
\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 | |
\frac{\partial \varphi }{\partial x}+\frac{\partial \varphi }{\partial y}=\frac{\partial \Psi }{\partial x}=\frac{\partial \Psi }{\partial y}= 0 |
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.
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
degree 1 order 2 | |
degree 1 order 1 | |
degree 2 order 1 | |
degree 2 order 2 |
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.
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)?
\delta Q = dU + \delta W | |
T dS = dU + pdV | |
T dS = dU + \delta W | |
\delta Q = dU + p dV |
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.
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.