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.

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.