Question 1 |

The partial differential equation \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2}
is a

linear equation of order 2 | |

non-linear equation of order 1 | |

linear equation of order 1 | |

non-linear equation of order 2 |

Question 1 Explanation:

A differential equation in the form \frac{d y}{d x}+P y=Q

where, P and Q are functions of x i.e., f(x) is said

to be linear equation.

\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\frac{\partial^{2} u}{\partial x^{2}}

The given equation is not complying with the definition of linear equation, therefore it is a nonlinear equation of order 2.

where, P and Q are functions of x i.e., f(x) is said

to be linear equation.

\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\frac{\partial^{2} u}{\partial x^{2}}

The given equation is not complying with the definition of linear equation, therefore it is a nonlinear equation of order 2.

Question 2 |

The eigenvalues of a symmetric matrix are all

complex with non-zero positive imaginary part. | |

complex with non-zero negative imaginary part. | |

real. | |

pure imaginary. |

Question 2 Explanation:

(i) The Eigen values of symmetric matrix

\left[A^{T}=A\right] are purely real

(ii) The Eigen value of skew-symmetric matrix

\left[A^{T}=-A\right] are either purely imaginary or zeros

\left[A^{T}=A\right] are purely real

(ii) The Eigen value of skew-symmetric matrix

\left[A^{T}=-A\right] are either purely imaginary or zeros

Question 3 |

Match the CORRECT pairs.

P-2, Q-1, R-3 | |

P-3, Q-2, R-1 | |

P-1, Q-2, R-3 | |

P-3, Q-1, R-2 |

Question 4 |

A rod of length L having uniform cross-sectional area A is subjected to a tensile force P as shown in the figure below. If the Young's modulus of the material varies linearly from E_{1}
to E_{2}
along the length of the rod, the normal stress developed at the section-SS is

\frac{P}{A} | |

\frac{P(E_{1}-E_{2})}{A(E_{1}+E_{2})} | |

\frac{PE_{2}}{AE_{1}} | |

\frac{PE_{1}}{AE_{2}} |

Question 4 Explanation:

Normal stress at any section is independent of modulus of elasticity.

Question 5 |

Two threaded bolts A and B of same material and length are subjected to identical tensile load. If the elastic strain energy stored in bolt A is 4 times that of bolt B and the mean diameter of bolt A is 12 mm, the mean diameter of bolt B in mm is

16 | |

24 | |

36 | |

48 |

Question 5 Explanation:

Given:

\begin{aligned} P_{1}&=P_{2}=P\\ &\text{(identical tensile load on bolt A \& B )} \\ &\qquad \text{(same length)}\\ L_{1} &=L_{2}=L \\ d_{A} &=12 \mathrm{mm} \\ U_{A} &=\text { strain energy in bolt } A \\ U_{B} &=\text { Strain energy in bolt } B \\ U_{A} &=4 U_{B}(\text { Given }) \qquad \cdots(i)\\ \therefore \quad d_{B} &=? \end{aligned}

Strain energy is given by

U_{A}=\frac{1}{2} P \times \delta=\frac{1}{2} \frac{P^{2} L}{A E}

\therefore Eq. (i)

\begin{aligned} \frac{1 P_{1}^{2} L_{1}}{2} &=4 \times \frac{1}{2} \frac{P_{2}^{2} L_{2}}{A_{2} E} \\ \frac{1}{A_{1}} &=4 \times \frac{1}{A_{2}} \\ \frac{4}{\pi(12)^{2}} &=4 \times \frac{4}{\pi\left(d_{B}\right)^{2}} \\ d_{B}^{2} &=576 \\ d_{B} &=24 \mathrm{mm} \end{aligned}

\begin{aligned} P_{1}&=P_{2}=P\\ &\text{(identical tensile load on bolt A \& B )} \\ &\qquad \text{(same length)}\\ L_{1} &=L_{2}=L \\ d_{A} &=12 \mathrm{mm} \\ U_{A} &=\text { strain energy in bolt } A \\ U_{B} &=\text { Strain energy in bolt } B \\ U_{A} &=4 U_{B}(\text { Given }) \qquad \cdots(i)\\ \therefore \quad d_{B} &=? \end{aligned}

Strain energy is given by

U_{A}=\frac{1}{2} P \times \delta=\frac{1}{2} \frac{P^{2} L}{A E}

\therefore Eq. (i)

\begin{aligned} \frac{1 P_{1}^{2} L_{1}}{2} &=4 \times \frac{1}{2} \frac{P_{2}^{2} L_{2}}{A_{2} E} \\ \frac{1}{A_{1}} &=4 \times \frac{1}{A_{2}} \\ \frac{4}{\pi(12)^{2}} &=4 \times \frac{4}{\pi\left(d_{B}\right)^{2}} \\ d_{B}^{2} &=576 \\ d_{B} &=24 \mathrm{mm} \end{aligned}

There are 5 questions to complete.