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.